JP2009246927A - Encoding method, encoder, and decoder - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To obtain excellent reception quality when an LDPCP-CC is generated, and an information sequence is sent after subjected to an error-correction encoding using the LDPC-CC. <P>SOLUTION: In an encoding method for generating low-density parity-check convolutional codes (LDPC) of a time-variant period 3g (g: a positive integer), an LDPC-CC code word is obtained through linear operation between first to 3g-th parity check polynomials and input data for LDPC-CC defined based upon a parity check polynomial expressed by expression (1-k) [Here, X<SB>1</SB>(D), X<SB>2</SB>(D), ..., X<SB>n-1</SB>(D) are polynomial representations (n: an integer not less than 2) of information sequences X<SB>1</SB>, X<SB>2</SB>, ..., X<SB>n-1</SB>, P(D) is a polynomial representation of a parity sequence. Further, a<SB>#k.p.1</SB>, a<SB>#k.p.2</SB>, a<SB>#k.p.3</SB>(k=1, 2, 3, ..., 3g: p=1, 2, 3, ..., n-1) are integers (provided that a<SB>#k.p.1</SB>≠a<SB>#k.p.2</SB>≠a<SB>#k.p.3</SB>), and b<SB>#k.1</SB>, b<SB>#k.2</SB>, and b<SB>#k.3</SB>are integers (provided that b<SB>#k.1</SB>≠b<SB>#k.2</SB>≠b<SB>#k.3</SB>). Furthermore, c%d represents the remainder after (c) is divided by (d)]. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to an encoding method, an encoder, and a decoder for Low-Density Parity-Check Convolutional Codes (LDPC-CC).

  In recent years, attention has been focused on a low-density parity check (LDPC) code as an error correction code that exhibits high error correction capability with a feasible circuit scale. Since the LDPC code has a high error correction capability and is easy to implement, the LDPC code is adopted in an error correction coding system such as an IEEE802.11n high-speed wireless LAN system or a digital broadcasting system.

The LDPC code is an error correction code defined by a low-density parity check matrix H. Also, the LDPC code is a block code having a block length equal to the number N of columns of the check matrix H. For example, Non-Patent Document 1, Non-Patent Document 2, and Non-Patent Document 3 propose a random LDPC code, Array LDPC code, and QC-LDPC code (QC: Quasi-Cyclic).

However, many of the current communication systems are characterized in that transmission information is collectively transmitted for each variable-length packet or frame, as in Ethernet (registered trademark). When an LDPC code, which is a block code, is applied to such a system, for example, there is a problem of how a block of a fixed-length LDPC code corresponds to a variable-length Ethernet (registered trademark) frame. In IEEE802.11n, the length of the transmission information sequence and the block length of the LDPC code are adjusted by performing padding processing and puncture processing on the transmission information sequence. However, the coding rate changes depending on the padding and puncture. It is difficult to avoid transmitting redundant sequences.

An LDPC code of such a block code (hereinafter referred to as LDPC-BC: Low-Density Parity-Check Block Code) can be encoded / decoded for an information sequence of an arbitrary length. Possible LDPC-CC (Low-Density Parity-Check Convolutional Codes) has been studied (for example, see Non-Patent Document 1 and Non-Patent Document 2).

The LDPC-CC is a convolutional code defined by a low-density parity check matrix. For example, the parity check matrix H ^T [0, n] of the LDPC-CC with a coding rate R = 1/2 (= b / c). Is shown in FIG. Here, the element h ₁ ^(m) (t) of H ^T [0, n] takes 0 or 1. All elements other than h ₁ ^(m) (t) are 0. M represents the memory length in LDPC-CC, and n represents the length of the LDPC-CC codeword. As shown in FIG. 68, in the LDPC-CC parity check matrix, 1 is arranged only in the diagonal term of the matrix and its neighboring elements, the lower left and upper right elements of the matrix are zero, and a parallelogram It has the feature of being a type matrix.

Here, when h ₁ ⁽⁰⁾ (t) = 1, h ₂ ⁽⁰⁾ (t) = 1, the LDPC-CC encoder defined by the parity check matrix H ^T [0, n] T is It is represented in FIG. As shown in FIG. 69, the LDPC-CC encoder is composed of M + 1 shift registers with a bit length c and a mod2 adder. For this reason, the LDPC-CC encoder is realized by a very simple circuit compared to a circuit that performs multiplication of a generator matrix and an LDPC-BC encoder that performs an operation based on the backward (forward) substitution method. There is a feature that can be. Further, since FIG. 69 shows a convolutional code encoder, it is not necessary to divide an information sequence into fixed-length blocks, and an information sequence having an arbitrary length can be encoded.
RG Gallager, “Low-density parity check codes,” IRE Trans. Inform. Theory, IT-8, pp-21-28, 1962. DJC Mackay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol.45, no.2, pp399-431, March 1999. JL Fan, “Array codes as low-density parity-check codes,” proc. Of 2nd Int. Symp. On Turbo Codes, pp.543-546, Sep. 2000. Y. Kou, S. Lin, and MPC Fossorier, “Low-density parity-check codes based on finite geometries: A rediscovery and new results,” IEEE Trans. Inform. Theory, vol.47, no.7, pp2711-2736 , Nov. 2001. A. J. Felstorom, and K. Sh. Zigangirov, “Time-Varying Periodic Convolutional Codes With Low-Density Parity-Check Matrix,” IEEE Transactions on Information Theory, Vol. 45, No. 6, pp2181-2191, September 1999 . RM Tanner, D. Sridhara, A. Sridharan, TE Fuja, and DJ Costello Jr., “LDPC block and convolutional codes based on circulant matrices,” IEEE Trans. Inform. Theory, vol.50, no.12, pp.2966 -2984, Dec. 2004. G. Richter, M. Kaupper, and K. Sh. Zigangirov, “Irregular low-density parity-Check convolutional codes based on protographs,” Proceeding of IEEE ISIT 2006, pp1633-1637. RD Gallager, “Low-Density Parity-Check Codes,” Cambridge, MA: MIT Press, 1963. MPC Fossorier, M. Mihaljevic, and H. Imai, “Reduced complexity iterative decoding of low density parity check codes based on belief propagation,” IEEE Trans. Commun., Vol.47., No.5, pp.673-680, May 1999. J. Chen, A. Dholakia, E. Eleftheriou, MPC Fossorier, and X.-Yu Hu, “Reduced-complexity decoding of LDPC codes,” IEEE Trans. Commun., Vol.53., No.8, pp.1288 -1299, Aug. 2005. Yuichi Ogawa, “sum-product decoding of turbo codes,” Master's thesis, Nagaoka University of Technology, 2007. S. Lin, DJ Jr., Costello, “Error control coding: Fundamentals and applications,” Prentice-Hall. Tadashi Wadayama, “Low-density parity check code and its decoding method,” Trikes. A. Pusane, R. Smarandache, P. Vontobel, and DJ Costello Jr., “On deriving good LDPC convolutional codes from QC LDPC block codes,” Proc. Of IEEE ISIT 2007, pp.1221-1225, June 2007. A. Sridharan, D. Truhachev, M. Lentmaier, DJ Costello, Jr., K. Sh. Zigangirov, “Distance Bounds for an Ensemble of LDPC Convolutional Codes,” IEEE Transactions on Information Theory, vol.53, no.12, Dec. 2007. A. Pusane, KS Zigangirov, and DJ Costello Jr., “Construction of irregular LDPC convolutional codes with fast encoding,” Proc. Of IEEE ICC 2006, pp.1160-1165, June 2006. Hiroyuki Yajima, “Convolutional Codes and Viterbi Decoding,” Triqueps. R. Johannesson, and KS Zigangirov, “Fundamentals of convolutional coding,” IEEE Press.

Here, in the case of a block code, if the number of bits of transmission data is not an integral multiple of the block length of the code, it is necessary to transmit extra bits. Therefore, when LDPC-BC having a large block size or LDPC-CC having a large protograph size is used, the number of extra bits to be transmitted increases. In particular, in packet communication, when a packet having a small number of bits of transmission data is transmitted, there is a problem that the data transmission efficiency is remarkably reduced by transmitting extra bits. However, with the LDPC codes shown in Non-Patent Document 1 to Non-Patent Document 7, it is difficult to solve this problem. In order to solve this problem, it is required to design an LDPC-CC that can be configured with a protograph smaller in size than Non-Patent Document 1 to Non-Patent Document 7.

  In this regard, if the LDPC-CC is created from the convolutional code, the size of the protograph can be made smaller than those in Non-Patent Document 1 to Non-Patent Document 7.

  Then, when an LDPC-CC is created from a convolutional code and error correction coding using LDPC-CC is performed on the information sequence and transmitted, a device for improving reception quality is required.

  The present invention has been made in consideration of such points. Good reception is achieved when an LDPC-CC is generated from a convolutional code and error correction coding using LDPC-CC is performed on an information sequence and transmitted. An encoding method, an encoder, and a decoder for LDPC-CC capable of obtaining quality are provided.

One aspect of the encoding method of the present invention is an encoding method for creating a low-density parity-check convolutional code (LDPC-CC) having a time-varying period of 3 g (g is a positive integer). Among the parity check polynomials represented by the equation (168-1), (a _{# 1} , _1,1 % 3, a _{# 1,1} , ₂ % 3, a _{# 1,1,3} % 3) , (A _{# 1,2,1} % 3, a _{# 1,2,2} % 3, a _{# 1,2,3} % 3), ..., (a _{# 1, n-1,1} % 3, a _{# 1, n-1, 2} % 3, a _{# 1, n-1, 3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0), and (b _{# 1,1} % 3, b _{# 1,2} % 3, b _{# 1,3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0 2), (1, 2, 0), (2, 0, 1), (2, 1, 0), and a first parity check polynomial and a parity check polynomial expressed by equation (168-2) _{of, (a # 2,1,1% 3,} a # 2,1,2% 3, a # 2,1,3% 3), (a # 2,2,1% 3, a # 2, _2,2 % 3, a _{# 2,2,3} % 3), ..., (a _{# 2, n-1,1} % 3, a _{# 2, n-1,2} % 3, a _{# 2, n-1,3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (B _{# 2,1} % 3, b _{# 2,2} % 3, b _{# 2,3} % 3) becomes (0,1,2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0). A utility check polynomial, of the parity check polynomial represented by the formula (168-kk) (kk = 3,4, ···, 3g-1), (a # kk, 1,1% 3, a #kk, _{1, 2 , 3} %, a _{# kk, 1, 3} % 3), (a _{# kk, 2, 1} % 3, a _{# kk, 2, 2} % 3, a _{# kk, 2, 3} % 3), (A _{# kk, n-1, 1} % 3, a _{# kk, n-1, 2} % 3, a _{# kk, n-1, 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2 _{, 1} , 0), and (b #kk _{, 1} % 3, b _{# kk, 2} % 3, b _{# kk, 3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0), kk parity check polynomial When, among the parity check polynomial represented by equation _{(168-3g), (a # 3g} , 1,1% 3, a # 3g, 1,2% 3, a # 3g, 1,3% 3), ( a _{# 3g, 2,1} % 3, a _{# 3g, 2,2} % 3, a _{# 3g, 2,3} % 3), ..., (a _{# 3g, n-1,1} % 3, a _{# 3g, n-1,2} % 3, a _{# 3g, n-1,3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1 2, 0), (2, 0, 1), (2, 1, 0), and (b _{# 3g, 1} % 3, b _{# 3g, 2} % 3, b _{# 3g, 3} % 3 ) Is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0) In the LDPC-CC defined based on the third g parity check polynomial And providing a second 3g parity check polynomials from the first, and to have, acquiring LDPC-CC codeword by linear computation of the first 3g parity check polynomial and the input data from the first.

  One aspect of the encoder of the present invention is an encoder that creates a low-density parity-check convolutional code (LDPC-CC) from a convolutional code, and includes the above encoding method. A configuration including a parity calculation unit for obtaining a parity sequence is employed.

  One aspect of the decoder of the present invention is a decoder that decodes low-density parity-check convolutional codes (LDPC-CC) using reliability propagation (BP). A row processing operation unit that performs a row processing operation using a parity check matrix corresponding to the parity check polynomial used in the encoder, a column processing operation unit that performs a column processing operation using the parity check matrix, And a determination unit that estimates a code word using calculation results in the row processing calculation unit and the column processing calculation unit.

According to the present invention, attention is paid to a convolutional code as a small-sized protograph, and the parity check polynomial of the convolutional code is used as a protograph, whereby the reception quality is good and transmission is performed in the LDPC-CC design method. The number of extra bits can be reduced. Further, when creating an LDPC-CC from a convolutional code, “1” is added to a predetermined position of the approximate lower triangular matrix or upper trapezoidal matrix of the check matrix H, and the degree of the parity check polynomial of the convolutional code at this time is added. By increasing the size, if the receiving apparatus performs BP decoding or approximated BP decoding using the created LDPC-CC parity check matrix, good reception quality can be obtained.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

(Embodiment 1)
In Embodiment 1, a method for designing a new LDPC-CC from the (7, 5) convolutional code will be described in detail.

  FIG. 1 is a diagram illustrating a configuration of an encoder for a (7, 5) convolutional code. The encoder shown in FIG. 1 includes shift registers 101 and 102 and exclusive OR circuits 103, 104, and 105. The encoder shown in FIG. 1 outputs an output x and a parity p for an input x. This code is a systematic code.

  In the present invention, it is important to use a convolutional code that is a systematic code. This point will be described in detail in the second embodiment.

ここで、Ｄは、遅延演算子である。 Consider a convolutional code of coding rate 1/2 and generator polynomial G = [1 G ₁ (D) / G ₀ (D)] as an example. In this case, _{G 1} is feed-forward polynomial, _{G 0} represents a feedback polynomial. When the polynomial expression of the information sequence (data) is X (D) and the polynomial expression of the parity sequence is P (D), the parity check polynomial is expressed as the following equation (1).

Here, D is a delay operator.

FIG. 2 describes information relating to the (7, 5) convolutional code. The generation matrix of the (7, 5) convolutional code is expressed as G = [1 (D ² +1) / (D ² + D + 1)]. Therefore, the parity check polynomial is expressed by the following equation (2).

Here, the data at the time point i is represented as X _i , the parity is represented as P _i, and the transmission sequence W _i = (X _i , P _i ). Then, the transmission vector w = (X ₁ , P ₁ , X ₂ , P ₂ ,..., X _i , P _i ...) ^T. Then, from equation (2), the check matrix H can be expressed as shown in FIG. At this time, the following relational expression (3) is established.

Therefore, in the receiving apparatus, BP (Belief Propagation) (reliability propagation) decoding as shown in Non-Patent Document 8 to Non-Patent Document 10, min-sum decoding approximating BP decoding, using check matrix H, Decoding using reliability propagation such as offset BP decoding, Normalized BP decoding, and shuffled BP decoding can be performed.

Here, in the parity check matrix of FIG. 2, the lower left portion (the lower left portion of 201 in FIG. 2) from row number = column number “1” is defined as an approximate lower triangular matrix. The upper right part of row number = column number “1” is defined as an upper trapezoid matrix.

  Next, the LDPC-CC design method in the present invention will be described in detail.

  In order to realize the encoder with a simple configuration, in the present embodiment, “1” is added to the approximate lower triangular matrix of the parity check matrix H for the (7, 5) convolutional code shown in FIG. Take the technique.

<Encoding method>
Here, as an example, it is assumed that “1” added to the parity check matrix in FIG. 2 is one for each of data and parity. When “1” is added to the approximate lower triangular matrix of the parity check matrix H in FIG. 2 for each of data and parity, the parity check polynomial is expressed as in the following equation (4). However, in formula (4), α ≧ 3 and β ≧ 3.

Therefore, the parity P (D) is expressed as the following equation (5).

When “1” is added to the approximate lower triangular matrix of the parity check matrix, D ^β P (D), D ² P (D), and DP (D) are past data and are known values. Parity P (D) can be obtained.

<Location to add “1”>
Next, the position of “1” to be added will be described in detail with reference to FIG. In FIG. 3, reference numeral 301 is “1” related to decoding of data X _i at time point i, and reference numeral 302 is “1” related to parity P _i at time point i. A dotted line 303 is a protograph related to the propagation of external information for the data X _i and the parity P _i at the time point i when BP decoding is performed once. That is, the reliability from time point i-2 to time point i + 2 is involved in propagation.

A boundary line 305 is drawn on the vertical axis for “1” (304) on the rightmost side of the protograph 303. Then, a boundary line 307 is drawn with respect to the leftmost “1” (306) adjacent to the boundary line 305. Then, “1” is added to the area 308 so that the reliability after the boundary line 305 is propagated to the data X _i and the parity P _i at the time point i. Accordingly, it is possible to propagate a probability that was not obtained before adding “1”, that is, a reliability other than the time point i−2 to the time point i + 2. In order to propagate a new probability, it is necessary to add to the area 308 in FIG.

  Here, in each row of the parity check matrix H in FIG. 3, the width of the rightmost “1” and the leftmost “1” is L. So far, the position where “1” is added has been described in the column direction. Considering this in the row direction, in the parity check matrix of FIG. 2, “1” is added to the left position of L−2 or more from the leftmost “1”. Further, in the case of explanation using a check polynomial, in equation (4), α may be set to 5 or more and β may be set to 5 or more.

This is expressed by a general formula. A general expression of the parity check polynomial of the convolutional code is expressed as the following expression (6).

When “1” is added to the approximate lower triangular matrix of the parity check matrix H, one for each of the data and parity, the parity check polynomial is expressed as the following equation (7).

  In this case, α may be set to 2K + 1 or more, and β may be set to 2K + 1 or more. K ≧ 2.

FIG. 4 is a diagram illustrating an example when “1” is added to the approximate lower triangular matrix of the parity check matrix in FIG. 3. When “1” is added to the data and parity at all times, the check matrix is represented as shown in FIG. FIG. 5 is a diagram illustrating an example of a configuration of an LDPC-CC parity check matrix according to the present embodiment. In FIG. 5, “1” in the areas 501 and 502 is the added “1”, and the code having the check matrix H is the LDPC-CC in the present embodiment. At this time, the check polynomial is expressed by the following equation (8).

  As described above, the transmission apparatus adds “1” to the approximate lower triangular matrix of the parity check matrix H and creates an LDPC-CC from the convolutional code, so that the reception apparatus uses the created LDPC-CC parity check matrix. By using BP decoding or approximated BP decoding, good reception quality can be obtained.

In this embodiment, the case where one “1” is added to each of data and parity has been described. However, the present invention is not limited to this, and for example, “1” is set to either data or parity. The method of adding may be used. For example, “1” may be added to data and “1” may not be added to parity. As an example, let us consider a case where there is no ^Dβ in the above equation (7). At this time, if α is set to 2K + 1 or more, the receiving apparatus can obtain good reception quality. Conversely, consider the case where there is no ^Dα in equation (7). At this time, if β is set to 2K + 1 or more, the receiving apparatus can obtain good reception quality.

Also, the reception quality is greatly improved by a code in which a plurality of “1” s are added to both data and parity. For example, as an example of inserting a plurality, a parity check polynomial of a certain convolutional code is expressed by equation (9). In Equation (9), K ≧ 2.

When a plurality of “1” s are added to the approximate lower triangular matrix of the parity check matrix H for data and parity, the parity check polynomial is represented by the following equation (10).

In this case, when α ₁ ,..., Α _{n is set} to 2K + 1 or more and β ₁ ,..., Β _m are set to 2K + 1 or more, good reception quality can be obtained in the receiving apparatus. This point is important in the present embodiment.

However, even if one or more of α ₁ ,..., Α _n satisfy 2K + 1 or more, good reception quality can be obtained in the receiving apparatus. Further, even when one or more of β ₁ ,..., Β _m satisfy 2K + 1 or more, good reception quality can be obtained in the receiving apparatus.

In addition, when the LDPC-CC parity check polynomial is expressed as in the following equation (11), when α ₁ ,..., Α _n are set to 2K + 1 or more, good reception quality can be obtained in the receiver. Can do. This point is important in the present embodiment.

However, even when one or more of α ₁ ,..., Α _n satisfy 2K + 1 or more, good reception quality can be obtained in the receiving apparatus.

Similarly, when the LDPC-CC parity check polynomial is expressed by the following equation (12), if β ₁ ,..., Β _m are set to 2K + 1 or more, good reception quality is obtained in the receiving apparatus. be able to. This point is important in the present embodiment.

However, even when one or more of β ₁ ,..., Β _m satisfy 2K + 1 or more, good reception quality can be obtained in the receiving apparatus.

  Next, a method for designing an LDPC-CC from a parity check polynomial different from Equation (2) of the (7, 5) convolutional code will be described in detail. Here, as an example, a case where two “1” s are added to data and two “1” s are added to parity will be described as an example.

Non-Patent Document 11 shows a parity check polynomial different from Equation (2) of the (7, 5) convolutional code. An example of this is expressed by the following equation (13).

  In this case, the check matrix H can be represented as shown in FIG.

<Encoding method>
Here, a case will be described in which two “1” s are added to each of data and parity in the parity check matrix of FIG. 6. When two “1” s are added to each of the data and parity in the approximate lower triangular matrix of the parity check matrix H in FIG. 6, the parity check polynomial is expressed as the following equation (14).

Therefore, the parity P (D) can be expressed as the following equation (15).

Thus, when “1” is added to the approximate lower triangular matrix of the parity check matrix, D ^β1 P (D), D ^β2 P (D), D ⁹ P (D), D ⁸ P (D), D ³ Since P (D) and DP (D) are past data and are known values, the parity P (D) can be easily obtained.

<Location to add “1”>
In order to obtain the same effect as described above, when α ₁ and α ₂ are set to 19 or more and β ₁ and β ₂ are set to 19 or more, good reception quality can be obtained in the receiving apparatus. As an example, in the parity check matrix of FIG. 7, α ₁ = 26, α ₂ = 19, β ₁ = 30, and β ₂ = 24. Thereby, for the same reason as described above, the reception apparatus can obtain good reception quality.

  From the above example, a method for creating an LDPC-CC from a convolutional code is as follows. The following procedure is an example when the convolutional code is an encoding rate of 1/2.

  <1> Select a convolutional code that gives good characteristics.

  <2> A check polynomial for the selected convolutional code is generated (for example, Equation (6)). However, it is important to use the selected convolutional code as a systematic code. Further, the check polynomial is not limited to one as described above. It is necessary to select a check polynomial that gives good reception quality. At this time, it is better to use an equivalent check polynomial having a higher degree than the check polynomial generated from the generator polynomial (see Non-Patent Document 11).

  <3> A check matrix H of the selected convolutional code is created.

  <4> Probability propagation is considered for data or (and) parity, and “1” is added to the parity check matrix. The position where “1” is added is as described above.

  In the present embodiment, the method of creating the LDPC-CC from the (7, 5) convolutional code has been described. However, the present invention is not limited to the (7, 5) convolutional code, and other convolutional codes are used. Can be similarly implemented. At this time, the generating polynomial G of the convolutional code that gives good reception quality is described in detail in Non-Patent Document 12.

As described above, the transmitting apparatus according to this embodiment, in Formula _{(10), α 1, ···} , the alpha _n 2K + 1 or more, beta _1, · · ·, a beta _m is set to 2K + 1 or more LD from convolutional code
By creating the PC-CC, the receiving apparatus can obtain good reception quality by performing BP decoding or approximated BP decoding using the created LDPC-CC parity check matrix. Further, when the LDPC-CC is created from the convolutional code, the size of the protograph, that is, the check polynomial, is much smaller than the protographs shown in Non-Patent Document 6 and Non-Patent Document 7, so that the transmission data Therefore, it is possible to reduce the number of extra bits generated when a packet having a small number of bits is transmitted, and to suppress the problem that the data transmission efficiency is lowered.

(Embodiment 2)
In the second embodiment, a method for designing a new LDPC-CC from a (7, 5) convolutional code, in particular, a method for adding “1” to the upper trapezoid matrix of the parity check matrix will be described in detail.

  The structure of the (7, 5) convolutional code parity check polynomial and parity check matrix H is as described in the first embodiment.

  The LDPC-CC design method in the present invention will be described in detail.

  In order to realize the encoder with a simple configuration, in the invention in this embodiment, “1” is added to the upper trapezoid matrix of the check matrix H for the (7, 5) convolutional code shown in FIG. Take a technique.

<Encoding method>
Here, as an example, it is assumed that “1” to be added to the parity check matrix in FIG. 8 is added to each of data and parity. In this case, the check polynomial is expressed by the following equation (16). In the equation _{(16), α 1, ···} , α n ≦ -1, β 1, ···, a beta _m ≦ -1.

Therefore, the parity P (D) can be expressed as the following equation (17).

Here, in the equation (17), D ^α1 X (D),..., D ^αn X (D) is known because it is input data, but D ^β1 P (D) ^{,. βm} P (D) is an unknown value. Therefore, although it is possible to insert “1” for the upper trapezoid matrix of the parity check matrix H, the parity can be inserted even if “1” is inserted for the parity related one. It is difficult to find a bit. Therefore, “1” is inserted into the upper trapezoid matrix of the check matrix H related to the data. That is, when the parity check polynomial is expressed by equation (18), the parity P (D) can be expressed by the following equation (19), and the parity P (D) can be obtained.

Here, in the case of a non-systematic code, since only parity bits exist in the check polynomial, it is difficult to obtain parity bits when “1” is added to the upper trapezoid matrix of the check matrix H as described above. It is. Thus, in the present invention, it can be seen that it is important to use a systematic code convolutional code.

<Location to add “1”>
Next, the position of “1” to be added will be described in detail with reference to FIG. In FIG. 8, reference numeral 801 is “1” related to the decoding of the data X _{i at} the time point i, and reference numeral 802 is “1” related to the parity p _i at the time point i. A dotted line 803 is a protograph related to the propagation of external information for the data X _i and the parity P _i at the time point i when BP decoding is performed once. That is, the reliability from time point i-2 to time point i + 2 is involved in propagation.

Then, the boundary lines 804 and 805 are drawn as in the first embodiment. Then, “1” is added to one of the regions 806 so that the reliability before the boundary line 804 is propagated to the data X _i at the time point i. As a result, it is possible to propagate a probability that was not obtained before adding “1”, that is, a probability other than the time point i + 2 to the time point i + 2. In order to propagate a new probability, it is necessary to add to the area 806 in FIG.

Here, in each row of the parity check matrix H in FIG. 8, the width of the rightmost “1” and the leftmost “1” is L. So far, the position where “1” is added has been described in the column direction. Considering this in the row direction, in the parity check matrix of FIG. 2, “1” is added to a position to the right of L−2 or more from the rightmost “1”. Further, in the case of explanation using a check polynomial, in the equation (18), α ₁ ,..., Α _n may be set to −2 or less.

  This is expressed by a general formula. A general expression of the parity check polynomial of the convolutional code is expressed as Expression (6).

When “1” is added to the upper trapezoidal matrix of the parity check matrix H, the parity check polynomial is represented by the following equation (20).

In this case, better reception quality can be obtained by setting α ₁ ,..., Α _n to −K−1 or less. However, even if one or more of α ₁ ,..., Α _n satisfy the condition of −K−1 or less, good reception quality can be obtained.

FIG. 9 is a diagram illustrating an example when “1” is added to the upper trapezoidal matrix of the parity check matrix in FIG. 3. FIG. 9 shows an example in which “1” is added to the area 806. Then, when “1” is added to the data at all points in the upper trapezoid matrix of the parity check matrix H, the parity check matrix is represented as shown in FIG. FIG. 10 is a diagram illustrating an example of a configuration of an LDPC-CC parity check matrix according to the present embodiment. In FIG. 10, “1” in the area 1001 is the added “1”, and the code having the matrix H shown in FIG. 10 as the check matrix is the LDPC-CC in the present embodiment. At this time, the check polynomial is expressed by the following equation (21).

As described above, according to the present embodiment, in the transmission device, α ₁ ,.
, α _n is set to −K−1 or less and LDPC-CC is created from the convolutional code, so that the receiving apparatus performs BP decoding or approximated BP decoding using the created LDPC-CC parity check matrix. Good reception quality can be obtained. In the present embodiment, an example has been described in which “1” is added to the upper trapezoid matrix of the parity check matrix for the data. However, the present invention is not limited to this, and the parity check matrix is combined with the first embodiment. In addition to adding “1” to the upper trapezoidal matrix, “1” may be added to the approximate lower triangular matrix of the parity check matrix. At this time, if the conditions described in Embodiment 1 are satisfied, better reception quality can be ensured.

  When “1” is added to the upper trapezoidal matrix of the check matrix, there is also an advantage that the convergence speed at the end of the third embodiment to be described later is improved. This point will be described in detail in Embodiment 3.

  Further, according to the present embodiment, LDPC-CC can be designed from a parity check polynomial different from equation (2) of the (7, 5) convolutional code, as in the first embodiment.

  In the present embodiment, the method of creating the LDPC-CC from the (7, 5) convolutional code has been described. However, the present invention is not limited to the (7, 5) convolutional code, and other convolutional codes can be used similarly. Can be implemented. At this time, the generating polynomial G of the convolutional code that gives good reception quality is described in detail in Non-Patent Document 12.

  Further, when the LDPC-CC is created from the convolutional code, the size of the protograph, that is, the check polynomial, is much smaller than the protographs shown in Non-Patent Document 6 and Non-Patent Document 7, so that the transmission data It is possible to reduce the number of extra bits to be transmitted that occur when a packet having a small number of bits is transmitted, and to suppress the problem that the data transmission efficiency is reduced.

(Embodiment 3)
In Embodiment 3, as described in Embodiment 1, when LDPC-CC is generated from a convolutional code, termination when “1” is added to the approximate lower triangular matrix of the parity check matrix
The problem and the method of solving the problem are described.

  FIG. 11 is a diagram illustrating an example of a parity check matrix at the time of termination. In FIG. 11, reference numeral 1100 denotes a boundary between information bits and termination bits. The “information bit” is a bit related to information that the transmitting apparatus wants to transmit to the receiving apparatus. On the other hand, the “termination bit” is an extra bit for accurately transmitting the information bit, and the termination bit itself does not belong to the information required by the receiving apparatus, and is necessary for accurately receiving the information bit. Is a bit.

Here, in the information bits, the last bit of the data bit is X _f , the last bit of the parity bit is P _f , and the time point is f. 11, 1101 corresponds to “1” regarding the data at the time point f, and 1102 corresponds to “1” regarding the parity at the time point f.

A check polynomial corresponding to the protograph of FIG. 11 is expressed as the following equation (22).

In FIG. 11, reference numeral 1104 denotes a check matrix for the termination bit. The data bit to be transmitted with the termination bit is set to “0”. However, it is only an example that the data bit to be transmitted with the termination bit is set to “0”. If the transmitter / receiver is known information, the data bit to be transmitted with the termination bit is terminated regardless of the sequence. Can be a bit.

In FIG. 11, the time point f + 1, that is, the first check polynomial of the end bit is expressed as the following Expression (23).

Further, the check polynomial at the terminal bit at the time point f + 2 is expressed as the following Expression (24).

  As described above, in the present embodiment, as shown in FIG. 11, with the termination bit, the position of the added “1” is shifted with time, and the order of the check polynomial is reduced with time. It is characterized by. In FIG. 11, reference numeral 1105 indicates an example of the configuration of “1” related to the addition of the termination bit. In FIG. 11, the added “1” exists in the parity.

  Since the terminal bit sequence transmitted from the transmitter is known, the receiver can set the likelihood of the terminal bit to be known when performing BP decoding.

  As described above, according to the present embodiment, the speed at which the trellis diagram is stabilized (converges) is improved by decreasing the order of the check polynomial with time in the termination bit. Therefore, the number of bits transmitted for termination can be reduced, and the data transmission efficiency can be improved.

  FIG. 12 is a diagram illustrating an example of a check matrix configuration related to “information bits” and “termination bits” different from those in FIG. In FIG. 12, reference numeral 1200 indicates a boundary between information bits and termination bits.

Here, in the information bits, the last bit of the data bit is X _f , the last bit of the parity bit is P _f , and the time point is f. The code 1201 in the area 1203 in FIG. 12 corresponds to “1” regarding the data at the time point f, and the code 1202 corresponds to “1” regarding the parity at the time point f.

The check polynomial corresponding to the protograph of FIG. 12 is expressed as the following equation (25).

In FIG. 12, reference numeral 1204 denotes a parity check matrix for terminal bits. The data bit to be transmitted with the termination bit is set to “0”. Note that the data bit to be transmitted by the termination bit is an example, and the transmission / reception apparatus may have any sequence of data bits to be transmitted by the termination bit as long as it is known information. .

In FIG. 12, the time point f + 1, that is, the first check polynomial of the terminal bit is expressed as the following Expression (26).

Further, the check polynomial at the terminal bit at the time point f + 2 is expressed as the following equation (27).

  Thus, as shown in FIG. 12, the termination bit is characterized in that the position of the added “1” is shifted with time, and the order of the check polynomial is reduced with time. In FIG. 12, reference numeral 1205 is an example of the configuration of “1” related to the addition of the termination bit. In FIG. 12, the added “1” exists in the data.

Further, in order to prevent the degradation of the reception quality of information bits, as described in Embodiment 1, the order of the check polynomial is reduced so as to satisfy the condition “2K + 1 or more”. Therefore, the final check bit polynomial of the end bit is expressed as the following Expression (28), for example.

  Since the terminal bit sequence transmitted from the transmitter is known, the receiver can set the likelihood of the terminal bit to be known when performing BP decoding.

  As described above, according to the present embodiment, the speed at which the trellis diagram stabilizes (converges) can be improved by reducing the order of the parity check polynomial with time in the terminal bit. Therefore, the number of bits transmitted for termination can be reduced, and the data transmission efficiency can be improved.

  Next, an example of a termination method using the method of adding “1” to the upper trapezoid matrix of the parity check matrix described in the second embodiment will be described. FIG. 13 is a diagram illustrating an example of a configuration of a check matrix related to “information bits” and “termination bits” different from those in FIGS. 11 and 12. In FIG. 13, reference numeral 1300 indicates a boundary between information bits and termination bits.

Here, in the information bits, the last bit of the data bit is X _f , the last bit of the parity bit is P _f , and the time point is f. In the area 1303 in FIG. 13, the code 1301 corresponds to “1” regarding the data at the time point f, and the code 1302 corresponds to “1” regarding the parity at the time point f.

The parity check polynomial corresponding to the protograph of FIG. 13 is assumed to be expressed by the following equation (29).

  In FIG. 13, reference numeral 1304 denotes a parity check matrix for terminal bits. The data bit to be transmitted with the termination bit is set to “0”. Note that the data bit to be transmitted by the termination bit is an example, and the transmission / reception apparatus may have any sequence of data bits to be transmitted by the termination bit as long as it is known information. .

In FIG. 13, the time point f + 1, that is, the first check polynomial of the end bit is expressed as the following equation (30).

Further, the check polynomial for the terminal bit at the time point f + 2 is expressed by the following equation (31).

  Thus, as shown in FIG. 13, in the termination bit, the position of the added “1” is shifted with time, and the order of the check polynomial is decreased with time (reference numeral 1305 in FIG. 13). Equivalent). In FIG. 13, reference numeral 1304 is an example of the configuration of a protograph of terminal bits. In FIG. 13, the added “1” exists in data and parity.

  Further, in order to prevent the degradation of the reception quality of information bits, as described in Embodiment 1, the order of the check polynomial is reduced so as to satisfy the condition “2K + 1 or more”.

  Another feature of the last bit in FIG. 13 is that the number of “1” additionally inserted in the parity check matrix is changed from 2 to 1 as shown in FIG. 13 from 1305 to 1306. is there. As a result, the speed at which the trellis diagram is stabilized (converged) is improved.

  In this embodiment, the number of “1” added to the check matrix is described as being reduced to 2 when transmitting information bits and then 1 when transmitting end bits. For example, the same effect can be obtained even if the number of “1” added to the check matrix is reduced to M when transmitting information bits and then to N (M> N) when transmitting termination bits. .

In FIG. 13, an advantage when “1” is added to the upper trapezoidal matrix of the check matrix (“1” of reference numeral 1307 in FIG. 13) will be described. For example, the data X _f-2 (1308) and the parity P _f-2 (1309) at the time point f-2 in FIG. 13 are affected by the termination bit 1310. As a result, the reception device improves the reception quality of the data X _f-2 (1308) at the time point f-2. Similarly, the same effect can be obtained for data after time point f-2. Thus, when “1” is added to the upper trapezoidal matrix of the check matrix, the speed at which the trellis diagram is stabilized (converges) can be improved due to the above-described effects.

  In the present embodiment, the order of the check polynomial is regularly decreased (the order is decreased every time the number of rows is increased by one), but the present invention has the same effect even if it is not regular. For example, the same effect can be obtained even if the order is reduced every few rows.

(Embodiment 4)
In the first and second embodiments, the method of designing the LDPC-CC from the (7, 5) convolutional code, that is, the feedback type convolutional code has been described. In this embodiment, a case will be described in which the LDPC-CC design method described in Embodiments 1 and 2 is applied to a feedforward type convolutional code. The advantage of using a feedforward type convolutional code is that if the constraint length is the same, the parity check matrix of the feedforward type convolutional code has lower row weights and column weights than the parity check matrix of the feedback type convolutional code. In addition, there is a feature that the number of loops of length 4 when a Tanner graph is drawn is small. A loop is a circuit (path that circulates) that starts at a certain node and ends at that node. If the number of loops having a length of 4 is large, the reception quality deteriorates (see Non-Patent Document 13). Therefore, when a feedforward type convolutional code is used, it is highly possible that reception quality is improved when BP decoding is performed. Therefore, the LDPC-CC designed from the feedforward type convolutional code has a feature that the performance is better than the LDPC-CC designed from the feedback type convolutional code.

Non-Patent Document 12 describes a feedforward type convolutional code that is a systematic code. Below, the case where the convolutional code of (1,1547) is used is demonstrated as an example. The check polynomial of the (1,1547) convolutional code is expressed as follows.

そして、ＬＤＰＣ−ＣＣ用の検査多項式として、以下式より与えられるＰ（Ｄ）を考える。

Then, the following equation is used as an example of a parity check polynomial different from equation (32) of the convolutional code (1,1547).

Then, P (D) given by the following equation is considered as a parity check polynomial for LDPC-CC.

In this case, α _1, ···, α _e is 15 or more integer, β _1, ···, β _f is 15 or more integer, γ _1, ···, γ _g is set to -1 or less integer . At this time, as described in the first embodiment and the second embodiment, at least one of α ₁ ,..., Α _e is set to an integer of 29 or more, and β _{1 ,.} At least one of them is set to an integer of 29 or more, and at least one of γ ₁ ,..., Γ _g is set to an integer of −15 or less. In addition, α _1, ···, α _e all the settings to 29 an integer equal to or greater than, β _1, ···, β all _f is set to 29 or more of integer, γ _1, ···, γ _g all It is more effective to set the value to -15 or less. With this setting, the reception quality (decoding performance) can be greatly improved.

For example, in equation (40), γ ₁ = −25, γ ₂ = −55, and γ ₃ = −95, or in equation (40), γ ₁ = −25 and γ ₂ = −65. Or, in equation (39), when β ₁ = 35, γ ₁ = −40, and γ ₂ = −90, the reception quality (decoding performance) is greatly improved.

However, in Formula (36), Formula (37), and Formula (39), at least one of α ₁ ,..., Α _e is set to an integer of 29 or more, or β ₁ ,. , β _f is set to an integer greater than or equal to 29, or at least one of γ ₁ ,..., γ _g is −
Even when it is set to 15 or less, the reception quality (decoding performance) can be greatly improved.

(Embodiment 5)
In the present embodiment, a method of creating an LDPC-CC with a coding rate of 1/3 from an LDPC-CC with a coding rate of 1/2 will be described in detail. Below, it demonstrates using the convolutional code | cord | chord which is a feedforward type | system | group code | symbol shown by the nonpatent literature 12, and a systematic code | cord | chord, and the (1,1547) convolutional code | symbol. The check polynomial of the (1,1547) convolutional code is expressed as follows.

Then, the following equation is used as an example of a parity check polynomial different from the equation (41) of the convolutional code of (1,1547).

Here, as a new parity sequence polynomial, consider Pn (D) given by the following equation.

The data at the time point i is X _i , the parity for P (D) of the equation (42) at the time point i is P _i , the equation (43) at the time point i, the equation (44), or the Pn of the equation (45) Assuming that the parity Pn _{i for} D), the transmission sequence W _i = (X _i , P _i , Pn _i ) can be expressed.

Then, the following equation is considered as a check polynomial for X (D) and P (D) for LDPC-CC, as in the fourth embodiment.

In this case, α _1, ···, α _e is 15 or more integer, β _1, ···, β _f is 15 or more integer, γ _1, ···, γ _g is set to -1 or less integer . At this time, as described in the first embodiment and the second embodiment, at least one of α ₁ ,..., Α _e is set to an integer of 29 or more, and β _{1 ,.} At least one of them is set to an integer of 29 or more, and at least one of γ ₁ ,..., Γ _g is set to −15 or less. However, α _1, ···, α _e all the set to 29 or more of integer, β _1, ···, β all _f set to 29 or more of integer, γ _1, ···, γ _g all It is more effective to set the value to -15 or less. With this setting, the reception quality (decoding performance) is greatly improved.

Then, the parity check polynomial for the new parity sequence Pn (D) for LDPC-CC is any one of Equations (43) to (45). At this time, at least one of a ₁ ,..., A _v is an integer of 29 or more, or at least one of a ₁ ,..., A _v is set to an integer of −15 or less. . Then, at least one of b _w ,..., B _w is set to an integer of 29 or more. However, if all of a ₁ ,..., A _v are set to an integer of 29 or more or an integer of −15 or less, reception quality (decoding performance) is greatly improved. Further, even if all of b ₁ ,..., B _w are set to integers of 29 or more, the reception quality (decoding performance) is greatly improved. Here, c _1, · · ·, but do not give a constraint for c _y, c _1, · · ·, at least one of c _y is effective when 29 or more integer, and generally thereof include, c _1, · · ·, in the c _y is made "0" is present one.

  Hereinafter, a method for generating an LDPC-CC of a convolutional code with a coding rate of 1/3 from a convolutional code with a coding rate of 1/2 will be summarized.

A parity check polynomial of a coding rate 1/2 convolutional code is expressed by the following equation: + K _x (data X
The maximum value of the maximum order of the term (D) and K ₁ (the maximum order of the term of the parity P (D)) is defined as K _max .

Then, as in Embodiments 1 to 4 and other Embodiments 1, X (D) and P (D) LDPC-CC parity check polynomials are created. Thereafter, Pn (D) obtained from the equations (54) to (56) is considered as a new parity sequence polynomial.

At this time, at least one of a ₁ ,..., A _v is an integer greater than or equal to 2K _max +1, or at least one of a ₁ , ..., a _v is −K _max −1 or less. Set to an integer. Then, at least one of b ₁ ,..., B _w is set to an integer equal to or greater than 2K _max +1. A ₁ ,..., A _v are all integers greater than or equal to 2K _max +1, or −K _max
When set to an integer of -1 or less, the reception quality (decoding performance) is greatly improved. B ₁ ,.
· ·, _{B w} reception quality set all the _{2K max} +1 or more integer (decoding performance) is greatly improved. Here, c _1, · · ·, but do not give a constraint for _{c y,} c _1, · · ·, at least one of _{c y} is effective when a _{2K max} +1 or greater, also generally, c _1, · · ·, in the c _y is made "0" is present one.

As described above, according to the present embodiment, the coding rate 1 is obtained by using the polynomial Pn (D) obtained from the equations (54) to (56) as the new parity sequence for the coding rate 1/3. An LDPC-CC with a coding rate of 1/3 is generated from a / 2 convolutional code. In this _{case, a 1, ···, a v} , b 1, ···, to b _w, by setting a limit as described above, check polynomial for encoding rate 1/2 P (D) Without changing, it is possible to expand the range in which the reliability is propagated, and to improve the reception quality (decoding performance).

  The method for generating LDPC-CC with a coding rate of 1/3 from a convolutional code with a coding rate of 1/2 has been described above. In addition, when generating an LDPC-CC with a coding rate of 1/4 or less, if a parity check polynomial for a new parity is generated under the same conditions as when generating an LDPC-CC with a coding rate of 1/3, An LDPC-CC with a coding rate of 1/4 or less can be generated.

(Embodiment 6)
A modification of the method for generating LDPC-CC from the convolutional code described in Embodiment 4 will be described in detail.

Any of the following polynomials is considered as the parity check polynomial for LDPC-CC described in the fourth embodiment.

In this case, α _1, ···, α _e is 15 or more integer, β _1, ···, β _f is 15 or more integer, γ _1, ···, γ _g is set to -1 or less integer . At this time, as described in the first embodiment and the second embodiment, at least one of α ₁ ,..., Α _e is set to an integer of 29 or more, and β _{1 ,.} At least one of them is set to an integer of 29 or more, and at least one of γ ₁ ,..., Γ _g is set to an integer of −15 or less. In addition, α _1, ···, α _e all the settings to 29 an integer equal to or greater than, β _1, ···, β all _f is set to 29 or more of integer, γ _1, ···, γ _g all It is more effective to set the value to -15 or less.

Then, the terms excluding the original convolutional code “1” and the maximum degree “D ¹⁴ ”, that is, the terms 1401 (D ¹⁰ ) and 1402 (D ⁵ , D ⁴ , D ³ , D ¹ ) in FIG. Select some terms from For example, as shown in FIG. 14, the term 1401 (D ¹⁰ ) is selected. Then, the selected terms are removed, and check polynomials such as the polynomial 1403 and the polynomial 1404 in FIG. 14 are considered. In polynomial 1403 in FIG. ^{14, D 10} is deleted, section ^{D z} is added relates X (D). In polynomial 1404 in FIG. ^{14, D 10} is deleted, section ^{D z} is added relates P (D). At this time, similarly to the description of the fourth embodiment, the reception quality is improved by setting z to an integer equal to or greater than 2K _max +1 with respect to the maximum order K _max (= 14). In the case of the polynomial 1403 in FIG. 14, the reception quality is improved even if z is an integer equal to or smaller than −K _max −1.

In FIG. 14, the case where the term 1401 (D ¹⁰ ) is selected and the term of D ^z is added has been described. However, the present invention is not limited to this, and a plurality of terms 1401 and 1402 are selected and the terms are removed. In addition, a plurality of D ^z terms may be added to create an LDPC-CC parity check polynomial.

Thus, in the present embodiment, at least one term is deleted except for the maximum degree D ^K of the original convolutional code, and at least one term of D ^z satisfying z ≧ 2K _max +1, Added for (D) or P (D). The LDPC-CC is configured using the parity check polynomial having the above configuration. Note that the term ^Dz may be added to X (D) and P (D).

  Moreover, although Formula (57) was demonstrated to the example, it is not restricted to this, In any case of Formula (58)-Formula (63), it can implement similarly. By performing such an operation, it is possible to delete a loop having a length of 4 in the Tanner graph described in Non-Patent Document 13 or a loop having a small length, for example, a loop having a length of 6, so that the reception quality is improved. This leads to a significant improvement.

(Embodiment 7)
In this embodiment, a configuration of a time-varying LDPC-CC that can easily perform puncturing and that has a simple encoder configuration will be described. In particular, in this embodiment, an LDPC-CC capable of periodically puncturing data will be described. In the LDPC code, a puncturing method for periodically puncturing data has not been sufficiently studied so far, and a method for performing puncturing in particular has not been sufficiently discussed. In LDPC-CC in the present embodiment, if data can be punctured periodically and regularly instead of being randomly punctured, degradation of reception quality can be suppressed. In the following, a configuration method of time-varying LDPC-CC capable of realizing the coding rate R = 1/2 will be described.

When the coding rate is 1/2 and the polynomial expression of the information sequence (data) is X (D) and the polynomial expression of the parity sequence is P (D), the parity check polynomial is expressed as follows.

In formula (64), a1, a2,..., An are integers of 1 or more (where a1 ≠ a2 ≠ ... ≠ an, and a1 to an are all different from each other). Note that when “X ≠ Y ≠... ≠ Z”, X, Y,..., Z are all different from each other. Further, b1, b2,... Bm are integers of 1 or more (where b1 ≠ b2 ≠ ... ≠ bm). Here, in order to enable easy encoding, it is assumed that the terms D ⁰ X (D) and D ⁰ P (D) (D ⁰ = 1) exist. Therefore, P (D) is expressed as follows.

As can be seen from the equation (65), since D ⁰ = 1 exists and past parity terms, that is, b1, b2,..., Bm are integers of 1 or more, the parity P is sequentially changed. Can be sought.

Next, a parity check polynomial with a coding rate of 1/2 different from the equation (64) is expressed as follows.

In formula (66), A1, A2,..., AN are integers of 1 or more (provided that A1 ≠ A2 ≠... AN). Further, B1, B2,..., BM are integers of 1 or more (where B1 ≠ B2 ≠ ... ≠ BM). Here, it is assumed that the terms D ⁰ X (D) and D ⁰ P (D) (D ⁰ = 1) exist in order to enable easy encoding. At this time, P (D) is expressed as shown in Expression (67).

Hereinafter, data X and parity P at time 2i are represented by X _2i and P _2i , respectively, and data X and parity P at time 2i + 1 are represented by X _{2i + 1} and P _{2i + 1} , respectively (i: integer).

In the present embodiment, the parity P _2i at the time point 2i is calculated (encoded) using the equation (65), and the parity P _{2i + 1} at the time point _{2i + 1} is calculated (encoded) using the equation (67). An LDPC-CC with a period of 2 is proposed. Similar to the above-described embodiment, there is an advantage that the parity can be easily obtained sequentially.

Below, it demonstrates using Formula (68) and Formula (69) as an example of Formula (64) and Formula (66).

  At this time, the check matrix H can be represented as shown in FIG. In FIG. 15, (Ha, 11) is a portion corresponding to the equation (68), and (Hc, 11) is a portion corresponding to the equation (69). Hereinafter, (Ha, 11) and (Hc, 11) are defined as sub-matrices.

  In this way, the LDPC-CC parity check matrix H of the proposed time-varying period 2 is divided into the first sub-matrix representing the parity check polynomial of equation (64) and the second sub-matrix representing the parity check polynomial of equation (66). Matrix. Specifically, in the check matrix H, the first sub-matrix and the second sub-matrix are alternately arranged in the row direction. When the coding rate is 1/2, the sub-matrix is shifted to the right by two columns in the i-th row and the i + 1-th row (see FIG. 15).

In the case of time-varying LDPC-CC with a time-varying period of 2, the i-th row sub-matrix and the i + 1-th row sub-matrix are different sub-matrices. That is, one of the sub-matrices (Ha, 11) or (Hc, 11) is the first sub-matrix, and the other is the second sub-matrix. Assuming that the transmission vector u is u = (X ₀ , P ₀ , X ₁ , P ₁ ,..., X _k , P _k ,...) ^T , Hu = 0 holds. This is the same as described in the first embodiment (see formula (3)).

  When the BP decoding is performed using the parity check matrix of FIG. 15, that is, the time-varying periodicity 2 parity check matrix, the data reception quality is greatly improved as compared with the LDPC-CC described in the first to sixth embodiments. Confirmed to do.

  Although the case where the time varying period is 2 has been described above, the time varying period is not limited to 2. However, if the time-varying period is too large, it is difficult to puncture periodically, and for example, it is necessary to puncture at random, and reception quality may deteriorate. Hereinafter, the advantage that the reception quality is improved by reducing the time-varying period will be described.

ここで、送信系列ベクトルｖ＝（ｖ１、ｖ２、ｖ３、ｖ４、ｖ５、ｖ６、・・・、ｖ2i、ｖ2i+1、・・・）^Ｔである。 FIG. 16 shows an example of a puncturing method in the case of time varying period 1. In the figure, H is an LDPC-CC parity check matrix, and when a transmission sequence vector is represented by v, a relational expression of Expression (70) is established.

Here, the transmission sequence vector v = (v1, v2, v3, v4, v5, v6,..., V2i, v2i + 1,...) ^T.

  FIG. 16 shows an example in which a transmission sequence with a coding rate R = 1/2 is punctured to obtain a coding rate R = 3/4. When puncturing is performed periodically, first, a block cycle for selecting puncture bits is set. FIG. 16 shows an example in which the block period is set to 6 and the block is set as indicated by a dotted line (1602). Of the 6 bits constituting one block, 2 bits are selected as puncture bits, and the selected 2 bits are set as bits that are not transmitted. In FIG. 16, a bit 1601 surrounded by a circle is a bit that is not transmitted. In this way, a coding rate of 3/4 can be realized. Therefore, the transmission data series are v1, v3, v4, v5, v7, v9, v11, v13, v15, v16, v17, v19, v21, v22, v23, v25,.

  In FIG. 16, “1” surrounded by a square frame is set to 0 because the initial log likelihood ratio does not exist at the time of reception due to puncturing.

  In BP decoding, row operations and column operations are repeated. Therefore, if two or more bits (erasure bits) for which no initial log likelihood ratio exists (log likelihood ratio is 0) are included in the same row, the initial log likelihood ratio is calculated by column operation in that row. The log-likelihood ratio is not updated by the row operation alone until the log-likelihood ratio of bits for which there is no (the log-likelihood ratio is 0) is updated. That is, the reliability is not propagated by the row operation alone, and in order to propagate the reliability, it is necessary to repeat the row operation and the column operation. Therefore, if there are a large number of such rows, there is a limit to the number of iteration processes in BP decoding, or the reliability is not propagated even if the iteration process is repeated many times, which causes deterioration in reception quality. It becomes. In the example shown in FIG. 16, the bit corresponding to 1 surrounded by a square frame indicates an erasure bit, and the row 1603 is a row in which reliability is not propagated by a row operation alone, that is, a cause of deterioration of reception quality. Line.

  Therefore, as a method for determining puncture bits (bits not to be transmitted), that is, a method for determining a puncture pattern, it is necessary to search for a method that minimizes the number of rows whose reliability is not propagated by puncture alone. Hereinafter, a search for a puncture bit selection method will be described.

When 2 bits out of 6 bits constituting one block are set as puncture bits, there are _{3 × 2} C ₂ selection methods for 2 bits. Among these, the selection method that is cyclically shifted within 6 bits of the block period can be regarded as the same. Hereinafter, supplementary explanation will be given with reference to FIG. 18A. As an example, FIG. 18A shows six puncture patterns in the case of continuously puncturing 2 bits out of 6 bits. As shown in FIG. 18A, the puncture patterns # 1 to # 3 become the same puncture pattern by changing the block delimiter. Similarly, puncture patterns # 4 to # 6 become the same puncture pattern by changing the block delimiter. As described above, the selection methods that are cyclically shifted in the 6 bits of the block period can be regarded as the same. Therefore, there are _{3 × 2} C ₂ × 2 / (3 × 2) = 5 ways to select puncture bits.

In addition, when one block is composed of L × k bits and k bits of L × k bits are punctured, there are the number of puncture patterns obtained by Expression (71).

FIG. 18B shows the relationship between the encoded sequence and the puncture pattern when attention is paid to one puncture pattern. Although not shown, puncture bits are also used for Xi + 3 and Pi + 3. For this reason, as can be seen from FIG. 18B, when 2 bits out of 6 bits constituting one block are punctured, the existing check expression pattern is (3 × 2) × 1/2 for one puncture pattern. It becomes. Similarly, when one block is composed of L × k bits and k bits of L × k bits are punctured, there are the number of check equations obtained by Equation (72) for one puncture pattern.

Therefore, in the puncture pattern selection method, it is necessary to check whether or not the reliability is propagated independently for the number of check expressions (rows) obtained from Expression (73).

From the above relationship, when the coding rate is 1/2 and the coding rate is 3/4, when the k bits are punctured from the L × k bit block, the number of check equations obtained from Equation (74) ( It is necessary to check whether or not the reliability is propagated by itself.

  If no good puncture pattern is found, L and k need to be increased.

  Next, consider the case where the time-varying period is m. Also in this case, as in the case where the time-varying period is 1, m different inspection formulas represented by the formula (64) are prepared. Hereinafter, m inspection formulas are named “inspection formula # 1, inspection formula # 2,..., Inspection formula #m”.

なお、式（７５）において、ＬＣＭ｛α，β｝は、自然数αと自然数βとの最小公倍数をあらわす。 Then, the parity P _{mi + 1} at the time point mi + 1 is obtained using the “check equation # 1”, the parity P _{mi + 2} at the time point mi + 2 is obtained using the “check equation # 2”, and the parity P _{mi + m} at the time point mi + m is obtained as “ Consider the LDPC-CC obtained using the “check formula #m”. At this time, when considered in the same manner as in FIG. 15, the parity check matrix is represented as shown in FIG. 17. Then, in the case where the coding rate is 1/2 to the coding rate 3/4, for example, when puncturing 2 bits from a 6-bit block, when considered similarly to Equation (73), Equation (75) It is necessary to check whether or not the number of check formulas (rows) obtained from (1) alone is a row whose reliability is not propagated.

In Expression (75), LCM {α, β} represents the least common multiple of the natural number α and the natural number β.

  As can be seen from equation (75), as m increases, the number of check equations that must be checked increases. Therefore, a puncturing method that periodically punctures is not suitable, and for example, a method of puncturing at random is used, so that reception quality may be deteriorated.

  It should be noted that in FIG. 18C, when puncturing is performed to generate a code sequence of coding rate R = 2/3, 3/4, and 5/6 by puncturing from L × k bits to k bits, this must be checked. Indicates the number of parity check polynomials that should not be.

  In reality, the time-varying period during which the best puncture pattern can be searched is about 2 to 10. In particular, the time-varying cycle 2 is suitable in consideration of the time-varying cycle in which the best puncture pattern can be searched and the improvement in reception quality. Further, when the time-varying cycle is 2, and the check equations such as the equations (64) and (66) are periodically repeated, there is an advantage that the encoder / decoder can be configured very easily.

  In the case of time-varying periods 3, 4, 5,..., The configuration of the code / decoder is slightly larger than when the time-varying period is 2. Similarly to the above, when a plurality of parity check expressions based on the expressions (64) and (66) are periodically repeated, a simple configuration can be adopted.

In addition, when the time-varying cycle is semi-infinite (extremely long cycle) or when an LDPC-CC is created based on LDPC-BC, the time-varying cycle is generally very long. It is difficult to search for the best puncture pattern by adopting a method of selecting puncture bits. For example, it is conceivable to adopt a method of selecting puncture bits at random, but there is a possibility that the reception quality at the time of puncture is greatly deteriorated.

Incidentally, formula (64), (66), (68), it is also possible to express check polynomial by multiplying the ^{D n} in, on both sides (69). In the present embodiment, there are D ⁰ X (D) and D ⁰ P (D) terms (D ⁰ = 1) in formulas (64), (66), (68), and (69). It was.

In this way, since the parity can be calculated sequentially, the configuration of the encoder is simplified, and in the case of the systematic code, considering the reliability propagation to the data at the time point i, the data When the term D ⁰ exists in both the parity and the parity, since the reliability propagation to the data can be easily understood, the code design can be easily performed. If the ease of code design is not taken into consideration, D ⁰ X (D) does not need to exist in the equations (64), (66), (68), and (69).

  FIG. 19A shows an example of an LDPC-CC parity check matrix with a time varying period of 2. As shown in FIG. 19A, in the case of time-varying period 2, two parity check expressions, a parity check expression 1901 and a parity check expression 1902, are used alternately.

  FIG. 19B shows an example of an LDPC-CC parity check matrix with a time varying period of 4. As shown in FIG. 19B, in the case of time-varying period 4, four parity check expressions, that is, a parity check expression 1901, a parity check expression 1902, a parity check expression 1903, and a parity check expression 1904 are repeatedly used.

As described above, according to the present embodiment, a parity sequence is obtained using a parity check polynomial (64) and a parity check polynomial (66) different from equation (64), which is a time-varying period 2 parity check matrix. I did it. Note that the time-varying period is not limited to 2, and for example, a parity sequence may be obtained using a parity check matrix having a time-varying period 4 as shown in FIG. 19B. However, if the time-varying period m is too large, it is difficult to puncture periodically, and, for example, punctures at random, so that reception quality deteriorates. Realistically, the time-varying period during which an optimum puncture pattern can be searched is about 2 to 10. In this case, reception quality can be improved and puncturing can be performed periodically, so that an LDPC-CC encoder can be configured easily.

  It has been confirmed that good reception quality can be obtained when the row weight in the parity check matrix H, that is, the number of elements that is 1 among the row elements constituting the parity check matrix is 7 to 12. As described in Non-Patent Document 12, when a code having an excellent minimum distance in a convolutional code is considered, the row weight increases as the constraint length increases. For example, in a feedback convolutional code with a constraint length of 11, Considering that the weight is 14, the point where the row weight is 7 to 12 can be considered as a value peculiar to the LDPC-CC of the present proposal. Further, when considering the merit of code design, the design is facilitated by making the row weights of the rows of the LDPC-CC parity check matrix equal.

  In the above description, the case of coding rate 1/2 has been described. However, the present invention is not limited to this, and a parity sequence is obtained using a check matrix having a time-varying period m even in cases other than coding rate 1/2. The same effect can be obtained when the time varying period is 2 to about 10 time varying periods.

  In particular, when the coding rate R = 5/6, 7/8 or more, in the LDPC-CC having the time varying period 2 or the time varying period m described in the present embodiment, only by a row including two or more erasure bits. Select a puncture pattern that is not configured. In other words, selecting a puncture pattern in which there are 0 or one erasure bit is a good reception when the coding rate is high, such as coding rate R = 5/6, 7/8 or higher. It is important in obtaining quality.

(Embodiment 8)
In this embodiment, a time-varying LDPC- that uses a check equation in which “1” exists in the upper trapezoid matrix of the check matrix described in Embodiment 2 and that can easily configure an encoder. CC will be described. In the following, a configuration method of time-varying LDPC-CC capable of realizing the coding rate R = 1/2 will be described.

When the coding rate is 1/2 and the polynomial expression of the information sequence (data) is X (D) and the polynomial expression of the parity sequence is P (D), the parity check polynomial is expressed as follows.

In the formula (76), a1, a2,..., An are integers of 1 or more (where a1 ≠ a2 ≠ ... ≠ an). In addition, b1, b2,..., Bm are integers of 1 or more (however, b1 ≠ b2 ≠ ... ≠ bm. Also, c1, c2,..., Cq are −1 or less. As an integer, c1 ≠ c2 ≠... ≠ cq, where P (D) is expressed as follows.

  Similar to the second embodiment, the parity P can be obtained sequentially.

Next, Equation (78) and Equation (79) are considered as parity check polynomials with a coding rate of 1/2 different from Equation (76).

In the expressions (78) and (79), A1, A2,..., AN are integers of 1 or more (provided that A1 ≠ A2 ≠ ... ≠ AN). Further, B1, B2,..., BM are integers of 1 or more (where B1 ≠ B2 ≠ ... ≠ BM). C1, C2,..., CQ are integers equal to or less than −1 (where C1 ≠ C2 ≠ ... ≠ CQ). At this time, P (D) is expressed as follows.

Hereinafter, data X and parity P at time 2i are represented by X _2i and P _2i , respectively, and data X and parity P at time 2i + 1 are represented by X _{2i + 1} and P _{2i + 1} , respectively (i: integer).

At this time, the parity P _2i at the time point 2i is obtained by using the equation (77), and the parity P _{2i + 1} at the time point 2i + 1 is obtained by using the equation (80). The LDPC-CC having the time varying period of 2 or the parity at the time point 2i P _2i is found using equation (77), parity _{P 2i + 1} of point in time 2i + 1 is considered the varying period of 2 LDPC-CC when determined using equation (81).

Such LDPC-CC codes are:
The encoder can be easily configured, and the parity can be obtained sequentially. The puncture bit can be set periodically.
-It has the advantage that it is possible to reduce the number of termination bits and improve the reception quality at the time of puncturing at the termination.

Next, consider LDPC-CC with a time-varying period of m. As in the case of the time-varying cycle 2, “check expression # 1” represented by the expression (78) is prepared, and from “check expression # 2” represented by either the expression (78) or the expression (79), “ “Inspection formula #m” is prepared. The data X and the parity P at the time point _{mi + 1} are represented as X _{mi + 1} and P _{mi + 1} , respectively, the data X and the parity P at the time point mi + 2 are respectively represented as X _{mi + 2} , P _{mi + 2 ,.} , P _{mi + m} (i: integer).

At this time, the parity P _{mi + 1} at the time point mi + 1 is obtained using the “check equation # 1”, the parity P _{mi + 2} at the time point mi + 2 is obtained using the “check equation # 2,” and the parity P _{mi + m} at the time point mi + m is obtained. Consider an LDPC-CC obtained using “checking formula #m”. Such LDPC-CC codes are:
The encoder can be easily configured, and the parity can be obtained sequentially. Advantages include reduction of termination bits and improvement in reception quality at the time of puncturing at the termination.

  As described above, according to the present embodiment, a parity sequence is obtained from a parity check matrix (76) and a parity check polynomial (78) different from equation (76), and a parity check matrix having a time-varying period of 2. I did it.

  Thus, when using a check equation in which “1” exists in the upper trapezoidal matrix of the check matrix, a time-varying LDPC-CC encoder can be easily configured. The time varying period is not limited to 2. However, in the same manner as in the seventh embodiment, when the method of periodically puncturing is employed, the time-varying period during which the best puncture pattern can be searched is about 2 to 10.

  In the case of time-varying periods 3, 4, 5,..., The configuration of the code / decoder is slightly larger than when the time-varying period is 2. Similarly to the above, when the inspection formulas of the formulas (78) and (79) are periodically repeated, a simple configuration can be adopted.

Incidentally, formula (76), (78), it is also possible to express check polynomial by multiplying the ^{D n} in, on both sides (79). In this embodiment, it is assumed that the terms (D ⁰ = 1) of D ⁰ X (D) and D ⁰ P (D) exist in formulas (76), (78), and (79). .

In this way, since the parity can be calculated sequentially, the configuration of the encoder is simplified, and in the case of the systematic code, considering the reliability propagation to the data at the time point i, the data In addition, if there is a term of D ⁰ in both the parity and the parity, code design can be easily performed. If the ease of code design is not taken into consideration, D ⁰ X (D) does not need to exist in the equations (76), (78), and (79).

  Further, it has been confirmed that good reception quality can be obtained when the row weight in the parity check matrix H, that is, the number of elements which is 1 among the row elements constituting the parity check matrix is 7-12. As described in Non-Patent Document 12, when a code having an excellent minimum distance in a convolutional code is considered, the row weight increases as the constraint length increases. For example, in a feedback convolutional code with a constraint length of 11, Considering that the weight is 14, the point where the row weight is 7 to 12 can be considered as a value peculiar to the LDPC-CC of the present proposal. Further, when considering the merit of code design, the design is facilitated by making the row weights of the rows of the LDPC-CC parity check matrix equal.

(Embodiment 9)
In this embodiment, a method for creating an LDPC-CC with a coding rate of 1/3 from an LDPC-CC with a coding rate of 1/2 (time-varying period m) described in Embodiments 7 and 8. explain in detail. As an example, an LDPC-CC with a time varying period of 2 will be described as an example.

Data X and parity P at time _2i are represented as X _2i and P _2i , respectively, and data X and parity P at time 2i + 1 are represented as X _{2i + 1} and P _{2i + 1} (i: integer), respectively. At this time, parity _{P 2i} of point in time 2i is found using Equation (64), parity _{P 2i + 1} of point in time 2i + 1 is the varying period when determined using Equation (66) Consider the second LDPC-CC.

Here, a new parity sequence polynomial is Pn (D), and any one of the equations (82) to (84) is considered.

  It should be noted that a1, a2,..., Ay are integers of 1 or more (where a1 ≠ a2 ≠ ... ≠ ay). In addition, b1, b2,..., Bw are integers of 1 or more (where b1 ≠ b2 ≠ ... ≠ bw). In addition, c1, c2,..., Cy (where c1.noteq.c2.noteq.cy) are integers of 0 or more.

  Then, different check polynomials “check formula # 1” and “check formula # 2” configured by any one of formulas (82) to (84) are prepared.

The data at the time point 2i is determined as X _2i and the parity P2i at the time point 2i using the equation (64), and the parity Pn _{, 2i} (the parity for the coding rate 1/3) at the time point 2i is expressed as “check equation # 1 ". At this time, the transmission sequence can be expressed as W _2i = (X _2i , P _2i , Pn _2i ).

Similarly, _{X 2i + 1} data at the time 2i + 1, and the parity _{P 2i + 1} at time 2i + 1, calculated using equation (66), the parity _{Pn 2i + 1} at point in time 2i + 1 of the (parity for a coding rate of 1/3) It calculates | requires using "inspection formula # 2." At this time, the transmission sequence can be expressed as W _{2i + 1} = (X _{2i + 1} , P _{2i + 1} , Pn _{2i + 1} ). In general, there is one of c1,..., Cy that is “0”.

  Consider terms corresponding to X (D), P (D), and Pn (D) in equations (82), (83), and (84), respectively. The check expression for the coding rate 1/2 is composed of Expressions (64) and (66). At this time, X (D) and P (D) each have a plurality of terms (a plurality of “1” s in the check matrix). When the coding rate is 1/3, a check expression configured by any one of the expressions (82), (83), and (84) is added.

Consider the column weight at this time. According to the check equation when the coding rate is 1/2, the data X and the parity P have a certain column weight, for example, a weight of about 5. In this state, if a check expression configured by any one of the expressions (82), (83), and (84) is added to make the coding rate 1/3, the column weight of the data X and the parity P increases. However, if the BP decoding is performed unless the column weight is suppressed to some extent, the reception quality cannot be improved. Therefore, when the coding rate is set to 1/3, the number of increases in the column weights of the data X and the parity P when the check equation configured by any one of the equations (82), (83), and (84) is added. Must be limited to 1 or 2. Therefore, Formula (82) becomes either of Formula (85)-Formula (88).

Moreover, Formula (83) becomes either Formula (89) or (90).

Moreover, Formula (84) becomes either Formula (91) or (92).

  Hereinafter, the relationship between the number of terms of the parity check polynomial with a coding rate of 1/2 and the number of terms of the parity check polynomial added to achieve the coding rate of 1/3 will be described. In the following description, an LDPC-CC with a coding rate of ½ is created using a parity check polynomial expressed by Equation (64) and Equation (66), and an LDPC-CC with a coding rate of ３ is used. Will be described by taking as an example a case in which a parity check polynomial expressed by the equations (82) to (92) is added.

The number of terms of the data X (D) in the equation (64) is n + 1 because D ^{a1 to} D ^an and D ⁰ exist. Further, the number of terms of the parity P (D) in the equation (64) is m + 1 because D ^{b1 to} D ^bm and D ⁰ exist.

Similarly, the number of terms of the data X (D) in the equation (66) is N + 1 because D ^{A1 to} D ^AN and D ⁰ exist. Further, the number of terms of the parity P (D) in the equation (66) is M + 1 because D ^{B1 to} D ^BM and D ⁰ exist.

Similarly, the number of terms of the parity Pn (D) in the equations (82) to (92) is γ + 1 because D ^{c1 to} D ^cγ and D ⁰ exist.

  Here, Z is the minimum value of the number of terms, n + 1, m + 1, N + 1, and M + 1 in Expression (64) and Expression (66). At this time, it has been confirmed that good reception quality can be obtained when the relationship of γ + 1 <Z is established with respect to the number of terms γ + 1 of the parity Pn (D) in the equations (82) to (92).

  In the case of time-varying period 2, two different parity check polynomials are added to create an LDPC-CC with a coding rate of 1/3. Therefore, in these two parity check polynomials, γ + 1 <Z A relationship needs to be established.

  In this way, good reception quality can be obtained because the parity check matrix with a coding rate of 1/2 is “1” so that the reception characteristics are good when the coding rate is 1/2. This is because the influence on the reception quality is reduced by preventing the number of “1” s from being increased so much.

  As described above, in this embodiment, the parity check polynomial (82) is added to the parity check matrix (64) and the parity check polynomial (66) different from the equation (64) with the time-varying period 2 check matrix. A check matrix having a time-varying period 2 composed of (84) is added, and a parity sequence is obtained using the added check matrix.

  By doing in this way, LDPC-CC of the coding rate 1/3 of the time-varying period 2 can be produced | generated from the convolutional code of the coding rate 1/2 of the time-varying period 2. Also, when generating an LDPC-CC with a coding rate of ¼ or less, it can be generated in the same way as when generating an LDPC-CC with a coding rate of ３.

  Note that the time-varying cycle is not limited to 2, and the same can be applied to the case of the time-varying cycle m described in the seventh and eighth embodiments. Of course, the same can be applied to the case of the time varying period 2 of the eighth embodiment. Also, in the above description, in order to set the coding rate to 1/3, a parity check equation of time-varying period 2 configured by any one of Equations (82) to (84) is newly used as a parity check equation. Although the case where it is used has been described, it can be similarly implemented even if a parity check polynomial having a time-varying period n is used. In the formula (82) to the formula (84), as in the eighth embodiment, a1, a2,.

  It has also been confirmed that good reception quality can be obtained when the relationship n = Km or m = Kn (K: natural number) is established for the time varying period n and the time varying period m.

  Also, among m parity check polynomials of coding rate 1/2 (time-varying period m), the minimum number of terms Z of terms X (D) and P (D) and coding rate 1/3 (hours) It is preferable that the relationship of γ + 1 <Z is established in all n parity check polynomials with the number of terms γ + 1 of the Pn (D) of n parity check polynomials added for the variable period n). Receive quality can be obtained.

(Other embodiment 1)
In the above description, the convolutional code when the coding rate is 1/2 has been described as an example. In the present embodiment, a configuration method of LDPC-CC when the coding rate is 1 / n will be described.

When the coding rate is 1 / n, the polynomial representation of the information sequence (data) is X (D), the polynomial representation of the parity 1 sequence is P ₁ (D), and the polynomial representation of the parity 2 sequence is P ₂ (D). ,..., P _n−1 (D) is a polynomial expression of the parity n−1 sequence, the parity check polynomial can be expressed as the following equation (93).

At this _{time, Kx, K 1, K 2} , ···, K n-1 is set to an integer of 0 or _{more, Kx, K 1, K 2} , ···, maximum value _{K max} of _{K n-1} And

Here, data at time point i is represented as X _i , parity 1 is represented as P _{1, i} , parity 2 is represented as P _{2, i} ,..., And parity n−1 is represented as P _{n−1, i} . The transmission vector _{w = (X 1, P 1,1} , P 2,1, ···, P n-1,1, X 2, P 1,2, P 2,2, ···, P n _1,2, representing _{···, X i, P 1,} i, P 2, i, ···, P n-1, i, ···) and. In this case, when the check matrix is H, the above equation (3) is established.

Here, as in the first embodiment, “1” is added to the parity check matrix in consideration of probability propagation for data or (and) parity. At this time, one or more of the terms 2001_0, 2001_1, 2001_2,..., 2001_n−1 in FIG. 20 are selected and changed to the terms 2002_0, 2002_1, 2002_2,. For example, when 2001_0 in FIG. 20 is selected, 2001_0 is changed to 2002_0, and the others are not changed. When 2001_0 and 2001_n-1 in FIG. 20 are selected, 2001_0 is changed to 2002_0, and 2001_n-1 is changed to 2002_n-1, and the others are not changed. Naturally, all of the terms 2001_0, 2001_1, 2001_2,..., 2001_n−1 in FIG. That is, the check polynomial is expressed by the following equation (94).

At this time, s _x , s ₁ , s ₂ ,..., S _n−1 in FIG. 20 and Expression (94) are 1 or more, and h1, h2,..., Hs _k ≧ 2K _max +1 are set. (K = x, 1, 2,..., N−1). Thereby, good reception quality can be obtained. Further, h1, h2, ···, one or more of hs _k can obtain good reception quality even if it meets the above _{2K max} +1.

  Next, a method of adding “1” to the upper trapezoid matrix of the parity check matrix with the coding rate of 1 / n will be described in detail.

When the coding rate is 1 / n, the polynomial representation of the information sequence (data) is X (D), the polynomial representation of the parity 1 sequence is P ₁ (D), and the polynomial representation of the parity 2 sequence is P ₂ (D). ,..., P _n−1 (D) is a polynomial expression of the parity n−1 sequence, the parity check polynomial is expressed by Equation (32).

Here, data at time point i is represented as X _i , parity 1 is represented as P _{1, i} , parity 2 is represented as P _{2, i} ,..., And parity n−1 is represented as P _{n−1, i} . The transmission vector _{w = (X 1, P 1,1} , P 2,1, ···, P n-1,1, X 2, P 1,2, P 2,2, ···, P n _1,2, representing _{···, X i, P 1,} i, P 2, i, ···, P n-1, i, ···) and. In this case, when the check matrix is H, the above equation (3) is established.

Here, as in the second embodiment, probability propagation is considered for data or (and) parity, and “1” is added to the parity check matrix. At this time, the term 15_0 in FIG. 21 is changed to the term 15_1, 0. That is, the check polynomial is expressed by the following equation (95).

At this time, s _x in FIG. 21 is 1 or more, and is set to h1, h2,... Hs _x ≦ −K _max −1. Thereby, good reception quality can be obtained. Also, good reception quality can be obtained even when one or more of h1, h2,..., Hs _x satisfy −K _max −1 or less.

  As described above, the method described in Embodiments 1 and 2 may be extended to a method for generating an LDPC-CC from a convolutional code having a coding rate of 1 / n as in the present embodiment. it can. Also, when creating an LDPC-CC from a convolutional code with a coding rate other than those described above, the LDPC-CC can be created in the same manner by extending the method described so far.

  In the present invention, even when data is transmitted by performing puncturing as described in Non-Patent Document 12, data can be obtained by performing BP decoding in the receiving device. At this time, since the LDPC-CC described in the embodiment is represented by a simple parity check matrix, data can be easily punctured as compared with the LDPC-BC.

In the present embodiment, an example in which “1” is added to the upper trapezoid matrix of the parity check matrix as shown in FIG. 21 has been described. However, the present invention is not limited to this, and the case of FIG. In addition to adding “1” to the upper trapezoidal matrix of the combination and parity check matrix, “1” may be added to the approximate lower triangular matrix of the parity check matrix. As a result, further improvement in reception quality can be expected. The check polynomial in this case is represented by the following equation (96).

  Also, the termination method at the coding rate 1/2 described in the third embodiment can be similarly performed at the coding rate 1 / n as in the present embodiment.

(Other embodiment 2)
Here, the configuration of the encoder of the present invention will be described. FIG. 22 is a diagram illustrating an example of the configuration of the encoder of Expression (15).

22, the parity calculation unit 2202 includes data x (2201) (that is, X (D) of (Expression 15)), stored data 2205 (that is, D ^α1 X (D) and D of ( ^Expression 15)). ^α2 X (D), D ⁹ X (D), D ⁶ X (D), D ⁵ X (D)), stored parity 2207 (ie, D ^β1 P (D), D ^{β2 of} ( ^Equation 15)) P (D), D ⁹ P (D), D ⁸ P
(D), D ³ P (D), DP (D)) are input, the calculation of Expression (15) is performed, and the parity 2203 (that is, P (D) of Expression (15)) is output.

  The data storage unit 2204 receives the data x (2201) and stores the value. Similarly, the parity storage unit 2206 receives the parity 2203 and stores the value.

FIG. 23 is a diagram illustrating an example of the configuration of the encoder of Expression (19). 23 that operate in the same manner as in FIG. 22 are given the same reference numerals. The storage unit 2302 stores the data 2301 and outputs the stored data 2303 (D ^α1 X (D)... D ^αn X (D) in Expression (19)).

The data storage unit 2204 outputs the stored data 2205 (that is, D ² X (D) in Expression (19)).

The parity storage unit 2206 outputs the stored parity 2207 (that is, D ² P (D), DP (D) in Expression (19)).

  The parity calculation unit 2202 receives each signal as input, calculates the parity of Equation (19), and outputs it.

  As described above, basically, an encoder can be configured by a shift register and exclusive OR.

  Next, sum-product decoding will be described as an example of a decoder algorithm. The algorithm for sum-product decoding is as follows.

sum-product decoding A binary M × N matrix H = {H _mn } is set as a parity check matrix of an LDPC code to be decoded. Subsets A (m) and B (n) of the set [1, N] = {1, 2,..., N} are defined as the following equations (97) and (98).

  A (m) means a set of column indexes that are “1” in the m-th row of the parity check matrix H, and B (n) is a row index that is “1” in the n-th row of the parity check matrix H. It is a set.

Step A ・ 1 (Initialization)
The log likelihood ratio β ⁽⁰⁾ _mn = λ _n is set for all pairs (m, n) satisfying H _mn = 1. The loop variable (number of iterations) is set to l _sum = 1, and the maximum number of loops is set to l _{sum and mux} .

Step A ・ 2 (line processing)
Logarithm using the following update formulas (99), (100), and (101) for all pairs (m, n) satisfying H _mn = 1 in the order of m = 1, 2,... The likelihood ratio α ⁽ⁱ⁾ _mn is updated. Here, i represents the number of iterations, and f is a Gallager function.

Step A ・ 3 (column processing)
For all pairs (m, n) that satisfy H _mn = 1 in the order of n = 1, 2,..., N, the log likelihood ratio β ⁽ⁱ⁾ _mn using the following update equation (102): Update.

Step A ・ 4 (Calculation of log-likelihood ratio)
The log likelihood ratio L ⁽ⁱ⁾ _{n is obtained} by the following equation (103) for n∈ [1, N].

Step A ・ 5 (Counting the number of iterations)
If l _sum <l _{sum, mux} , increment l _sum and return to step A · 2. On the other hand, if l _sum = l _{sum, mux} , codeword w is estimated as shown in equation (104), and sum-product decoding is terminated.

By the way, as shown in FIG. 3, the transmission sequence (data after encoding) is n ₁ , n ₂ , n ₃ , n ₄ ,..., And u = (n ₁ , n ₂ , n ₃ , n ₄ , ..) And the generator matrix is G, the following relational expression (105) is established.

If the information sequence vector i = (i ₁ , i ₂ ,...), The following relational expression (106) is established.

  By using the relational expressions of the expressions (105) and (106), a transmission sequence is obtained.

FIG. 24 is a diagram illustrating an example of a configuration when sum-product decoding is used as a decoder. 24 includes a log likelihood ratio storage unit 2403, a row processing calculation unit 2405, a row processing data storage unit 2407, a column processing calculation unit 2409, a column processing data storage unit 2411, a control unit 2413, a log likelihood. A degree ratio calculation unit 2415 and a determination unit 2417 are provided.

  The log likelihood ratio storage unit 2403 receives the log likelihood ratio signal 2401 and the timing signal 2402 and stores the log likelihood ratio of the data section based on the timing signal 2402. Then, the log likelihood ratio storage unit 2403 outputs the stored log likelihood ratio as a signal 2404 to the row processing calculation unit 2405.

The row processing calculation unit 2405 receives the log likelihood ratio signal 2404 and the column-processed signal 2412 as input, and performs the above Step A · 2 (row processing) calculation at a position where “1” exists in the check matrix H. Do. Since the decoder performs iterative decoding, the row processing operation unit 2405 performs row processing using the log likelihood ratio signal 2404 at the time of the first decoding (corresponding to the processing of Step A · 1 described above). In the second decoding, row processing is performed using the signal 2412 after column processing. Then, the row processing operation unit 2405 outputs the signal 2406 after the row processing to the post-row processing data storage unit 2407.

  The post-row processing data storage unit 2407 receives the post-row processing signal 2406 and stores all post-row processing values (signals). Then, the post-row processing data storage unit 2407 outputs the post-row processing signal 2408 to the column processing calculation unit 2409 and the log likelihood ratio calculation unit 2415.

The column processing operation unit 2409 receives the signal 2408 and the control signal 2414 after row processing as input, confirms from the control signal 2414 that it is not the last iterative operation, and at the position where “1” exists in the check matrix H, Perform Step A • 3 (column processing). The column processing calculation unit 24
09 outputs the post-column processing signal 2410 to the post-column processing data storage unit 2411.

  The post-column processing data storage unit 2411 receives the post-column processing signal 2410 and stores all post-column processing values (signals). Then, the post-column processing data storage unit 2411 outputs the post-column processing signal 2412 to the row processing calculation unit 2405.

  The control unit 2413 receives the timing signal 2402, counts the number of iterations, and outputs the number of iterations as a control signal 2414 to the column processing computation unit 2409 and the log likelihood ratio computation unit 2415.

The log likelihood ratio calculation unit 2415 receives the row-processed signal 2408 and the control signal 2414 as input, and determines that the last iterative calculation is based on the control signal 2414, the check matrix H has “1”. Step A · 4 (logarithmic likelihood ratio calculation) is performed on the position to obtain a log likelihood ratio signal 2416. Then, the log likelihood ratio calculation unit 2415 outputs the log likelihood ratio signal 2416 to the determination unit 2417.

  The determination unit 2417 receives the log likelihood ratio signal 2416, estimates a codeword, and outputs an estimated bit 2418.

Here, sum-product decoding has been described as BP decoding, but decoding can also be performed by using min-sum decoding approximating BP decoding, offset BP decoding, Normalized BP decoding, shuffled BP decoding, and the like.

(Other embodiment 3)
In the LDPC-CC described in the embodiments so far, when puncturing is performed, there arises a problem of whether data or parity should be punctured preferentially (selected as a bit not to be transmitted).

When a row of the parity check matrix is considered, that is, when a parity check polynomial is considered, the number of 1s in the position corresponding to the data in the row of the parity check matrix is Nx, and the number of 1s in the position corresponding to the parity is Np may be selected, and a bit to be punctured preferentially (selected as a bit not to be transmitted) may be selected as in 1) or 2) below, depending on the comparison result between Np and Nx.
1) When Np <Nx: data is punctured preferentially 2) When Nx <Np: parity is preferentially punctured

  In this way, it is possible to suppress deterioration in reception quality when puncturing is performed.

(Other embodiment 4)
In this embodiment, a transmission device and a reception device that realize the puncturing method described in the above embodiments will be described. The transmission apparatus and the reception apparatus according to the present embodiment can support a plurality of coding rates.

  FIG. 25 is a diagram illustrating a configuration of the transmission apparatus according to the present embodiment. 25 includes an LDPC-CC encoding unit (LDPC-CC encoder) 2510, a puncturing unit 2520, an interleaving unit 2530, and a modulation unit 2540.

  The LDPC-CC encoding unit 2510 performs encoding on the data X using an LDPC-CC parity check matrix having a coding rate specified by the control signal. For example, when the control signal specifies an encoding rate of 1/2 or more, the LDPC-CC encoding unit 2510 encodes the data X using an LDPC-CC parity check matrix with an encoding rate of 1/2. The data X and the parity P are output to the puncture unit 2520. In addition, when the control signal specifies the coding rate 1/3, the LDPC-CC coding unit 2510 performs coding on the data X using the LDPC-CC parity check matrix with the coding rate 1/3. The data X, the parity P, and the parity Pn are output to the puncturing unit 2520.

  Puncturing section 2520 performs puncturing on data X, parity P, or parity Pn output from LDPC-CC encoding section 2510 according to the coding rate specified by the control signal. In the present embodiment, puncturing section 2520 does not puncture at random, but periodically punctures bits periodically. Puncturing section 2520 outputs the punctured transmission sequence to interleaving section 2530.

  Specifically, when the coding rate specified by the control signal exceeds 1/2, the puncturing unit 2520 punctures the parity P periodically to obtain a predetermined coding rate.

  On the other hand, when the coding rate specified by the control signal is 1/2 or 1/3, puncturing section 2520 outputs the transmission sequence to interleaving section 2530 without performing puncturing.

Interleaving section 2530 rearranges the order of the transmission sequences and outputs the rearranged transmission sequences to modulation section 2340.

  Modulation section 2540 modulates the interleaved transmission sequence using the modulation scheme specified by the control signal.

  FIG. 26 shows an example of the transmission format of the transmission sequence. The transmission sequence is composed of control information symbols and data symbols. The control information symbol is a symbol for notifying the communication partner of the coding rate and modulation method.

  FIG. 27 is a diagram illustrating a configuration of the reception apparatus according to the present embodiment. 27 includes a receiving unit 2710, a log likelihood ratio generating unit 2720, a control information generating unit 2730, a deinterleaving unit 7540, a depuncturing unit 2750, and a BP decoding unit 2760.

  The reception unit 2710 receives a reception signal transmitted from the transmission device 2500, performs radio demodulation processing such as RF (Radio Frequency) filter processing, frequency conversion, A / D (Analog to Digital) conversion, orthogonal demodulation, and the like. The demodulated baseband signal is output to log likelihood ratio generation section 2720. In addition, the reception unit 2710 estimates a channel variation in the wireless transmission path between the transmission device 2500 and the reception device 2700 using a known signal included in the baseband signal, and uses the estimated channel estimation signal as a log likelihood ratio. The data is output to the generation unit 2720.

  Log likelihood ratio generation section 2720 obtains the log likelihood ratio of each transmission sequence using the baseband signal, and outputs the obtained log likelihood ratio to deinterleave section 2740.

  The control information generation unit 2730 extracts control information from control information symbols included in the baseband signal. The control information symbol includes coding rate and modulation scheme information. Control information generation section 2730 outputs the extracted control information as a control signal to log likelihood ratio generation section 2720, deinterleave section 2740, depuncture section 2750, and BP decoding section 2760.

  Deinterleaving section 2740 rearranges the order of log-likelihood ratio sequences in the original order, using the reverse process of the rearrangement process performed by interleaving section 2530 of transmitting apparatus 2500, and the log likelihood after rearrangement. The ratio is output to the depuncture unit 2750.

  Depuncturing section 2750 performs depuncturing on the log likelihood ratio output from deinterleaving section 2740 using the inverse processing of puncturing performed by puncturing section 2520 of transmitting apparatus 2500. That is, in puncturing section 2520 of transmitting apparatus 2500, when the coding rate exceeds 1/2, parity P is periodically punctured, and in this case, deinterleaving section 2740 has bits punctured by puncturing section 2520. 0 is inserted as the log likelihood ratio of. Puncturing section 2520 does not perform puncturing when the coding rate is ½ or ３, and therefore outputs the log likelihood ratio to BP decoding section 2560 without performing the depuncturing process.

  The BP decoding unit 2760 switches the LDPC-CC parity check matrix according to the coding rate indicated by the control signal, and performs BP decoding. Specifically, BP decoding section 2760 includes an LDPC-CC parity check matrix corresponding to coding rate 1/2 and coding rate 1/3, and when the control signal indicates coding rate 1/3, BP decoding is performed using a parity check matrix with a coding rate of 1/3. On the other hand, when the control signal indicates a coding rate other than coding rate 1/3, BP decoding is performed using a parity check matrix of coding rate 1/2.

  In addition, in FIG. 28, the structural example of the LDPC-CC encoding part 2510 in case coding rate R = 1/2 is shown as an example. As illustrated in FIG. 28, the LDPC-CC encoding unit 2510 includes shift registers 2511-1 to 2511 -M and 2514-1 to 2514 -M, weight multipliers 2512-0 to 2512 -M, and 2513-0 to 2513. -M, a weight control unit 2316, and a mod2 adder 2515.

Shift registers 2511-1 to 2511 -M and 2514-1 to 2514 -M are registers that hold v _{1, ti} , v _{2, ti} (i = 0,..., M), respectively. Is input to the shift register on the right and the value output from the shift register on the left is newly stored. The initial state of the shift register is all zero.

The weight multipliers 2512-0 to 2512-M and 2513-0 to 2513-M set the values of h ₁ ^(m) and h ₂ ^(m) to 0/1 according to the control signal output from the weight control unit 2316. Switch. The weight control unit 2516 outputs the values of h ₁ ^(m) and h ₂ ^{(m) at} the timing based on the check matrix held therein, and the weight multipliers 2512-0 to 2512 -M and 2513. Supply to −0 to 2513-M.

The mod2 adder 2515 performs mod2 addition on the outputs of the weight multipliers 2512-0 to 2512-M and 2513-0 to 2513-M, and calculates v2 _{, t} .

  By adopting such a configuration, LDPC-CC encoder (LDPC-CC encoder) 2510 can perform LDPC-CC encoding according to a parity check matrix.

  Note that, when the rows of the parity check matrix held by the weight control unit 2516 are different from row to row, the LDPC-CC encoding unit 2510 is a time varying convolutional encoder.

In FIG. 29, when D ^−K (X) (K: positive integer) is included in the parity check polynomial, that is, using a check matrix in which “1” is added to the upper trapezoid matrix of the check matrix, encoding is performed. The structural example of the LDPC-CC encoding part (LDPC-CC encoder) in case of rate R = 1/2 is shown. The LDPC-CC encoding unit 2910 in FIG. 29 is different from the LDPC-CC encoding unit (LDPC-CC encoder) 2510 in FIG. 28 in terms of shift registers 2911-1 to 2911 -K and a weight multiplier 2912-. The structure which added 1-292-K is taken.

The shift registers 2911-1 to 2911 -K have v _{1, ti} (i = −1,..., −K).
At the timing when the next input is received, the held value is output to the shift register on the right and the value output from the shift register on the left is newly stored. The initial state of the shift register is all zero.

Weight multipliers 2912-0 to 2912 ^-K switch the values of h ₁ ^(−k) and h ₂ ^(−k) to 0/1 in accordance with the control signal output from weight control section 2316.

The weight control unit 2516 outputs the values of h ₁ ^(m) and h ₂ ^{(m) at} the timing based on the check matrix held therein, and the weight multipliers 2512-0 to 2512 -M and 2513. Supply to −0 to 2513-M. Further, the weight control unit 2516 outputs the values of h ₁ ^(−k) and h ₂ ^{(−k) at} the timing based on the check matrix held therein, and the weight multipliers 2912-1 to 2912. -Supply to K.

The mod2 adder 2515 performs mod2 addition on the outputs of the weight multipliers 2512-0 to 2512-M, 2513-0 to 2513-M, 2912-0 to 2912-K, and v _{2
, t} is calculated.

With the configuration as shown in FIG. 29, the LDPC-CC encoder (LDPC-CC encoder) 2510 includes D ^−K (X) (K: positive integer) in the parity check polynomial. Can also respond.

  In addition, the LDPC-CC encoding part (LDPC-CC encoder) corresponding to coding rate R = less than 1/2 can be comprised by the structure similar to FIG. 28, FIG. For example, when the coding rate R = 1/3, a shift register, a weight multiplier, and a mod 2 adder for generating the parity sequence Pn may be added to FIGS. 28 and 29.

  Further, in the above description, the LDPC-CC encoding unit 2510 generates a coding sequence according to the case where the coding rate R = 1/2 or more and the case where the coding rate R = 1/3. However, regardless of the coding rate, the LDPC-CC coding unit 2510 generates all the transmission sequences (including the parity Pn), and the coding rate R = 1/2. The parity Pn may not be output. In this way, the LDPC-CC encoder (LDPC-CC encoder) can be made to correspond to the coding rate R = 1/2 and the coding rate R = 1/3.

  Further, in the above description, when BP decoding section 2760 switches the decoding method of the encoded sequence according to the case where coding rate R = 1/2 or more and the case where coding rate R = 1/3. However, regardless of the coding rate, the BP decoding unit 2760 performs BP decoding using a parity check matrix with a coding rate of 1/3, and the coding rate indicated by the control signal indicates other than 1/3. Alternatively, the log likelihood ratio for the obtained parity Pn may be replaced with 0. In this way, the BP decoding unit can be shared.

(Other embodiment 5)
In the present embodiment, a modification of the eighth embodiment will be described in detail. Hereinafter, a configuration method of time-varying LDPC-CC with a coding rate R = 1/2 will be described.

When the coding rate is 1/2 and the polynomial expression of the information sequence (data) is X (D) and the polynomial expression of the parity sequence is P (D), the parity check polynomial is expressed as follows.

In the formula (107), a1, a2,..., An are integers greater than or equal to 0 (where a1 ≠ a2 ≠ ... ≠ an). In addition, b1, b2,..., Bm are integers of 1 or more (where b1 ≠ b2 ≠ ... ≠ bm). C1, c2,..., Cq are integers of −1 or less (where c1 ≠ c2 ≠ ... ≠ cq). At this time, P (D) is expressed as follows.

  Therefore, the parity P can be obtained sequentially (see Embodiment 2 and Embodiment 8).

Next, equations (109) and (110) are considered as parity check polynomials of coding rate ½ different from equation (107).

  In the expressions (109) and (110), A1, A2,..., AN are integers greater than or equal to 0 (where A1 ≠ A2 ≠ ... ≠ AN). Further, B1, B2,..., BM are integers of 1 or more (where B1 ≠ B2 ≠ ... ≠ BM). Further, C1, C2,..., CQ are integers equal to or less than −1 (where C1 ≠ C2 ≠ ... ≠ CQ). At this time, P (D) is expressed as follows.

Data X and parity P at time _2i are represented by X _2i and P _2i , respectively, and data X and parity P at time 2i + 1 are represented by X _{2i + 1} and P _{2i + 1} , respectively (i: integer).

At this time, the parity P _2i at the time point 2i is obtained by using the equation (108), and the parity P _{2i + 1} at the time point 2i + 1 is obtained by using the equation (111). The LDPC-CC having the time varying period of 2 or the parity at the time point 2i Consider an LDPC-CC with a time-varying period of 2 where P _2i is obtained using equation (108), and parity P _{2i + 1} at time 2i + 1 is obtained using equation (112).

In such LDPC-CC,
The encoder can be easily configured, and the parity can be obtained sequentially. The puncture bit can be set periodically.
-It has the advantage that it is possible to reduce the number of termination bits and improve the reception quality at the time of puncturing at the termination.

Next, consider LDPC-CC with a time-varying period of m. As in the case of the time-varying period 2, “inspection formula # 1” expressed by either formula (109) or formula (110) and either by formula (109) or formula (110), “ “Inspection Formula # 2”, “Inspection Formula # 3”... “Inspection Formula #m” are prepared. Data X and parity P at time mi + 1 are X _{mi + 1} , P _{mi + 1} , data X and parity P at time mi + 2 are X _{mi + 2} , P _{mi + 2} ,... Data X and parity P at time mi + m are X _{mi + m} , P _{mi + m} , respectively. (I: integer).

In this case, stop the parity _{P mi + 1} of point in time mi + 1 even "check equation # 1", stop the parity _{P mi + 2} of point in time mi + 2 even "check equation # 2",..., The parity _{P mi + m} of point in time mi + m "check equation # Consider the LDPC-CC obtained by “m”. Such LDPC-CC
The sign is
The encoder can be easily configured, and the parity can be obtained sequentially. Advantages include reduction of termination bits and improvement in reception quality at the time of puncturing at the termination.

  Hereinafter, a configuration example of the parity check matrix in the present embodiment will be shown.

FIG. 30 illustrates a configuration example of an LDPC-CC parity check matrix with a time varying period of 2. In FIG. 30, reference numeral 3001 indicates a portion corresponding to the data X _i and the parity P _i at the time point i. Similarly, reference numeral 3002 indicates a portion corresponding to the data X _{i + 1} and the parity P _{i + 1} at the time point i + 1.

A parity check polynomial (for example, a parity check polynomial at time point i) corresponding to the code 3003 in FIG. 30 can be expressed as follows.

  In the formula (113), a1, a2,..., An are integers other than −1 and 0 (provided that a1 ≠ a2 ≠ ... ≠ an). In addition, b1, b2,..., Bm are integers of 1 or more (where b1 ≠ b2 ≠ ... ≠ bm). In FIG. 30, a1, a2,..., An are positive integers. At this time, P (D) can be obtained sequentially.

Similarly, the parity check polynomial denoted by reference numeral 3004 in FIG. 30 (for example, the parity check polynomial at time point i + 1) can be expressed as follows.

  In formula (114), A1, A2,..., AN are integers other than 1 and 0 (where A1 ≠ A2 ≠ ... ≠ AN). Further, B1, B2,..., BM are integers of 1 or more (where B1 ≠ B2 ≠ ... ≠ BM). At this time, P (D) can be obtained sequentially.

That is, in the case of time 2j, the parity P is obtained based on the equation (113), and in the case of time 2j + 1, the parity P is obtained based on the equation (114) (j is an integer). For varying LDPC-CC when varying period 2 when taking the structure shown in FIG. 30, the reliability of the parity _{P i} at time i is propagated to data _{X i + 1} at time i + 1, as a result, data at point in time i + 1 _{X i + 1} Is decrypted. Then, the reliability of the parity P _{i + 1 at} the time point i + 1 is propagated to the data X _i at the time point i, and as a result, the data X _i at the time point _i is decoded.

In the seventh embodiment, data and parity at the same time are related and the data is decoded, whereas in this embodiment, a parity check expression in which data and parity at different time are related is provided. Will exist. Then, considering the positional relationship between the related data and the parity except for the case of the same time point, in the example shown in FIG. 30, the data Xi + 1 and the parity Pi at the time point i and the time point i + 1 are related. Therefore, in the time-varying LDPC-CC with the time-varying period 2, the positional relationship is closest in time. Therefore, there is an advantage that it is less necessary to consider the temporal positional relationship between related data and parity during decoding. As described above, the LDPC-CC having the time varying period of 2 is configured so that the data Xi + 1 and the parity Pi at the time point i and the time point i + 1 are related to each other, and are related within the time varying period 2. .

  Similar characteristics can be provided to time-varying LDPC-CCs other than the time-varying period 2. That is, the LDPC-CC can be configured so that the data and the parity are related within the time varying period m. Hereinafter, the case of the time varying period 7 will be described with reference to FIG.

FIG. 31 illustrates a configuration example of an LDPC-CC parity check matrix with a time varying period of 7. In FIG. 31, reference numeral 3101 denotes a portion corresponding to data Xi and parity P _i at time point i. Reference numeral 3102 denotes a portion corresponding to the data X _{i + 1} and the parity P _{i + 1} at the time point i + 1. Reference numeral 3103 denotes a portion corresponding to the data X _{i + 2} and the parity P _{i + 2} at the time point i + 2. Reference numeral 3104 denotes a portion corresponding to the data X _{i + 3} and the parity P _{i + 3} at the time point i + 3. Reference numeral 3105 denotes a portion corresponding to the data X _{i + 4} and the parity P _{i + 4} at the time point i + 4. Reference numeral 3106 denotes a portion corresponding to the data X _{i + 5} and the parity P _{i + 5} at the time point i + 5. Reference numeral 3107 denotes a portion corresponding to the data X _{i + 6} and the parity P _{i + 6} at the time point i + 6.

For varying LDPC-CC when varying period 7 when employing a configuration shown in FIG. 31, the reliability of the parity _{P i} at point in time i is (are arranged "1"), propagated to data _{X i + 6} at point in time i + 6 (“1” is arranged in the same row as the parity P _i ) As a result, the data X _{i + 6} at the time point _{i + 6} is decoded.

The reliability of the parity P _{i + 1 at} the time point i + 1 is propagated to the data X _{i + 1} at the time point i + 1. As a result, the data X _{i + 1} at the time point _{i + 1} is decoded.

The reliability of the parity P _{i + 2 at} the time point i + 2 propagates to the data X _{i + 5} at the time point i + 5, and as a result, the data X _{i + 5} at the time point _{i + 5} is decoded.

The reliability of the parity P _{i + 3 at} the time point i + 3 is propagated to the data X _{i + 4} at the time point i + 4, so that the data X _{i + 4} at the time point _{i + 4} is decoded.

The reliability of the parity P _{i + 4 at} the time point i + 4 is propagated to the data X _{i + 3} at the time point i + 3, and as a result, the data X _{i + 3} at the time point _{i + 3} is decoded.

The reliability of the parity P _{i + 5 at} the time point i + 5 is propagated to the data X _{i + 2} at the time point i + 2, and as a result, the data X _{i + 2} at the time point _{i + 2} is decoded.

The reliability of the parity P _{i + 6 at} the time point i + 6 propagates to the data X _i at the time point i, so that the data X _i at the time point _i is decoded.

  As shown in FIG. 31, an LDPC-CC with a time-varying period of 7 is configured so that data and parity from time point i to time point i + 6 are related to each other and are related within time-varying period of time 7. .

  Thus, it is necessary to consider the temporal positional relationship between data and parity in decoding by configuring the LDPC-CC so that the parity and data are related within a time-varying period. Has the advantage of low.

(Other embodiment 6)
In the present embodiment, the method for creating a time-varying LDPC-CC with a coding rate of 1/2 explained in Embodiment 7 is expanded, and a method for creating a time-varying LDPC-CC with a coding rate of 1/3. explain.

Data X time 2i, parity P, and parity Pn respectively _{X 2i,} P 2i, expressed in _{Pn 2i,} represents time 2i + 1 of data X, parity P, and parity Pn each _{X 2i + 1, P 2i +} 1, Pn 2i + 1 (i: integer). Here, the polynomial of data X is X (D), the polynomial of parity P is P (D), the polynomial of parity Pn is Pn (D), and the following parity check polynomial is considered.

  In formula (115), a1, a2,..., An are integers other than 0 (where a1 ≠ a2 ≠ ... ≠ an). In addition, b1, b2,..., Bm are integers of 1 or more (where b1 ≠ b2 ≠ ... ≠ bm). In addition, c1, c2,..., Cq are integers of 1 or more (where c1 ≠ c2 ≠ ... ≠ cq). Then, P (D) at time point 2i is obtained using the relational expression of Expression (115). At this time, P (D) can be obtained sequentially.

Next, Equation (116) is considered as a parity check polynomial.

  In the formula (116), A1, A2,..., AN are integers other than 0 (however, A1 ≠ A2 ≠ ... ≠ AN). Further, B1, B2,..., BM are integers of 1 or more (where B1 ≠ B2 ≠ ... ≠ BM). Further, C1, C2,..., CQ are integers of 1 or more (where C1 ≠ C2 ≠ ... ≠ Cq). Then, using the relational expression of Expression (116), Pn (D) at time 2i is obtained. At this time, Pn (D) can be obtained sequentially.

Next, Equation (117) is considered as a parity check polynomial.

  In equation (117), α1, α2,..., Αω are integers other than 0 (where α1 ≠ α2 ≠ ... ≠ αω). In addition, β1, β2,..., Βξ are integers of 1 or more (where β1 ≠ β2 ≠ ... ≠ βξ). .Gamma.1, .gamma.2,..., .Gamma..lamda. Are integers of 1 or more (where .gamma.1.noteq..gamma.2.noteq..gamma..lamda.). Then, P (D) at time point 2i + 1 is obtained using the relational expression of Expression (117). At this time, P (D) can be obtained sequentially.

Next, Equation (118) is considered as a parity check polynomial.

In equation (118), E1, E2,..., EΩ are integers other than 0 (where E1 ≠ E
2 ≠ ... ≠ EΩ). F1, F2,..., FZ are integers of 1 or more (where F1 ≠ F2 ≠ ... ≠ FZ). In addition, G1, G2,..., GΛ are integers of 1 or more (where G1 ≠ G2 ≠ ... ≠ GΛ). Then, Pn (D) at time point 2i + 1 is obtained using the relational expression of Expression (118). At this time, Pn (D) can be obtained sequentially.

  As described above, by creating an LDPC-CC code having a time-varying period of 2, an optimum puncture pattern can be easily obtained when a method of periodically selecting puncture bits is employed, as in the seventh embodiment. There is an advantage that can be selected.

  If the time variation period is within 10, it is easy to search for the best puncture pattern by adopting a periodic puncturing method.

  Next, consider LDPC-CC with a time-varying period of m.

  In the case of the time-varying period m, m different inspection formulas expressed by the equation (115) are prepared, and the m inspection formulas are expressed as “inspection formula A # 1, inspection formula A # 2,. A # m ”. Also, m different check expressions represented by Expression (116) are prepared, and the m check polynomials are referred to as “check expression B # 1, check expression B # 2,..., Check expression B # m”. Name it.

Then, data X, parity P, and parity Pn at time point mi + 1 are X _{mi + 1} , P _{mi + 1} , Pn _{mi + 1} , and data X, parity P, and parity Pn at time point mi + 2 are X _{mi + 2} , P _{mi + 2} , Pn _{mi + 2} ,. Data X, parity P, and parity Pn of _{mi + m} are represented by X _{mi + m} , P _{mi + m} , and Pn _{mi + m} , respectively (i: integer).

At this time, the parity P _{mi + 1} at the time point mi + 1 is obtained using the “check equation A # 1”, the parity Pn _{mi + 1} is obtained using the “check equation B # 1,” and the parity P _{mi + 2} at the time point _{mi + 2} is obtained from the “check equation A #”. 2 ”, the parity Pn _{mi + 2} is obtained using“ check equation B # 2 ”,..., The parity P _{mi + m} of the time point _{mi + m} is obtained using“ check equation A # m ”, and the parity Pn _{mi + m} Consider an LDPC-CC with a time-varying period m determined using "check equation B # m". Such an LDPC-CC code is a code with good reception quality and has the advantage that the parity can be obtained sequentially.

  Note that the coding rate is not limited to 1/3, and an LDPC-CC code having a coding rate of 1/3 or less can be similarly created.

(Other embodiment 7)
In the present embodiment, an example of a transmission apparatus and a puncture method for performing puncturing suitable for a transmission codeword sequence obtained by LDPC-CC coding will be described.

FIG. 32 is a diagram illustrating a configuration of a time-invariant LDPC-CC parity check matrix used in the present embodiment. Figure 32 is different from FIG. 68, instead of ^{H T,} shows the configuration of parity check matrix H. If the transmission codeword vector is represented by v, the relational expression of Hv = 0 holds.

  In the description of the puncturing method in the present embodiment, first, a problem when a general puncturing method is applied to the transmission codeword sequence v will be described. A general puncturing method is described in Non-Patent Document 12, for example. In the following description, an example in which the LDPC-CC is configured using a convolutional code having a coding rate R = 1/2 and (177, 131) will be described.

FIG. 33 is a diagram for explaining a general puncturing method. In the figure, v _{1, t}
, V _{2, t} (t = 1, 2,...) Indicate a transmission codeword sequence v. In a general puncturing method, the transmission codeword sequence v is divided into a plurality of blocks, and the same puncture pattern is used for each block, and transmission codeword bits are thinned out.

FIG. 33 shows that the transmission codeword sequence v is divided into blocks every 6 bits, and the same puncture pattern is used for all blocks, and transmission codeword bits are thinned out at a constant rate. ing. In the figure, bits surrounded by circles indicate bits that are punctured (bits that are not transmitted), and v is set so that the coding rate after puncturing is 3/4 for all the blocks 1 to 5. _{2,1, v 2,3, v 2,4,} v 2,6, v 2,7, v 2,9, v select _{2,10, v 2,12, v 2,13,} v 2,15 Then, puncture is performed (the bit is not transmitted).

  Next, the influence of the reception side (decoding side) when a general puncture as shown in FIG. 33 is applied to a transmission codeword sequence obtained by encoding using LDPC-CC will be considered. In the following, the case where BP decoding is used on the reception side (decoding side) will be considered. In BP decoding, decoding processing is performed based on an LDPC-CC parity check matrix. FIG. 34 shows the correspondence between transmission codeword sequence v and LDPC-CC parity check matrix H. In FIG. 34, bits surrounded by circles are transmission codeword bits that are thinned out by puncturing. As a result, in check matrix H, the bit corresponding to 1 surrounded by a square frame is not included in the transmission codeword sequence. As a result, when performing BP decoding, the log likelihood ratio is set to 0 because there is no initial log likelihood ratio for the bit corresponding to 1 surrounded by a square frame.

  In BP decoding, row operations and column operations are repeated. Therefore, if two or more bits (bits corresponding to 1 surrounded by a square frame in FIG. 34) for which there is no initial log likelihood ratio (the log likelihood ratio is 0) are included in the same row, In a row, the log-likelihood ratio is updated in the row operation alone in the row until the log-likelihood ratio of the bit where the initial log-likelihood ratio does not exist (log-likelihood ratio is 0) is updated by the column operation. Will not be. That is, the reliability is not propagated by the row operation alone, and in order to propagate the reliability, it is necessary to repeat the row operation and the column operation. Therefore, if there are a large number of such rows, if the number of iteration processes is limited in BP decoding, the reliability is not propagated, causing deterioration in reception quality. In the example illustrated in FIG. 34, the row 3410 is a row in which reliability is not propagated by a row operation alone, that is, a row that causes deterioration in reception quality.

  On the other hand, when the puncturing method according to the present embodiment is used, the number of rows in which the reliability is not propagated by the row operation alone can be reduced. In the present embodiment, for each processing unit of transmission codeword bits on the reception side (decoding side), a first puncture pattern and a second puncture pattern that thins out more bits than the first puncture pattern Is used to puncture the transmitted codeword bits. This will be described below with reference to FIGS. 35 and 36.

FIG. 35 is a diagram for explaining a puncturing method according to the present embodiment. Similarly to FIG. 33, v _{1, t} , v _{2, t} (t = 1, 2,...) Indicate a transmission codeword sequence v. In the following, as in FIG. 33, a case where one block is composed of 6 bits will be described. Further, it is assumed that the processing units of transmission codeword bits on the reception side (decoding side) are block 1 to block 5. In the example shown in FIG. 35, the first puncture pattern that is not punctured is used for the first block 1, and the second puncture pattern that is punctured is used for blocks 2 to 5. As a _{result, v 2,1, v 2,3, v} 2,4, v 2,6, v 2,7, v 2,9, v 2,10, v 2,12, v 2,13, v _It shows how _2,15 is punctured. Thus, in the present embodiment, puncture patterns with different coding rates are used to provide a range in which the number of bits to be thinned out is small within the processing unit of transmission codeword bits.

  FIG. 36 shows the correspondence between transmission codeword sequence v and LDPC-CC parity check matrix H in this case. In FIG. 36, although three lines including two or more 1 surrounded by a square frame are generated in the same line, it can be seen that the number of lines is reduced compared to the case of FIG. This is because block 1 is not punctured.

  In this way, by providing blocks that are not punctured, it is possible to reduce the number of rows that cause deterioration in reception quality during BP decoding. As a result, in the lines up to line 3610, log likelihood exists initially, and the reliability is reliably updated in BP decoding, and the updated reliability is propagated to line 3610. Deterioration can be suppressed. Thus, the reliability of the row obtained by the row operation alone is propagated sequentially by performing iterative decoding a plurality of times from the characteristics of the structure of the parity check matrix of the convolutional code (LDPC-CC), and the reception quality by puncture is transmitted. Can be prevented. In addition, since the number of rows whose reliability is not propagated is reduced by the row operation alone, the number of iterations necessary for propagating the reliability can be reduced.

  By the way, in the example shown in FIG. 35, by providing a block that is not punctured, transmission codeword bits to be transmitted increase and the transmission rate decreases. However, if a relationship of N << M is established between the number N of bits in which the first puncture pattern is used and the number of bits M in which the second puncture pattern is used, a decrease in transmission rate is suppressed. However, the reception quality can be improved. FIG. 35 shows an example in which N = 6 and M = 24, and the number of additional transmission codeword bits is as small as 2 bits. Can be reduced to lines.

  Hereinafter, the configuration of the transmission apparatus according to the present embodiment will be described. FIG. 37 is a block diagram showing a main configuration of the transmission apparatus according to the present embodiment. In the description of the present embodiment, the same components as those in FIG. The transmission apparatus 3700 in FIG. 37 includes a puncture unit 3710 in place of the puncture unit 2520 with respect to the transmission apparatus 2500 in FIG. The puncture unit 3710 includes a first puncture unit 3711, a second puncture unit 3712, and a switching unit 3713.

  Puncturing section 3710 performs puncturing on a transmission codeword sequence including a transmission information sequence and a termination sequence, and outputs the punctured transmission codeword sequence to interleaving section 2530.

  Specifically, puncturing section 3710 punctures a transmission codeword sequence using a first puncture pattern and a second puncture pattern that thins out more bits than the first puncture pattern. The first puncture pattern and the second puncture pattern are different in the ratio of bits to be punctured. Puncturing section 3710 punctures the transmission codeword sequence using, for example, a puncture pattern as shown in FIG. In FIG. 38, (N + M) bits are a processing unit on the receiving side (decoding side).

  First puncturing section 3711 performs puncturing on a transmission codeword sequence using the first puncture pattern. Second puncturing section 3712 performs puncturing on the transmission codeword sequence using the second puncture pattern.

When the puncture pattern of FIG. 38 is used, the first puncturing unit 3711 does not perform puncturing on the N-bit transmission codeword sequence from the beginning of the processing unit on the receiving side (decoding side), and the first puncturing unit 3711 The input transmission codeword sequence is output to switching section 3713. Second puncturing section 3712 performs puncturing on a transmission codeword sequence of (N + 1) to (N + M) bits, and outputs the punctured transmission codeword sequence to switching section 3713.

  Note that the first puncturing unit 3711 and the second puncturing unit 3712 may determine whether to puncture the transmission codeword sequence based on the control information from the control information generating unit 1050. The switching unit 3713 transmits a transmission codeword sequence output from the first puncture unit 3711 or a transmission codeword output from the second puncture unit 3712 in accordance with control information from a control information generation unit (not shown). One of the sequences is output to the interleave unit 2530.

  Hereinafter, the operation of transmitting apparatus 3700 configured as described above will be described mainly with respect to the puncturing process of puncturing section 3710. Hereinafter, a case will be described as an example where LDPC-CC encoding section 2510 performs LDPC-CC encoding using a convolutional code with encoding rate R = 1/2 and (177, 131).

In LDPC-CC encoding unit 2510, transmission information sequence u _t (t = 1,..., N) is subjected to LDPC-CC encoding processing, and v = (v _{1, t} , v _{2, t} ) is obtained. To be acquired. In the case of an organized code, v _{1, t} is a transmission information sequence u _t , and v _{2, t} indicates parity. The parity v _{2, t} is obtained based on the transmission information sequence v _{1, t} and the check expression of each row in FIG.

Puncturing section 3710 performs puncturing processing on transmission codeword sequence v with coding rate R = 1/2. For example, when the puncture shown in FIG. 35 is used by the puncture unit 3710, the block 1 is not punctured, and the blocks 2 to 5 are regularly spaced at predetermined intervals. Be drawn. That is, for block 2, the bits v _2,4 and v _2,6 are thinned out, and for block 3, the bits v _2,7 and v _2,9 are thinned out, and block 4 V ₂ , ₁₀ and v _2,12 are thinned out, and v ₂ , ₁₃ and v _2,15 are thinned out for block 5. In this way, a transmission codeword sequence having a coding rate R = 3/4 is acquired for blocks 2 to 5.

The punctured transmission codeword sequence is transmitted to the reception side (decoding side) via interleave section 2530 and modulation section 2540. At this time, when the puncture pattern shown in FIG. 35 _{used, v 2,4, v 2,6, v} 2,7, v 2,9, v 2,10, v 2,12, v 2,13 , V _2,15 will not be transmitted.

In this way, when the puncture pattern shown in FIG. 35 is used, blocks that are not punctured are generated every predetermined period. As shown in FIG. 35, by preventing the block 1 from being punctured, v ₂ , ₁ , v _2,3 that were not transmitted when the general puncturing method of FIG. 33 was used. Will be sent. In this way, when the BP decoding is used, the rows whose reliability is not propagated by the row operation alone are three rows shown in the row 3610 in FIG. As can be seen from the comparison between FIG. 33 and FIG. 35, by adding 2 transmission bits, the number of rows in which the reliability is not propagated by the row operation alone is reduced from 6 rows to 3 rows. As a result, the number of rows in which the log likelihood is initially present increases, and the initial reliability is reliably updated by BP decoding. Further, this reliability is propagated to the row 3610 in FIG. It becomes like this.

  Thereafter, due to the structure of the parity check matrix of the convolutional code (LDPC-CC), the reliability existing at the head of the parity check matrix is propagated sequentially by performing iterative decoding multiple times, and reception by puncture is performed. Quality degradation can be suppressed.

In the example of FIG. 35, since the increased number of bits to be transmitted is as small as 2 bits, the decrease in transmission speed is small and the deterioration in reception quality can be suppressed. Such an effect can be obtained because the LDPC-CC adopts a type in which the locations where 1 exists are concentrated in the range of the parallelogram as shown in FIG. Therefore, even if it is applied to the case of LDPC-BC, it is unlikely that the same effect can be obtained.

  Thus, by providing blocks that are not punctured, it is possible to reduce the number of rows that adversely affect BP decoding. At this time, in consideration of transmission efficiency, it is important that a relationship of N << M is established between the bit M constituting the block that is not punctured and the bit N constituting the block to be punctured. By setting N << M, it is possible to suppress deterioration in reception quality while suppressing deterioration in transmission efficiency.

  It should be noted that the puncturing unit 3710 may puncture the blocks 2 to 5 to which the second puncture pattern is applied, according to a predetermined rule, instead of puncturing at random. Compared with puncturing at random, when puncturing is performed according to a predetermined rule, puncture calculation processing is simplified.

(Other puncture patterns)
The puncture pattern used by the puncture unit 3710 is not limited to FIG. For example, as shown in FIG. 39, the puncture unit 3710 uses a puncture pattern with a coding rate R1 = 2/3 as the first puncture pattern, and uses a coding rate R2 = as the second puncture pattern. A 5/6 puncture pattern may be used.

  Also, as shown in FIGS. 40A and 40B, puncturing may be performed with n frames as processing units on the receiving side (decoding side). As shown in FIG. 40A, the first puncture pattern that does not perform puncturing is used for N bits from the beginning of n frames (n is an integer of 1 or more), and (N + 1) to (N + M) bits are used. Thus, a second puncture pattern for performing puncturing may be used.

  Also, as shown in FIG. 40B, the first puncture pattern with coding rate R1 = 2/3 is used for N bits from the beginning of n frames, and for (N + 1) to (N + M) bits. The second puncture pattern with the coding rate R2 = 5/6 may be used. As a result, the number of bits to be punctured can be reduced in N bits from the beginning of the n frame.

  Also, as shown in FIGS. 41A and 41B, a pattern may be used in which the number of bits punctured by puncturing is reduced toward the rear of the processing unit on the receiving side (decoding side). By reducing the number of bits thinned out by puncturing toward the rear of the processing unit on the reception side (decoding side), reception quality can be improved in BP decoding.

  In consideration of the decoding processing timing, for example, if puncturing is performed to reduce the number of bits punctured by puncturing at the rear part of a received data sequence consisting of n frames, This is because the row where the reliability is propagated is included in both and the rear, so that the reliability can be efficiently propagated.

  As in the case of FIG. 38, the relationship of N << M is established between the number N of bits in which the first puncture pattern is used and the number of bits M in which the second puncture pattern is used. For example, it is possible to improve reception quality while suppressing a decrease in transmission speed.

Further, as shown in FIG. 42A, the first puncture is not performed on the N1 bits from the beginning of n frames (n is an integer of 1 or more) as a processing unit on the reception side (decoding side).
Are used for the (N1 + 1) to (N1 + M) bits, and the first puncture is not performed for the (N1 + M + 1) to (N1 + M + N2) bits. A puncture pattern may be used.

  Also, as shown in FIG. 42B, the coding rate R1 = 2/3 for the N1 bits from the beginning of n frames (n is an integer equal to or greater than 1) as a processing unit on the receiving side (decoding side). For the (N1 + 1) to (N1 + M) bits, the second puncture pattern with the coding rate R2 = 5/6 is used, and for the (N1 + M + 1) to (N1 + M + N2) bits. May use the first puncture pattern with the coding rate R1 = 2/3.

  Compared to the case where the first puncture pattern in which the number of bits thinned out by puncturing is small is used in one processing unit on the receiving side (decoding side) (see FIGS. 40 and 41), 42), the number of test rows with high reliability increases, so that the convergence speed at the time of BP decoding is high, and a decoding result can be obtained with a small number of iterations.

  Note that the number of places where the first puncture pattern with a small number of bits thinned out by puncturing is used in the processing unit is not limited to two, and may be three or more.

  In addition, when the number of bits where the first puncture pattern with a small number of bits punctured by the puncture is used is two or more, the total number N of bits where the first puncture pattern is used and the second If the relationship of N << M is established with the total number M of bits in which the puncture pattern is used, reception quality can be improved while suppressing a decrease in transmission rate.

  40, 41, and 42, the case where the first puncture pattern and the second puncture pattern are used for n frames has been described. However, n may be an integer of 1 or more. It can also be applied to the case of one frame.

  In the following, a puncture pattern suitable for a transmission codeword sequence obtained by LDPC-CC coding will be considered in consideration of the relationship with the decoding processing timing.

  FIG. 43 is a diagram for explaining the decoding processing timing. In FIG. 43, each received data sequence is composed of n frames (for example, n OFDM (Orthogonal Frequency Division Multiplexing) symbols: OFDM symbols), and the OFDM scheme is composed of 32 subcarriers. In the case where the modulation signal is configured, the symbol is composed of all carriers (32 subcarriers). This received data sequence length is a processing unit on the receiving side (decoding side), and the n frames (or n OFDM symbols) are delivered to the upper layer as one unit. In general, there is a time lag until the upper layer takes in the next n frames of data, so the timing of t3, t6, t9 in FIG. 43, that is, the timing of receiving the last part of the n frames is BP decoded. It is realistic to end the period.

  Since LDPC-CC has the nature of a convolutional code, in order to make the data estimated by BP decoding effective from the timing of t2 (data that is highly likely to be correct), the BP before the timing of t2 Decoding needs to be started. For example, in the example shown in FIG. 43, in order to make the estimated data obtained by BP decoding between t2 and t5 valid data, it is necessary to perform BP decoding between t1 and t6. Similarly, in order to make the estimated data obtained between t5 and t8 valid data, it is necessary to perform BP decoding between t4 and t9.

  In consideration of such decoding processing timing, for example, when puncturing is performed so that the number of bits punctured by puncturing is reduced at the rear part of a received data sequence composed of n frames, in the BP decoding processing period, Since both the front side and the rear side include the row where the reliability is propagated, the reliability can be efficiently propagated.

  As described above, according to the present embodiment, puncturing section 3710 has a second puncture pattern that thins out more bits than the first puncture pattern and the first puncture pattern for each processing unit of transmission codeword bits. The transmission codeword bits are punctured using the puncture pattern.

  By using the first and second puncture patterns having different coding rates after puncturing instead of puncturing the transmission codeword sequence at a certain rate, it is possible to suppress degradation of decoding characteristics due to BP decoding. It becomes like this.

  As long as puncturing is performed, a line that causes degradation in reception quality occurs, but a method for suppressing degradation in reception quality while suppressing a decrease in transmission speed as in the puncturing method in the present embodiment is This is very important for building a system with good performance.

  Each of the first and second puncture patterns may be composed of the same plurality of sub-patterns. That is, as shown in FIG. 35, transmission codeword bits may be regularly thinned out by using the same subpuncture pattern for each of blocks 2 to 5. Thereby, puncture calculation processing can be simplified.

  Also, the first puncture pattern with a low coding rate is not necessarily arranged at the last part of the n frame, and as shown in FIG. 43, is provided between t1 to t3, t4 to t6, and t7 to t9. You can do it. In addition, since the periods t1 to t3, t4 to t6, and t7 to t9 are specified by the relationship between the BP decoding process period and the period during which valid data is obtained, the first period is changed when the BP decoding process period changes. The position suitable for arranging the puncture pattern varies.

  In the above description, as an example, the puncturing method in the case where BP decoding is performed on the convolutional code has been described. However, the puncturing method is not limited to this, and the puncturing method of the present invention is not disclosed in Non-Patent Document 5 to Non-Patent Document. 7 and Non-Patent Document 14 as described above can be similarly applied to time-invariant LDPC-CC and time-variant LDPC-CC.

  Further, the puncturing method in the present embodiment can be used for the time-invariant LDPC-CC and the time-varying LDPC-CC described in each of the embodiments of the present invention and other embodiments, and the reception quality deteriorates. The effect of suppressing is obtained.

(Time-invariant and time-variant LDPC-CC (coding rate 1/2) based on convolutional code)
The outline of the time-invariant / time-variant LDPC-CC based on the convolutional code will be described below.

Consider a systematic convolutional code with coding rate R = 1/2 and generator matrix G = [1 G1 (D) / G0 (D)]. At this time, G1 represents a feedforward polynomial, and G0 represents a feedback polynomial. When the polynomial expression of the information sequence is X (D) and the polynomial expression of the parity sequence is P (D), the parity check polynomial is expressed as follows.

Here, the parity check polynomial of equation (120) that satisfies equation (119) is considered.

  In the formula (120), ai (i = 1, 2,..., R) is an integer other than 0, and bj (j = 1, 2,..., S) is an integer of 1 or more. A code defined by a parity check matrix based on the parity check polynomial of equation (120) is referred to herein as time invariant LDPC-CC.

M different parity check polynomials based on Expression (120) are prepared (m is an integer of 2 or more). The parity check polynomial is expressed as follows.

ここで、「ｊ ｍｏｄ ｍ」は、ｊをｍで除算した余りである。 At this time, i = 0, 1,..., M−1. The information and parity at the time point j are represented by X _j and P _j , and u _j = (X _j , P _j ). At this time, the information at the time point j and the parities X _j and P _j satisfy the parity check polynomial of Expression (122).

Here, “j mod m” is a remainder obtained by dividing j by m.

  A code defined by a parity check matrix based on the parity check polynomial of Equation (122) is referred to herein as time-varying LDPC-CC. At this time, the time-invariant LDPC-CC defined by the parity check polynomial of Equation (121) and the time-varying LDPC-CC defined by the parity check polynomial of Equation (122) sequentially register the parity and exclusive OR. It has the feature that it can be obtained easily.

In the decoding unit, a time-invariant LDPC-CC creates a check matrix H from the equation (121), and a time-variant LDPC-CC from the equation (122), and the vector u = (u ₀ , u ₁ ,..., U _i , )), The following relational expression is established.

  Then, BP decoding is performed from the relational expression of Expression (123) to obtain a data series.

(Specification or proposal draft)
The following is a description example when a specification or proposal is created.

1.
It is proposed to use LDPC-CC (Low-Density Parity-Check Convolutional Codes), which is an error correction code that can handle multiple coding rates, as a FEC (Forward Error Correction) Scheme. LDPC-CC is a convolutional code defined by a low-density parity check matrix and, like CTC (Convolutional Turbo Code) and LDPC-BC (Block Code), is a code class that has a correction capability approaching the Shannon Limit ( Non-patent document 12 and Non-patent document 15).

LDPC-CC has the following advantages over CTC.
(1) No interleaver is required for the encoder.-The encoder can be configured with only a shift register and an adder.-Since the length of the information sequence is not limited to the interleaver length, an information sequence of any length can be encoded. (2 ) Sum-Product decoding, which allows parallel processing to the decoding algorithm, can be used.
Processing delay can be reduced compared to CTC decoding that requires serial processing. Also, it has the following advantages over LDPC-BC standardized by IEEE802.11n.
(3) Since the length of the information sequence is not limited by the block length of the parity check matrix, it is possible to encode an information sequence of an arbitrary length. (4) Encoding is performed with an operation scale proportional to the memory length (constraint length). Compared to LDPC-BC, which requires a computation scale proportional to the information sequence length, the encoder configuration is simple (memory length <information sequence length)
(5) Decoding processing delay can be reduced by applying a decoding algorithm that uses the parity check matrix structure unique to LDPC-CC

2.
2-1. FEC Encoding
FIG. 44 shows a block diagram of an error correction code scheme (FEC Scheme).
The error correction code system is composed of an LDPC-CC Encoder and a puncture unit (Puncture).
The payload data length to be encoded is k bits, and the code word data length obtained after encoding is n bits.

2-2. LDPC-CC Encoding
Payload data is encoded by the LDPC-CCEncoder.
FIG. 45 shows the configuration of the LDPC-CC Encoder. The LDPC-CC Encoder outputs k-bit systematic bits and k-bit parity bits for k-bit payload data. The coding rate R of the LDPC-CC Encoder is 1/2.

The LDPC Convolutional Encoding process is shown below.
(1) The input of k-bit information bits is branched into two, one is output as k-bit systematic bits, and the other is input to a constituent encoder.
(2) The Constituent Encoder performs an encoding process on k information bits and outputs k bit parity bits.
The LDPC-CC Encoder outputs Code Bits bit by bit in the order {d1, p1}, {d2, p2}, {d3, p3}, ..., {dk, pk}.

検査行列Hは、k×2kの行列である。検査行列Hの各列は、d1,p1,d2,p2,…dt, pt, …,dk,pkという並びでSystematic Bits（d1、…、dk）とParity Bits（p1、…、pk）に対応す
る。また、Mは、LDPC-CCのメモリ長を表す。 LDPC-CC is defined by the parity check matrix given by Equation (124).

The check matrix H is a k × 2k matrix. Each column of the check matrix H corresponds to Systematic Bits (d1, ..., dk) and Parity Bits (p1, ..., pk) in the order of d1, p1, d2, p2, ... dt, pt, ..., dk, pk To do. M represents the memory length of the LDPC-CC.

Each row of the parity check matrix H represents a parity check polynomial.
h _d ⁽ⁱ⁾ (t) (i = 0, ..., M) is the weight of the systematic bit in the t-th parity check polynomial (
1 or 0), and h _p ⁽ⁱ⁾ (t) (i = 0,..., M) represents a parity bit weight (1 or 0) in the t-th parity check polynomial.
In the check matrix H, all elements other than h _d ⁽ⁱ⁾ (t) and h _p ⁽ⁱ⁾ (t) are 0. As shown in Expression (1), the LDPC-CC parity check matrix H is a matrix having one element only in the diagonal term of the matrix and its vicinity.

ただしn=0,1,2,…である。 The inspection formula used in FEC Scheme is shown in Formula (125) and Formula (126).

However, n = 0, 1, 2,.

ここで、X(D)は、Systematic Bits（d1,…,dk），P(D)はParity Bits（p1,…,pk）を表す。 Moreover, the polynomial expression of Formula (125) and Formula (126) is as follows.

Here, X (D) represents Systematic Bits (d1,..., Dk), and P (D) represents Parity Bits (p1,..., Pk).

The proposed LDPC-CC Encoder is a time-varying LDPC-CC Encoder with a period of 2 and a memory length of 421, which uses two polynomials, the polynomial of equation (125) and the polynomial of equation (126), switched at each time point.

The LDPC-CC Encoder takes an arbitrary configuration for performing the calculation of Expression (129).

である。 The initial state of the LDPC-CC Encoder is all zero. That is,

It is.

LDPC-CC supports encoding of Information Bits of arbitrary length k with the same encoder configuration. LDPC-CC also supports multiple memory lengths.

3. Encoding Termination
Termination is necessary to make the LDPC-CC Encoder state unique at the end of encoding. Termination is performed by Zero-tailing.

Zero-tailing is realized by LDPC-CC Encoding a tail-bit consisting of M with a memory length of M bits. In addition, when termination is performed, the tail-bit is a known bit string on the receiving side, so it is not included in the systematic bits and transmitted, but only the M-bit parity bits obtained when the tail-bit is encoded are transmitted. To do.

4). Puncturing
Puncture is a process of discarding several systematic bits and / or parity bits from the output of LDPC-CC Encoder in order to obtain a code with a coding rate higher than 1/2 with a single encoder configuration. . Table 1 shows the coding rates supported by Puncturing. The coding rates to be supported are 1/2, 2/3, and 3/4, and 4/5 and 5/6 are optional.

Table 2 shows puncture patterns used at each coding rate. P and p of the puncture pattern represent Systematic Bit and Parity Bit, respectively, and when the value in the pattern is 0, the bit is punctured. LPunc represents the length of the puncture pattern.

Use regular rotated puncturing for puncturing. Each systematic bit and parity bit is divided for each LPunc bit, and puncturing is performed regularly according to the puncture pattern shown in Table 1. In the case of coding rates of 3/4, 4/5, and 5/6, Systematic Bits is also punctured, and the resulting code is a non-systematic code.

5.
As mentioned above, it was proposed to use LDPC-CC as the FEC Scheme.
LDPC-CC Encoder configuration, polynomial and puncture pattern are shown, and it can be used as FEC Scheme.

6).
6-1. An Example of LDPC-CC Encoder
The LDPC-CC encoding can be realized by an arbitrary encoder that realizes the equation (129).
A configuration shown in FIG. 46 as an example of the LDPC-CC Encoder is shown in Non-Patent Document 12.

As shown in FIG. 46, the LDPC-CC Encoder includes M1 shift registers that hold ut, M2 shift registers that hold pt, and h _d ⁽ⁱ⁾ (t), h of each column of the check matrix H. _p ⁽ⁱ⁾ A weight controller that outputs weights according to the sequence of (t) and a mod2 adder.

By adopting such a configuration, the LDPC-CC Encoder performs LDPC-CC encoding processing according to the equation (125). As shown in FIG. 46, an LDPC-CC encoder can be configured with only a shift register, an adder, and a weight multiplier.

(Other embodiment 8)
In the present embodiment, the method for creating a time-varying LDPC-CC with a coding rate of 1/2 described in Embodiment 7 is expanded, and a time-varying LDPC-CC with a coding rate greater than the coding rate of ½. How to create In the following, as an example, a method for creating a time-varying LDPC-CC such as a coding rate of 3/4 will be described.

Data X1 of time 2i, data X2, data X3, and parity P, respectively _{X 1,2i, X 2,2i, X 3,2i} , expressed in _{P 2i,} time 2i + 1 of the data X1, data X2, data X3, parity P _{Are represented} by X _{1,2i + 1} , X _{2,2i + 1} , X _{3,2i + 1} , and P _{2i + 1} (i: integer). Here, the polynomial of data X1 is X1 (D), the polynomial of data X2 is X2 (D), the polynomial of data X3 is X3 (D), the polynomial of parity P is P (D), and the following parity check polynomial is Think.

  In formula (131), a1, a2,..., Ar are integers other than 0 (however, a1 ≠ a2 ≠ ... ≠ ar). In addition, b1, b2,..., Bs are integers other than 0 (where b1 ≠ b2 ≠ ... ≠ bs). In addition, c1, c2,..., Bv are integers other than 0 (where c1 ≠ c2 ≠ ... ≠ cv). Further, e1, e2,..., Ew are integers of 1 or more (provided that e1 ≠ e2 ≠ ... ≠ ew). Then, P (D) at time point 2i is obtained using the relational expression of Expression (131). At this time, P (D) can be obtained sequentially.

Next, Equation (132) is considered as a parity check polynomial.

  In the formula (132), A1, A2,..., AR are integers other than 0 (however, A1 ≠ A2 ≠ ... ≠ AR). B1, B2,..., BS are integers other than 0 (B1 ≠ B2 ≠ ... ≠ BS). In addition, C1, C2,..., CV are integers other than 0 (where C1 ≠ C2 ≠ ... ≠ CV). Further, E1, E2,..., EW are integers of 1 or more (where E1 ≠ E2 ≠ ... ≠ EW). Then, P (D) at time 2i + 1 is obtained using the relational expression of Expression (132). At this time, P (D) can be obtained sequentially.

As described above, by creating an LDPC-CC code having a time-varying period of 2, an optimum puncture pattern can be easily obtained when a method of periodically selecting puncture bits is employed, as in the seventh embodiment. There is an advantage that can be selected.

  If the time variation period is within 10, it is easy to search for the best puncture pattern by adopting a periodic puncturing method.

Next, consider an LDPC-CC in which the time-varying period is m (an integer greater than or equal to 2).
In the case of the time-varying period m, m different inspection formulas expressed by the equation (131) are prepared, and the m inspection formulas are expressed as “inspection formula # 1, inspection formula # 2,. ".

Then, the data X1 of time mi + 1, data X2, data X3, represent parity P at _{X 1, mi + 1, X} 2, mi + 1, X 3, mi + 1, P mi + 1 respectively, data X1 of time mi + 2, data X2, data X3, represents a parity P, respectively _{X 1, mi + 2, X} 2, mi + 2, X 3, mi + 2, P mi + 2, data X1 of time mi + m, data X2, data X3, _X 1 and parity P, _{respectively, mi + m, X 2,} mi + m, X _{3, mi + m} , P _{mi + m} (i: integer)

At this time, the parity P _{mi + 1} at the time point mi + 1 is obtained using the “check equation # 1”, the parity P _{mi + 2} at the time point mi + 2 is obtained using the “check equation # 2,” and the parity P _{mi + m} at the time point _{mi + m} is Consider an LDPC-CC having a time-varying period m determined using “check equation #m”. Such an LDPC-CC code is a code with good reception quality and has the advantage that the parity can be obtained sequentially.

In the above description, the time-varying LDPC-CC based on the formula (131) and the formula (132) has been described. However, the formula (133) is used instead of the formula (131), and the formula (132) is used. Thus, the LDPC-CC having the time varying period 2 or the time varying period m can be formed using the equation (134).

Note that the coding rate is not limited to 3/4, and an LDPC-CC code having a coding rate of n / n + 1 can be similarly generated. For example, in the case of time-varying cycle 2, data X1, data X2, data X3,..., Data Xn, and parity P at time 2i are respectively X _{1,2 i} , X _{2,2 i} , X _{3,2 i,.} , X _{n, 2i} , P _2i , and data X1, data X2, data X3,..., Data Xn, and parity P at time 2i + 1 are respectively X _{1,2 i + 1} , X _{2,2 i + 1} , X _{3,2 i + 1,.} .., _{Xn, 2i + 1} , _{P2i + 1} (i: integer). Here, the polynomial of data X1 is X1 (D), the polynomial of data X2 is X2 (D), the polynomial of data X3 is X3 (D),..., The polynomial of data Xn is Xn (D), and the parity P Let P (D) be a polynomial and consider the following parity check polynomial.

In formula (135), a _1,1 , a ₁ , ₂ ,..., A _{1, r1} are integers other than 0 (where a _1,1 ≠ a _1,2 ≠... ≠ a _{1, r1} ). Further, a _2,1 , a _2,2 ,..., A _{2, r2} are integers other than 0 (provided that a _2,1 ≠ a _2,2 ≠... ≠ a _{2, r2} ). The same applies to X3 (D) to Xn-1. A _{n, 1} , a _{n, 2} ,..., A _{n, rn} are integers other than 0 (however, a _{n, 1} ≠ an _{, 2} ≠... ≠ an _{, rn} ). Further, e1, e2,..., Ew are integers of 1 or more (provided that e1 ≠ e2 ≠ ... ≠ ew). Then, P (D) at the time point 2i is obtained using the relational expression of the expression (135). At this time, P (D) can be obtained sequentially.

Next, Equation (136) is considered as a parity check polynomial.

In formula (136), A _1,1 , A _1,2 ,..., A _{1, R1} are integers other than 0 (provided that A _1,1 ≠ A _1,2 ≠... ≠ A _{1, R1} ). A _2,1 , A _2,2 ,..., A _{2, R2} are integers other than 0 (where A _2,1 ≠ A _2,2 ≠... ≠ A _{2, R2} ). The same applies to X3 (D) to Xn-1. A _{n, 1} , A _{n, 2} ,..., A _{n, Rn} are integers other than 0 (where A _{n, 1} ≠ A _{n, 2} ≠ ... ≠ A _{n, Rn} ). Further, E1, E2,..., EW are integers of 1 or more (where E1 ≠ E2 ≠ ... ≠ EW). Then, P (D) at the time point 2i + 1 is obtained using the relational expression of the expression (136). At this time, P (D) can be obtained sequentially.

  As described above, by creating an LDPC-CC code having a time-varying period of 2, an optimum puncture pattern can be easily obtained when a method of periodically selecting puncture bits is employed, as in the seventh embodiment. There is an advantage that can be selected.

  If the time variation period is within 10, it is easy to search for the best puncture pattern by adopting a periodic puncturing method.

  Next, consider an LDPC-CC in which the time-varying period is m (an integer greater than or equal to 2).

  In the case of the time-varying period m, m different inspection formulas expressed by the equation (135) are prepared, and the m inspection formulas are expressed as “inspection formula # 1, inspection formula # 2,. ".

Then, the data X1 of time mi + 1, data X2, data X3, · · ·, data Xn, parity P, respectively _{X 1, mi + 1, X} 2, mi + 1, X 3, mi + 1, ···, X n, mi + 1, P _{mi + 1} , data X1, data X2, data X3,..., data Xn, and parity P of time point mi + 2 are X1, _{mi + 2} , X2 _{, mi + 2} , X3 _{, mi + 2} ,..., _{Xn, mi + 2, respectively.} , expressed as _{P mi + 2,} data X1 of time mi + m, data X2, data X3, · · ·, data Xn, parity P, respectively _{X 1, mi + m, X} 2, mi + m, X 3, mi + m, ···, X n _{, Mi + m} and P _{mi + m} (i: integer).

At this time, the parity P _{mi + 1} at the time point mi + 1 is obtained using the “check equation # 1”, the parity P _{mi + 2} at the time point mi + 2 is obtained using the “check equation # 2,” and the parity P _{mi + m} at the time point _{mi + m} is Consider an LDPC-CC having a time-varying period m determined using “check equation #m”. Such an LDPC-CC code is a code with good reception quality and has the advantage that the parity can be obtained sequentially.

In the above description, the time-varying LDPC-CC based on the formula (135) and the formula (136) has been described. However, instead of the formula (135), the formula (137) is used, and the formula (136) is substituted. An LDPC-CC having a time-varying period 2 or a time-varying period m can also be formed using Expression (138).

(Other embodiment 9)
Bits corresponding to the maximum orders of a plurality of parity check polynomials for the reason that the degradation of reception quality when puncturing time-varying LDPC-CC based on Expressions (68) and (69) described in Embodiment 7 is small Will be described from the viewpoint of a condition that becomes a parity check polynomial (hereinafter abbreviated as “polynomial”) that are not punctured at the same time.

  FIG. 47 is a block diagram showing a main configuration of the transmission apparatus according to the present embodiment. In the description of the present embodiment, the same components as those in FIG. The transmitting apparatus 4700 of FIG. 47 is configured to include an LDPC-CC encoding unit 4710 and a puncturing unit 4720 instead of the LDPC-CC encoding unit 2510 and the puncturing unit 3710 with respect to the transmitting apparatus 3700 of FIG.

  The LDPC-CC encoding unit 4710 generates parity bits for the input information bits according to a check matrix H described later. LDPC-CC encoding section 4710 outputs codeword bits including information bits and parity bits to puncturing section 4720.

  Puncturing section 4720 punctures codeword bits. The puncture pattern will be described later.

Next, a case will be described as an example where the parity check matrix used in LDPC-CC encoding section 4710 is composed of polynomials of equations (139) and (140).

  Table 3 shows parameters of the parity check matrix in which the polynomials (139) and (140) are alternately repeated.

図４８に、二つの多項式の最大次数α１，α２，β１，β２、及び、第２次数Α１，Α２，Β１，Β２の関係を示す。

FIG. 48 shows the relationship between the maximum orders α1, α2, β1, β2 of the two polynomials and the second orders Α1, Α2, Β1, Β2.

  As shown in FIG. 48, the maximum orders α1 and α2 of the information bits of the two polynomials are even and odd pairs. Hereinafter, it is expressed as [α1: even, α2: odd]. The second order ビ ッ ト 1 and Α2 of the information bits, the maximum orders β1 and β2 of the parity bits, and the second orders パ リ テ ィ 1 and Β2 of the parity bits are pairs of even numbers or odd numbers. Hereinafter, [Α1: even, Α2: even], [β1: odd, β2: odd], and [Β1: even, Β2: even].

  These relationships will be described using the parity check matrix of FIG. FIG. 49 is a parity check matrix with a time-varying period 2 configured using polynomials (139) and (140). In FIG. 49, positions 4910-1 and 4910-2 indicate the positions of bits corresponding to the maximum orders α1 and α2 of the information bits of the two polynomials. Positions 4920-1 and 4920-2 indicate the positions of bits corresponding to the maximum orders β1 and β2 of the parity bits of the two polynomials. Positions 4930-1 and 4930-2 indicate the positions of bits corresponding to the second orders Α1 and Α2 of the information bits of the two polynomials. Positions 4940-1 and 4940-2 indicate the positions of bits corresponding to the second orders Β1 and Β2 of the parity bits of the two polynomials. In FIG. 49, all of the shaded elements are 1.

  As can be seen from positions 4910-1 and 4910-2 in FIG. 49, when a certain degree of two polynomials is an even and odd pair (for example, [α1: even, α2: odd]), the degree is checked. Appears in the same column on the matrix. Further, as can be seen from the positions 4920-1, 4920-2, 4930-1, 4930-2, 4940-1, 4940-2 in FIG. 49, the order of the two polynomials is an even pair or an odd pair. In some cases (eg, [Α1: even, Α2: even], [β1: odd, β2: odd], [Β1: even, Β2: even]), the orders appear in different columns on the parity check matrix.

When the maximum orders α1 and α2 of the information bits are even and odd pairs ([α1: even, α2: odd]) as in the equations (139) and (140), the bits corresponding to the maximum order are in the same sequence. Appear in For example, in the example of FIG. 49, bits corresponding to the maximum orders α1, α2 of information bits appear in the columns of information bits v _1,1 , v _1,3 , v _1,5 , v _1,7,. Therefore, when these bits are punctured, the bits corresponding to the maximum degree of the two polynomials are both punctured, and the constraint length in the polynomial is shortened, so that the error correction capability is reduced.

  As a method of avoiding a decrease in error correction capability by puncturing all the bits corresponding to the maximum degree of the two polynomials, the maximum degrees of the two polynomials are both even or odd. It is mentioned to use LDPC-CC having a polynomial as follows. That is, the maximum orders α1 and α2 of information bits are either [α1: even, α2: even] or [α1: odd, α2: odd], and the maximum orders β1 and β2 of parity bits are [β1. : Polynomial, β2: even], or [β1: odd, β2: odd].

In other words, in the present embodiment, the maximum order α1 and α2 of the information bits satisfy the expression (141-1), while the maximum order β1 and β2 of the parity bit satisfy the expression (141-2). It is characterized by using LDPC-CC having

  However, when the maximum orders α1 and α2 of the information bits have the same value, or when the maximum orders β1 and β2 of the parity bits have the same value, the bit corresponding to the maximum order is used regardless of the pattern. Will be punctured. Therefore, these maximum orders need to be an even pair or an odd pair while taking different values.

  That is, the maximum orders α1 and α2 of the information bits are [α1: even, α2: even, α1 ≠ α2] or [α1: odd, α2: odd, α1 ≠ α2]. Similarly, the maximum orders β1 and β2 of the parity bits are [β1: even, β2: even, β1 ≠ β2] or [β1: odd, β2: odd, β1 ≠ β2].

Equations (141-1) and (141-2) show the time varying period T = 2, that is, the condition of the maximum order of the LDPC-CC composed of two types of polynomials. The time-varying period T is not limited to 2, and may be 3 or more. When the time-varying period T is 3 or more, with respect to the maximum orders α1, α2,..., ΑT of the information bits, while satisfying the expression (142-1), the maximum orders β1, β2, .., Βt,..., ΒT may be LDPC-CC having a polynomial satisfying the equation (142-2).

  Next, returning to the case where the time-varying period is 2 again, the maximum orders [α1, α2] and [β1, β2] of the two polynomials are both or one of the expressions (141-1) and (141-2). A case where the above is not satisfied will be described.

  Hereinafter, the case where only the expression (141-2) is satisfied without using the expression (141-1) will be described using the polynomials (139) and (140). In this case, as shown at positions 4910-1 and 4910-2 in FIG. 49, the bits corresponding to the maximum orders α1 and α2 regarding the information bits of the two polynomials appear in the same column. For this reason, depending on the puncture pattern, the bits corresponding to the maximum orders α1 and α2 are punctured in both polynomials, and the range in which the reliability is propagated is reduced, which may reduce the error correction capability.

  When the bits corresponding to the maximum orders α1 and α2 are punctured, the range in which the reliability propagates depends on the second orders Α1 and Α2. Therefore, it is necessary to prevent the bits corresponding to the second orders Α1 and Α2 from being punctured. That is, in this embodiment, when the maximum order α1 and α2 of information bits are a combination of an even number and an odd number ([α1: even, α2: odd] or [α1: odd, α2: even]), the information bits And the second order Α1 and Α2 of an even pair ([Α1: even, Α2: even, Α1 ≠ Α2]) or odd pairs ([Α1: odd, Α2: odd, Α1 ≠ Α2]). The LDPC-CC having a polynomial is used.

  Similarly, when the maximum orders β1 and β2 of parity bits are a combination of an even number and an odd number ([β1: even, β2: odd] or [β1: odd, β2: even]), the second order of parity bits Β1 , Β2 is an even pair ([Β1: even, Β2: even, Β1 ≠ Β2]) or an odd pair ([Β1: odd, Β2: odd, Β1 ≠ Β2]). -Use CC.

For example, the LDPC-CC encoding unit 4710 performs LDPC-CC encoding using the polynomials shown in Equations (143) and (144), so that LDPC-CC having high error correction capability even when puncturing is applied. Can be provided.

  In the equations (143) and (144), the maximum orders α1 and α2 of the information bits are even and odd, whereas the second orders 情報 1 and Α2 of the information bits are both even. As a result, the maximum orders α1 and α2 of the information bits of the two polynomials appear in the same column, but the second orders 情報 1 and Α2 of the information bits do not appear in the same column, so the second order of the information bits In the range depending on Α1 and Α2, reliability propagation can be secured. Thereby, it is possible to avoid a decrease in error correction capability.

  In equations (143) and (144), since the maximum orders β1 and β2 of the parity bits are both odd numbers, it is possible to ensure reliability propagation within a range depending on the maximum orders β1 and β2 of the parity bits. it can.

  In the equations (68) and (69) described in the seventh embodiment, the maximum orders α1 and α2 of information bits are a combination of even and even numbers, and α1 and α2 are different ([α1: even, α2: Even], α1 ≠ α2), and bits corresponding to the maximum orders α1 and α2 of information bits do not appear in the same column due to puncturing.

In equations (68) and (69), the parity bit maximum orders β1 and β2 are a combination of an even number and an odd number ([β1: even, β2: odd]), and the parity bit second order Β1 , Β2 is an odd pair ([Β1: odd, Β2: odd, Β1 ≠ Β2]), so that the maximum orders β1 and β2 of the parity bits of the two polynomials appear in the same column, Since the second orders B1 and B2 of the parity bits do not appear in the same column, the reliability propagation can be ensured in a range depending on the second orders B1 and B2 of the parity bits.

  As a result, the LDPC-CC having the time varying period 2 defined by the equations (68) and (69) can avoid a decrease in error correction capability even when puncturing is applied.

In the following, an LDPC-CC having a time-varying period of 2 using Equation (145) and Equation (146) instead of Equation (68) and Equation (69) will be considered.

  In the formula (145), a1, a2,..., Ar are integers (where a1 ≠ a2 ≠ ... ≠ ar). Further, e1, e2,..., Ew are integers (where e1 ≠ e2 ≠ ... ≠ ew).

  In the equation (146), A1, A2,..., AR are integers (where A1 ≠ A2 ≠ ... ≠ AR). Further, E1, E2,..., EW are integers (where E1 ≠ E2 ≠ ... ≠ EW).

  In this case, when there are three or more even numbers in each order a1, a2,..., Ar of the information bits of the equation (145), each order appears in the same column on the parity check matrix. Loop 6 will occur. If there is a short loop such as loop 6 (a loop having a length of 6, also referred to as “Girth 6”), reception quality deteriorates. Therefore, it is preferable not to include three or more even numbers in a1, a2,.

  Similarly, when there are three or more odd numbers in each order a1, a2,..., Ar of the information bits in Expression (145), each order appears in the same column on the parity check matrix. Since the loop 6 is generated, it is preferable that three or more odd numbers are not included in a1, a2,.

  Further, in consideration of the row weight described in the seventh embodiment, r ≦ 4 is preferable.

  Alternatively, it is preferable that each of the orders e1, e2,..., Ew of the parity bit in the formula (145) is not included in the number of three or more even numbers or three or more odd numbers. And w ≦ 4.

  Alternatively, it is preferable that each of the orders A1, A2,..., AR of the information bits in the equation (146) is not included in the number of three or more even numbers or three or more odd numbers. And R ≦ 4.

  Similarly, it is preferable that each of the orders E1, E2,... EW of the parity bit in the formula (146) does not include three or more even numbers or three or more odd numbers. And W ≦ 4.

  It should be noted that a better LDPC-CC having a time varying period of 2 can be designed by setting as follows.

That is, “in each of the orders a1, a2,... Ar of the information bits in the formula (145), 3 or more even numbers or 3 or more odd numbers are not included, and r ≦ 4.
In addition, it is made for each order e1, e2,. And
In addition, in each of the orders A1, A2,..., AR of the information bit of the formula (146), 3 or more even numbers or 3 or more odd numbers are not included, and R ≦ 4 And
In addition, in each of the orders E1, E2,. And ", An LDPC-CC having a better time-varying period of 2 and a coding rate of 1/2 is obtained.

In the above, the LDPC-CC with time-varying period 2 and coding rate 1/2 using Expression (145) and Expression (146) has been described. In the following, an LDPC-CC having a time-varying period of 2 and a coding rate of 1/3 using equations (147-1) to (147-4) will be considered.

  In the expression (147-1), a1, a2,..., An are integers (where a1 ≠ a2 ≠ ... ≠ an). In addition, b1, b2,..., Bm are integers (where b1 ≠ b2 ≠ ... ≠ bm). Further, c1, c2,..., Cq are integers (where c1 ≠ c2 ≠ ... ≠ cq).

  In the equation (147-2), A1, A2,..., AN are integers (however, A1 ≠ A2 ≠ ... ≠ AN). In addition, B1, B2,..., BM are integers (where B1 ≠ B2 ≠ ... ≠ BM). C1, C2,..., CQ are integers (where C1 ≠ C2 ≠ ... ≠ Cq). Then, P (D) and Pn (D) at time point 2i are obtained using the relational expressions of Expression (147-1) and Expression (147-2).

  In the equation (147-3), α1, α2,..., Αω are integers (where α1 ≠ α2 ≠ ... ≠ αω). In addition, β1, β2,..., Βξ are integers (where β1 ≠ β2 ≠ ... ≠ βξ). In addition, γ1, γ2,..., Γλ are integers (where γ1 ≠ γ2 ≠ ... ≠ γλ).

  In the equation (147-4), E1, E2,..., EΩ are integers (however, E1 ≠ E2 ≠ ... ≠ EΩ). In addition, F1, F2,..., FZ are integers (where F1 ≠ F2 ≠ ... ≠ FZ). G1, G2,..., GΛ are integers (where G1 ≠ G2 ≠ ... ≠ GΛ). Then, P (D) and Pn (D) at the time point 2i + 1 are obtained using the relational expressions of the expressions (147-3) and (147-4).

  In the case of the coding rate 1/3, as in the case of the coding rate 1/2, each order a1, a2,..., An in the equation (147-1) is also an even number of 3 or more, or It is preferable that three or more odd numbers are not included, and n ≦ 4, or b1, b2,..., Bm are also three or more even numbers, or three or more. Are preferably not included, and m ≦ 4, or c1, c2,..., Cq also include three or more even numbers, or three or more odd numbers. It is preferable that q is not more than 4, and q ≦ 4 is good.

  Similarly, it is preferable that each order A1, A2,..., AN of the formula (147-2) does not include three or more even numbers or three or more odd numbers, and It is preferable that N ≦ 4, or B1, B2,..., BM should not include three or more even numbers or three or more odd numbers, and M ≦ 4. Or C1, C2,..., CQ are preferably not included in three or more even numbers or three or more odd numbers, and Q ≦ 4 is preferable. .

  Similarly, it is preferable that each of the orders α1, α2,..., Αω in the formula (147-3) does not include three or more even numbers or three or more odd numbers, and It is preferable that ω ≦ 4, or β1, β2,..., βξ should not include three or more even numbers or three or more odd numbers, and ξ ≦ 4. Or γ1, γ2,..., Γλ are preferably not included in three or more even numbers or three or more odd numbers, and λ ≦ 4 is preferable. .

  Similarly, it is preferable that each order E1, E2,..., EΩ of the formula (147-4) does not include three or more even numbers or three or more odd numbers, and It is preferable that Ω ≦ 4, or F1, F2,..., FZ also be such that three or more even numbers or three or more odd numbers are not included, and Z ≦ 4. It is preferable that G1, G2,..., GΛ should not include three or more even numbers or three or more odd numbers, and Λ ≦ 4. .

  If the following setting is made, an LDPC-CC with a better time-varying period of 2 and a coding rate of 1/3 can be designed.

That is, “Each degree a1, a2,..., An of the formula (147-1) is also configured not to include three or more even numbers or three or more odd numbers, and n ≦ 4. Also, b1, b2,..., Bm should not include three or more even numbers or three or more odd numbers, and m ≦ 4, and c1, c2, .., Cq, so that three or more even numbers or three or more odd numbers are not included, and q ≦ 4.
Similarly, each order A1, A2,..., AN of the formula (147-2) is also configured not to include three or more even numbers or three or more odd numbers, and N ≦ 4 In addition, B1, B2,..., BM are not included in three or more even numbers or three or more odd numbers, and M ≦ 4. In addition, C1, C2,..., CQ are not included in three or more even numbers or three or more odd numbers, and Q ≦ 4.
Similarly, for the orders α1, α2,..., Αω in the formula (147-3), it is ensured that three or more even numbers or three or more odd numbers are not included, and ω ≦ 4 Also, for β1, β2,..., Βξ, three or more even numbers or three or more odd numbers are not included, and ξ ≦ 4. Γ1, γ2,..., Γλ also do not include three or more even numbers, or three or more odd numbers, and λ ≦ 4
And
Similarly, for each order E1, E2,..., EΩ of the equation (147-4), it is made not to include three or more even numbers or three or more odd numbers, and Ω ≦ 4 Also, F1, F2,..., FZ are set so as not to include three or more even numbers or three or more odd numbers and Z ≦ 4. In addition, G1, G2,..., GΛ are not included in three or more even numbers or three or more odd numbers, and Λ ≦ 4. ", An LDPC-CC having a better time-varying period of 2 and a coding rate of 1/3 is obtained.

Also, consider LDPC-CC with time-varying period 2 and coding rate n / n + 1 using equations (148-1) and (148-2).

In the formula (148-1), a _1,1 , a _1,2 ,..., A _{1, r1} are integers (where a _1,1 ≠ a _1,2 ≠... ≠ a _{1, r1} ). And Further, a _2,1 , a _2,2 ,..., A _{2, r2} are integers (provided that a _2,1 ≠ a _2,2 ≠... ≠ a _{2, r2} ). Further, a _{i, 1} , a _{i, 2} ,..., A _{i, ri} (i = 3,..., N−1) are integers (where a _{i, 1} ≠ a _{i, 2} ≠ ... ≠ a _{i, ri} ). A _{n, 1} , a _{n, 2} ,..., A _{n, rn} are integers (where a _{n, 1} ≠ an _{, 2} ≠... ≠ an _{, rn} ). Further, e1, e2,..., Ew are integers (where e1 ≠ e2 ≠ ... ≠ ew). For example, P (D) at the time point 2i is obtained using the relational expression of the expression (148-1).

In formula (148-2), A _1,1 , A _1,2 ,..., A _{1, R1} are integers (where A _1,1 ≠ A _1,2 ≠... ≠ A _{1, R1} ). And In addition, A _2,1 , A _2,2 ,..., A _{2, R2} are integers (however, A _2,1 ≠ A _2,2 ≠... ≠ A _{2, R2} ). In addition, A _{i, 1} , A _{i, 2} ,..., A _{i, ri} (i = 3,..., N−1) are integers (where A _{i, 1} ≠ A _{i, 2} ≠ ... ≠ A _{i, ri} ). In addition, A _{n, 1} , A _{n, 2} ,..., A _{n, Rn} are integers (where A _{n, 1} ≠ A _{n, 2} ≠... ≠ A _{n, Rn} ). Further, E1, E2,..., EW are integers (where E1 ≠ E2 ≠ ... ≠ EW). For example, it is assumed that P (D) at time point 2i + 1 is obtained using the relational expression of the expression (148-2).

In the case of the coding rate n / n + 1, as in the case of the coding rates 1/2 and 1/3, among the parity check polynomials of the LDPC-CC with the time-varying period 2 represented by the equation (148-1), [A _1,1 , a ₁ , ₂ ,..., A _{1, r1} ] does not include three or more even or odd numbers and satisfies r1 ≦ 4, or [a _{i, 1} , a _{i, 2} ,..., a _{i, ri} ] (i = 2, 3,..., n−1), three or more even or odd numbers are not included, and ri ≦ 4 is satisfied, or , [ _{An, 1} , a _{n, 2} ,..., A _{n, rn} ] do not include three or more even or odd numbers and satisfy rn ≦ 4, or [e1, e2, , Ew] does not include three or more even or odd numbers and satisfies w ≦ 4, the first parity check polynomial (14 -1), of the parity check polynomial of a convolutional code represented by the formula _(148-2), in the _{[A 1,1, A 1,2, ···} , A 1, R1], the even or odd Are not included and R1 ≦ 4 is satisfied, or [A _{i, 1} , A _{i, 2} ,..., A _{i, R i} ] (i = 2, 3,..., N -1) 3 or more of even or odd numbers are not included and Ri ≦ 4 is satisfied, or even or within [A _{n, 1} , A _{n, 2} ,..., A _{n, Rn} ] Three or more odd numbers are not included and Rn ≦ 4 is satisfied, or three or more even numbers or odd numbers are not included in [E1, E2,..., EW] and W ≦ 4 is satisfied. The parity check matrix defined based on the second parity check polynomial (148-2) may be used.

  In addition, in the parity check polynomial of the LDPC-CC with time-varying period 2 expressed in the form of the expressions (148-1) and (148-2), the first parity check polynomial that satisfies the following [Condition # 1] ( The LDPC-CC having a time-varying period of 2 is designed using a check matrix defined based on 148-1) and a second parity check polynomial (148-2) that satisfies the following [Condition # 2]. Thus, it is possible to obtain an LDPC-CC having a time-varying period of 2 and a coding rate of n / n + 1 with better characteristics.

[Condition # 1]
In formula (148-1), [a1, 1, a1, 2,..., A1, r1] does not include three or more even or odd numbers and satisfies r1 ≦ 4.
[Ai, 1, ai, 2,..., Ai, ri] (i = 2, 3,..., N−1), three or more even or odd numbers are not included, and ri ≦ 4 is satisfied.
[An, 1, an, 2,..., An, rn] does not include three or more even or odd numbers and satisfies rn ≦ 4.
[E1, e2,..., Ew] does not include three or more even or odd numbers and satisfies w ≦ 4.

[Condition # 2]
[A1, 1, A1, 2,..., A1, R1] does not include three or more even or odd numbers and satisfies R1 ≦ 4.
[Ai, 1, Ai, 2,..., Ai, Ri] (i = 2, 3,..., N−1) does not include three or more even or odd numbers, and Ri ≦ 4 is satisfied.
[An, 1, An, 2,..., An, Rn] does not include three or more even or odd numbers and satisfies Rn ≦ 4.
[E1, E2,..., EW] does not include three or more even or odd numbers and satisfies W ≦ 4.

  In the discussion of the loop 6, the condition is that the number of terms is 4 or less, because if there are 5 or more, there will always be 3 or more even numbers or 3 or more odd numbers. The important theorem regarding the loop 6 will be described in detail in another embodiment 14.

Table 4 shows a code list of Ak and Bk in a parity check polynomial with a time varying period of 2 and a coding rate of 1/2 based on the equation (122). Table 4 is an example of an LDPC-CC having a time-varying period of 2 and a coding rate of ½, which gives good reception characteristics when the maximum constraint length is 600 or less.

  In addition, in LDPC-CC with a time-varying period of 2 and a coding rate of 1/2, as a condition of LDPC-CC that gives good reception quality, the column weight is 10 or less in all columns of the parity check matrix. It is an important condition.

(Other embodiment 10)
In Embodiment 7, Embodiment 8, Other Embodiment 5, Other Embodiment 6, and Other Embodiment 8, when the time varying period of the time varying LDPC-CC is short, for example, the time varying period is The case of 2 to 10 has been described. Here, an LDPC-CC having a time varying period extended by applying an LDPC-CC having a time varying period 2 will be described. Hereinafter, as an example, a case where the coding rate is 1/2 will be described. In the seventh embodiment, the case where the coding rate is 1/2 has been described as an example, and the description will be made in comparison with the seventh embodiment.

In the seventh embodiment, the LDPC-CC having a time varying period of about 2 to 10 has been described. When a parity check polynomial is randomly generated, an LDPC-CC having a time-varying period of 2 can easily search for a code having a good characteristic, whereas an LDPC-CC having a long time-varying period has a good characteristic. It becomes difficult to search for a correct code. This is because when the parity check polynomial is randomly generated, the number of necessary parity check polynomials increases as the time-varying period increases, and therefore, a combination of parity check polynomials that can provide LDPC-CC with good characteristics is specified. Because it becomes difficult.

  Therefore, a method for generating an LDPC-CC having a long time-varying period by applying LDPC-CC having a time-varying period 2 will be discussed.

  As described in the seventh embodiment, when the coding rate is 1/2, the polynomial representation of the information sequence (data) is X (D) and the polynomial representation of the parity sequence is P (D). Is expressed as in equation (64).

In formula (64), a1, a2,..., An are integers other than 0 (where a1 ≠ a2 ≠ ... ≠ an). In addition, b1, b2,... Bm are integers of 1 or more (where b1 ≠ b2 ≠ ... ≠ bm). Here, in order to enable easy encoding, it is assumed that the terms D ⁰ X (D) and D ⁰ P (D) (D ⁰ = 1) exist. Therefore, P (D) is expressed as in Expression (65).

As can be seen from the equation (65), since D ⁰ = 1 exists and past parity terms, that is, b1, b2,..., Bm are integers of 1 or more, the parity P is sequentially changed. Can be sought.

  Next, a parity check polynomial with a coding rate of 1/2, which is different from Equation (64), is expressed as Equation (66).

In formula (66), A1, A2,..., AN are integers other than 0 (however, A1 ≠ A2 ≠ ... ≠ AN). Further, B1, B2,..., BM are integers of 1 or more (where B1 ≠ B2 ≠ ... ≠ BM). Here, it is assumed that the terms D ⁰ X (D) and D ⁰ P (D) (D ⁰ = 1) exist in order to enable easy encoding. At this time, P (D) is expressed as shown in Expression (67).

Hereinafter, data X and parity P at time _2i are represented by X _2i and P _2i , respectively, and data X and parity P at time 2i + 1 are represented by X _{2i + 1} and P _{2i + 1} (i: integer), respectively.

In an LDPC-CC with a time varying period of 2, the parity P _2i at the time point 2i is calculated using the equation (65), and the parity P _{2i + 1} at the time point _{2i + 1} is calculated using the equation (67).

  Here, consider an LDPC-CC with a time-varying period of 2Z (Z: an integer of 2 or more). At this time, a parity check polynomial of equation (65) and Z different parity check polynomials based on equation (67), that is, (Z + 1) different parity check polynomials are prepared. Note that Z different parity check polynomials based on equation (67) are named “check equation # 0”, “check equation # 1”,..., “Check equation # Z-1”.

  Then, the parity of the time point j is obtained according to the following case 1) or case 2).

Case 1) When j mod 2 (remainder of dividing j by 2) = 0 Parity at time point j is obtained using equation (65).

Case 2) j mod 2 (remainder of dividing j by 2) = 1 k is a quotient when j is divided by 2, and k = gZ + i (g: integer, i = 0, 1,... Z-1), the parity of the time point j is obtained using “check equation #i”.

  In this way, an LDPC-CC having a time-varying period 2Z can be generated by (Z + 1) different parity check polynomials. That is, the time-varying LDPC-CC is formed by (Z + 1) parity check polynomials, which is smaller than the time-varying period 2Z. In addition, although it demonstrated using the formula (64) and the formula (66) in the above, the format of a parity check polynomial is not restricted to this.

  In addition, the case of the coding rate of 1/2 has been described as an example, but the present invention is not limited to this, and has been described in other embodiment 5, other embodiment 6, other embodiment 8, etc. Even in cases other than coding rate 1/2, LDPC-CC having a long time varying period can be generated by applying LDPC-CC having time varying period 2 as in the case of coding rate 1/2. . That is, not limited to the coding rate, Z different parity check polynomials are expressed as “check equation # 0”, “check equation # 1”,..., “Check equation # Z-1”, and these “check equations” A check polynomial “polynomial #A” different from “# 0”, “check expression # 1”,..., “Check expression # Z-1” is prepared.

  Then, the parity of the time point j is obtained according to the following case 1) or case 2).

Case 1) When j mod 2 (remainder of dividing j by 2) = 0 Parity of time point j is obtained using “polynomial #A”.

Case 2) j mod 2 (remainder of dividing j by 2) = 1 k is a quotient when j is divided by 2, and k = gZ + i (g: integer, i = 0, 1,... Z-1), the parity of the time point j is obtained using “check equation #i”.

  As described above, the time-varying LDPC-CC can be formed with a smaller number of parity check polynomials than the time-varying period 2Z even at coding rates other than the coding rate 1/2.

  In addition to the method described above, the time-varying LDPC-CC can be formed using a smaller number of parity check polynomials than the time-varying period 2Z. For example, α different parity check polynomials are prepared, and some of the α parity check polynomials are used a plurality of times to form an LDPC-CC with a time varying period β (β> α). Also good. However, as in case 1), in the case of j mod 2 = 0, when the parity of the time point j is always obtained using the same “polynomial #A”, the parity with good characteristics is particularly good. There is an advantage that it is easy to search for a check polynomial.

(Other Embodiment 11)
Here, a method of creating a search for LDPC-CC having confidentiality by applying the LDPC-CC described in the other embodiment 10 will be described. Hereinafter, as an example, a case where the coding rate is 1/2 will be described.

  For example, α different parity check polynomials based on Expression (64) are prepared. Then, β (α ≧ β) parity check polynomials are extracted from α parity check polynomials to create an LDPC-CC with a time-varying period γ (γ ≧ β).

At this time, the parity at time j satisfying j mod γ = i is obtained using the same parity check polynomial. For example, β parity check polynomials are expressed as “polynomial # 1”, “polynomial # 2”.
... “Polynomial # β”, i = 0, 1,..., Γ−1, and “polynomial #k” (k = 1, 2,..., Β) is at least If used once, since γ ≧ β, all i parity check polynomials are used for i = 0, 1,..., Γ−1.

  At this time, there are a plurality of methods for selecting different β parity check polynomials and methods for setting the time-varying period γ. Therefore, it is difficult to correct errors unless the receiving side knows the method of selecting different β parity check polynomials determined on the transmitting side and the method of setting the time-varying period γ.

  Therefore, in the following, confidential communication including a configuration in which the transmission device can change the parity check polynomial selection method and the time-varying period, and the reception device uses the configuration of the encoder of the transmission device as an encryption key. Propose.

  FIG. 50 shows an example of a secret communication system using the above-described method. In the following, a wireless communication system will be described as an example, but the secret communication system is not limited to the wireless communication system.

  The wireless communication system 5000 in FIG. 50 includes a transmission device 5010 and a reception device 5020.

  The transmission device 5010 includes an LDPC-CC encoder 5012, a modulation unit 5014, an antenna 5016, a control unit 5017, and a key information generation unit 5019.

  The control unit 5017 selects β parity check polynomials. Note that the parity check polynomial constitutes a parity check matrix used in the LDPC-CC encoder 5012. The control unit 5017 outputs information on the encoding method including information on the selected β parity check polynomials to the LDPC-CC encoder 5012.

  For example, the control unit 5017 holds α different parity check polynomials based on Expression (64), and extracts (selects) β (α ≧ β) parity check polynomials from the α parity check polynomials. . The control unit 5017 outputs information on the extracted (selected) β parity check polynomials to the LDPC-CC encoder 5012 as information 5018 on the encoding method. Information 5018 regarding the encoding method is shown below. For example, first, α number parity check polynomials are numbered beforehand. Next, it is assumed that the numbers assigned to the α parity check polynomials are known in advance between the transmission device 5010 and the reception device 5020. The number assigned to the extracted (selected) β parity check polynomials is used as information 5018 regarding the encoding method.

  Also, the control unit 5017 sets a time-varying period γ, and outputs information related to the parity check polynomial used at time j among the selected β parity check polynomials to the LDPC-CC encoder 5012.

  LDPC-CC encoder 5012 receives information 5011 and information 5018 related to the encoding method output from control unit 5017 as input, and performs LDPC-CC encoding according to the encoding method specified by information 5018. .

Specifically, the LDPC-CC encoder 5012 obtains the parity at time j satisfying j mod γ = i using the same parity check polynomial. For example, β parity check polynomials are represented by “polynomial # 1”, “polynomial # 2”... “Polynomial # β”, and i = 0, 1
,..., Γ−1 so that “polynomial #k” (k = 1, 2,..., Β) is used at least once. In this case, since γ ≧ β, all i parity check polynomials are used for i = 0, 1,..., Γ−1. The LDPC-CC encoder 5012 outputs the encoded data 5013 to the modulation unit 5014.

  Modulation section 5014 receives encoded data 5013 as input, performs processing such as modulation, band limitation, frequency conversion, and amplification, and outputs obtained modulated signal 5015 to antenna 5016.

  The antenna 5016 emits the modulation signal 5015 as a radio wave.

  The key information generation unit 5019 receives information 5018 related to the encoding method in the LDPC-CC encoder 5012 as input, generates key information using this information 5018 as a key, and uses the generated key information using some communication means. Notify the receiving device 5020. As described above, for example, when α parity check polynomials are numbered in advance, the numbers assigned to the extracted (selected) β parity check polynomials may be used as keys. That is, key information generation section 5019 notifies reception apparatus 5020 of information related to the parity check polynomial used in LDPC-CC encoder 5012.

  The reception device 5020 includes an antenna 5021, a demodulation unit 5023, a decryption unit 5025, and a key information acquisition unit 5026.

  The key information acquisition unit 5026 receives the key information transmitted from the transmission device 5010 as an input and reproduces information related to the encoding method. For example, when the number of the parity check polynomial used in the LDPC-CC encoder 5012 of the transmission apparatus 5010 is a key, the key information acquisition unit 5026 reproduces the number of the parity check polynomial and includes the obtained number The encoded information 5027 is output to the decoding unit 5025.

  Demodulation section 5023 receives reception signal 5022 received by antenna 5021, performs processing such as amplification, frequency conversion, orthogonal demodulation, detection, etc., and outputs log-likelihood ratio 5024.

  Decoding section 5025 receives encoded information 5027 as input, creates a check matrix based on the encoding method, receives log likelihood ratio 5024 as input, performs decoding processing based on the check matrix, and outputs estimated information 5028 .

  As described above, according to the present embodiment, transmission apparatus 5010 selects a parity check polynomial that constitutes a parity check matrix used in LDPC-CC encoder 5012, and includes a code including information on the selected parity check polynomial. A control unit 5017 that outputs information on the encoding method to the LDPC-CC encoder 5012, an LDPC-CC encoder 5012 that performs encoding using the parity check polynomial selected by the control unit 5017, and a control unit 5017. And a key information generation unit 5019 for notifying the reception device 5020 of information related to the encoding method including the parity check polynomial selected by the reception device 5020, based on the information related to the encoding method notified from the transmission device 5010. Decoding is performed using the check matrix H.

  By doing in this way, it is possible to realize the confidential communication using the different selection methods of β parity check polynomials determined on the transmission side and the setting method of the time-varying period γ as keys.

  In FIG. 50, the case where the transmission device 5010 generates an encryption key, that is, information for specifying the check matrix H has been described. However, the present invention is not limited to this, and the reception device 5020 sets the encryption key and transmits You may make it notify to the apparatus 5010. FIG. A configuration example of the wireless communication system in this case is shown in FIG.

  The wireless communication system 5100 in FIG. 51 includes a transmission device 5110 and a reception device 5120. In FIG. 51, the same components as those in FIG. 50 are denoted by the same reference numerals, and the description thereof is omitted.

  Receiving device 5120 includes demodulator 5023, decryptor 5025, controller 5121, and key information generator 5123.

  Similar to the control unit 5017, the control unit 5121 generates information 5122 related to the encoding method, and outputs the information 5122 related to the generated encoding method to the decoding unit 5025.

  Similarly to the key information generation unit 5019, the key information generation unit 5123 receives the information 5122 related to the encoding method as input, generates key information using the information 5122 as a key, and uses the generated key information using some communication means. , Notify the transmission device 5110.

  The transmission apparatus 5110 includes an LDPC-CC encoder 5012, a puncture / error addition unit 5113, a modulation unit 5014, and a key information acquisition unit 5111.

  Key information acquisition section 5111 receives key information notified from receiving apparatus 5120 as input, reproduces information 5112 related to the encoding method, and outputs information 5112 to LDPC-CC encoder 5012.

  The LDPC-CC encoder 5012 performs encoding based on information 5112 regarding the encoding method.

  When LDPC-CC is a systematic code, data (information) can be obtained without performing error correction (decoding) on the receiving side in a good communication state such as a strong radio reception electrolysis strength. By extracting only the portion corresponding to X, data (information) X can be obtained by any receiving device. In other words, it may be possible to receive other people's information without permission. In order to avoid this, for example, the transmission apparatus 5110 is provided with a puncture / error adding unit 5113 as shown in FIG. 51, and the puncture / error adding unit 5113 punctures data (information) X or data For example, a part of the data may be intentionally replaced with erroneous data. Thus, by providing the puncture / error adding unit 5113, it is difficult to obtain data (information) X unless the receiving apparatus has a correct decoding function.

  In the above description, α different parity check polynomials based on Expression (64) are prepared. However, the present invention is not limited to Expression (64), and other parity check polynomials may be used.

(Time-invariant / time-varying LDPC-CC based on convolutional code (coding rate (n-1) / n) (n: natural number))
The outline of the time-invariant / time-variant LDPC-CC based on the convolutional code will be described below.

Information _X 1 coding rate R = (n-1) / n, X 2, ···, the polynomial representation of the _{X n-1 X 1 (D} ), X 2 (D), ···, X n ₋₁ (D), and a parity check polynomial expressed as shown in Equation (149), where P (D) is a polynomial expression of parity P.

In formula (149), at this time, a _{p, p} (p = 1, 2,..., N−1; q = 1, 2,..., Rp) is, for example, a natural number, and _{ap, 1} ≠ a _{p, 2} ≠... ≠≠ a _{p, rp} is satisfied. Further, b _q (q = 1, 2,..., S) is a natural number, and b ₁ ≠ b ₂ ≠ ·
.. ≠≠ b _s is satisfied. At this time, a code defined by a parity check matrix based on the parity check polynomial of Equation (149) is referred to herein as time invariant LDPC-CC.

ここで、ｉ＝０，１，・・・，ｍ−１である。 M different parity check polynomials based on Expression (149) are prepared (m is an integer of 2 or more). The parity check polynomial is expressed as follows.

Here, i = 0, 1,..., M−1.

ここで、「ｊ ｍｏｄ ｍ」は、ｊをｍで除算した余りである。 Then, the information X ₁ , X ₂ ,..., X _n−1 at the time point j is expressed as X _{1, j} , X _{2, j} ,..., X _{n−1, j,} and the parity P at the time point j is Pj and u _j = (X _{1, j} , X _{2, j} ,..., X _{n−1, j} , Pj) ^T. At this time, the information X _{1, j} , X _{2, j} ,..., X _{n−1, j} and the parity P _j at the time point j satisfy the parity check polynomial of Expression (151).

Here, “j mod m” is a remainder obtained by dividing j by m.

  A code defined by a parity check matrix based on the parity check polynomial of Equation (151) is referred to herein as time-varying LDPC-CC. At this time, the time-invariant LDPC-CC defined by the parity check polynomial of equation (149) and the time-varying LDPC-CC defined by the parity check polynomial of equation (151) sequentially register and exclude parity. It has the feature that it can be easily obtained with a logical OR.

  For example, FIG. 52 shows the configuration of LDPC-CC parity check matrix H of time varying period 2 based on equations (149) to (151) at a coding rate of 2/3. Two check polynomials having different time-varying periods 2 based on the formula (151) are named “check formula # 1” and “check formula # 2”. In FIG. 52, (Ha, 111) is a portion corresponding to “checking formula # 1”, and (Hc, 111) is a portion corresponding to “checking formula # 2”. Hereinafter, (Ha, 111) and (Hc, 111) are defined as sub-matrices.

Thus, the LDPC-CC parity check matrix H of the proposed time-varying period 2 is represented by the first sub-matrix representing the parity check polynomial of “check equation # 1” and the parity check polynomial of “check equation # 2”. The second sub-matrix can be defined. Specifically, in the check matrix H, the first sub-matrix and the second sub-matrix are alternately arranged in the row direction. In the case of a coding rate of 2/3, as shown in FIG. 52, the sub-matrix is shifted to the right by three columns in the i-th row and the i + 1-th row.

In the case of time-varying LDPC-CC with a time-varying period of 2, the i-th row sub-matrix and the i + 1-th row sub-matrix are different sub-matrices. That is, one of the sub-matrices (Ha, 11) or (Hc, 11) is the first sub-matrix, and the other is the second sub-matrix. The transmission vector u is expressed as u = (X _1,0 , X _2,0 , P ₀ , X _1,1 , X _2,1 , P ₁ ,..., X _{1, k} , X _{2, k} , P _k ,...) If ^T , Hu = 0 holds. This is the same as that described in the first embodiment (see formula (3)).

Next, consider an LDPC-CC in which the time-varying period is m when the coding rate is 2/3. As in the case of time-varying period 2, m parity check polynomials represented by Expression (149) are prepared. Then, “inspection formula # 1” represented by formula (149) is prepared. Similarly, “checking formula # 2” to “checking formula #m” represented by formula (149) are prepared. The data X and the parity P at the time point _{mi + 1} are represented as X _{mi + 1} and P _{mi + 1} , respectively. The data X and the parity P at the time point mi + 2 are represented as X _{mi + 2} and P _{mi + 2} , respectively. Are represented as X _{mi + m} and P _{mi + m} , respectively (i: integer).

At this time, the parity P _{mi + 1} at the time point mi + 1 is obtained using the “check equation # 1”, the parity P _{mi + 2} at the time point mi + 2 is obtained using the “check equation # 2,” and the parity P _{mi + m} at the time point mi + m is obtained. Consider an LDPC-CC obtained using “checking formula #m”. Such LDPC-CC codes are:
The encoder can be easily configured, and the parity can be obtained sequentially. Advantages include reduction of termination bits and improvement in reception quality at the time of puncturing at the termination.

FIG. 53A shows the configuration of the LDPC-CC parity check matrix with the coding rate 2/3 and the time varying period m described above. In FIG. 53A, (H ₁ , 111) is a portion corresponding to “check equation # 1,” (H ₂ , 111) is a portion corresponding to “check equation # 2,” (H) _m , 111) is a portion corresponding to “inspection formula #m”. Hereinafter, (H ₁ , 111) is defined as the first sub-matrix, (H ₂ , 111) is defined as the second sub-matrix,..., (H _m , 111) is defined as the m-th sub-matrix. To do.

  Thus, the LDPC-CC parity check matrix H of the proposed time-varying period m represents the first sub-matrix representing the parity check polynomial of “check formula # 1” and the parity check polynomial of “check formula # 2”. The second sub-matrix,..., And the m-th sub-matrix representing the parity check polynomial of “check equation #m”. Specifically, in the check matrix H, the first to m-th sub-matrices are periodically arranged in the row direction (see FIG. 53A). In the case of a coding rate of 2/3, the sub-matrix is shifted to the right by three columns in the i-th row and the i + 1-th row (see FIG. 53A).

The transmission vector u is expressed as u = (X _1,0 , X _2,0 , P ₀ , X _1,1 , X _2,1 , P ₁ ,..., X _{1, k} , X _{2, k} , P _k ,...) If ^T , Hu = 0 holds. This is the same as that described in the first embodiment (see formula (3)).

  In the above description, the case of the coding rate 2/3 has been described as an example of the time-invariant / time-varying LDPC-CC based on the convolutional code of the coding rate (n-1) / n. Thus, it is possible to create a time-invariant / time-variant LDPC-CC parity check matrix based on a convolutional code with a coding rate (n−1) / n.

That is, in the case of the coding rate 2/3, in FIG. 53A, (H ₁ , 111) is a portion (first sub-matrix) corresponding to “check equation # 1,” and (H ₂ , 111) is “check”. A part (second sub-matrix) corresponding to “Expression # 2”,..., (H _m , 111) is a part (m-th sub-matrix) corresponding to “check expression #m”, In the case of coding rate (n−1) / n, the result is as shown in FIG. 53B. That is, the portion (first sub-matrix) corresponding to “check equation # 1” is represented by (H ₁ , 11... 1), and “check equation #k” (k = 2, 3,... , M) (kth sub-matrix) is represented by (H _k , 11... 1). At this time, in the k-th sub-matrix, the number of “1” s in the portion excluding H _k is n−1. In the parity check matrix H, the sub-matrix is shifted to the right by n−1 columns in the i-th row and the i + 1-th row (see FIG. 53B).

The transmit vector _{u, u = (X 1,0,} X 2,0, ···, X n-1,0, P 0, X 1,1, X 2,1, ···, X n-1 _{, 1, P 1, ···,} X 1, k, X 2, k, ···, X n-1, k, P k, ····) When ^T, Hu = 0 is established. This is the same as that described in the first embodiment (see formula (3)).

Table 5 shows a code list of Ak and Bk in a parity check polynomial with a time varying period of 2 and a coding rate of 1/2 based on the equation (151). Table 5 is an example of an LDPC-CC having a time-varying period of 2, a coding rate of 2/3, 3/4, and 5/6 that gives good reception characteristics when the maximum constraint length is 600 or less.

(Other Embodiment 12)
Here, the relationship between the parity check polynomial and the check matrix H will be described in detail. In the following, a case where the time varying period is 2 will be described as an example.

FIG. 54A shows a parity check polynomial of “check equation # 1” and a corresponding first sub-matrix H ₁ (5405) used when obtaining the parity at time 2i. In the first sub-matrix H ₁ (5405) of FIG. 54A, a dotted line 5400-1 indicates the boundary between the time point 2i and the time point 2i + 1 in the parity check matrix H. The second element 5401 from the dotted line 5400-1 corresponds to “1” regarding the data (information) of the parity check polynomial, and the leftmost element 5402 of the dotted line 5400-1 is “1” regarding the parity of the parity check polynomial. ".

Similarly, FIG. 54B shows the parity check polynomial of “check equation # 2” used when obtaining the parity at time 2i + 1 and the corresponding second sub-matrix H ₂ (5406). In the second sub-matrix H ₂ (5406) in FIG. 54B, a dotted line 5400-2 indicates the boundary between the time point 2i + 1 and the time point 2i + 2 in the parity check matrix H. The second element 5403 from the dotted line 500-2 corresponds to “1” regarding the data (information) of the parity check polynomial, and the element 5404 on the left side of the dotted line 500-2 is “1” regarding the parity of the parity check polynomial. ".

Figure 55, comprised by the second sub-matrix _{H 2} of the first sub-matrix _{H 1} and Figure 54B in FIG. 54A, the coding rate of 1/2, when showing a check matrix H LDPC-CC parity of varying period 2. As can be seen from FIG. 55, in the case of the coding rate ½, in the 2i row and the 2i + 1 row, the boundary 5400-1 between the time point 2i and the time point 2i + 1 of the first sub matrix H ₁ and the second sub matrix H and ₂ time points 2i + 1 and point in time 2i + 2 of the boundary 5400-2 is, a configuration in which shift in two rows right. Further, in the first 2i + 1 row and the 2i + 2 rows, and time 2i + 1 and point in time 2i + 2 of the boundary 5400-2 of the second sub-matrix _{H 2,} the dotted line corresponding to the time 2i and the boundary time points 2i + 1 of first sub-matrix _{H 1} ( The boundary between the time point 2i + 2 and the time point 2i + 3) 5400-1 is shifted to the right by two columns.

  Note that the parity check polynomial and the check matrix H described above also apply to the LDPC-CC parity check matrix having the time varying period 2 or the time varying period m described in the above-described embodiment and other embodiments. It has the same relationship as

The transmission sequence u is expressed as u = (X ₀ , P ₀ , X ₁ , P ₁ ,..., X _i , P _i ...) ^T. Here, X _i is information, P _i is parity, and the transmission sequence u is a systematic code. In this case, the first sub-matrix H ₁ in FIG. 54A satisfies X _2i−3 + X _2i + P _2i−5 + P _2i−3 + P _2i = 0. Similarly, the second sub-matrix H ₂ in FIG. 54B satisfies X _{(2i + 1) −4} + X _{(2i + 1)} + X _{(2i + 1) +5} + P _{(2i + 1) −3} + P _{(2i + 1) −1} + P _{(2i + 1)} = 0.

  In the above description, the relationship between the parity check polynomial and the check matrix H has been described by taking the case of the coding rate 1/2 and the time varying period 2 as an example, but the relationship between the parity check polynomial and the check matrix H is It is not limited to the conversion rate and the time varying period. Hereinafter, a case where the coding rate is 2/3 and the time varying period is 2 will be described.

FIG. 56A shows a parity check polynomial of “check equation # 1” and a corresponding first sub-matrix H ₁ (5604) used when determining the parity at time 2i. In the first sub-matrix H ₁ (5604) in FIG. 56A, a dotted line 5600-1 indicates the boundary between the time point 2i and the time point 2i + 1 in the parity check matrix H. The third element 5601 from the dotted line 5600-1 corresponds to “1” regarding X1 (D), the second element 5602 from the dotted line 5600-1 corresponds to “1” regarding X2 (D), The leftmost element 5603 of the dotted line 5600-1 corresponds to “1” regarding the parity of P (D).

Similarly, FIG. 56B shows “check equation # 2” used when obtaining the parity at time 2i + 1.
, And a corresponding second sub-matrix H ₂ (5608). In the second sub-matrix H ₂ (5608) in FIG. 56B, a dotted line 5600-2 indicates the boundary between the time point 2i + 1 and the time point 2i + 2 in the parity check matrix H. The third element 5605 from the dotted line 5600-2 corresponds to “1” related to X1 (D), the second element 5606 from the dotted line 5600-1 corresponds to “1” related to X2 (D), The leftmost element 5607 of the dotted line 5600-1 corresponds to “1” regarding the parity of P (D).

Figure 57, comprised by the second sub-matrix _{H 2} of the first sub-matrix _{H 1} and Figure 56B in FIG. 56A, the coding rate of 2/3, when showing a check matrix H LDPC-CC parity of varying period 2. As can be seen from Figure 57, if the coding rate 2/3, in the first 2i row and the 2i + 1 row, the time 2i and point in time 2i + 1 of the boundary 5600-1 of the first sub-matrix _{H 1,} second sub-matrix H and ₂ time points 2i + 1 and point in time 2i + 2 of the boundary 5600-2 becomes a shifted arrangement in three columns right. Further, in the first 2i + 1 row and the 2i + 2 rows, and time 2i + 1 and point in time 2i + 2 of the boundary 5600-2 of the second sub-matrix _{H 2,} the dotted line corresponding to the time 2i and the boundary time points 2i + 1 of first sub-matrix _{H 1} ( The boundary between the time point 2i + 2 and the time point 2i + 3) 5600-1 is shifted to the right by three columns.

  Note that the parity check polynomial and the check matrix H described above also apply to the LDPC-CC parity check matrix having the time varying period 2 or the time varying period m described in the above-described embodiment and other embodiments. It has the same relationship as

The transmission sequence u is represented by u = (X ₁ , ₀ , X ₂ , ₀ , P ₀ , X ₁ , ₁ , X ₂ , ₁ , P ₁ ,..., X _{1, i} , X _{2, i} , It expressed as P _i, ^{···) T.} Here, X _{1, i} , X _{2, i} are information, P _i is parity, and the transmission sequence u is a systematic code. In this case, first sub-matrix _{H 1} of FIG. 56A _{satisfies X 1,2i-3 + X 1,2i +} X 2,2i-2 + X 2,2i + P 2i-5 + P 2i-3 + P 2i = 0. Similarly, the second sub-matrix H ₂ in FIG. 56B is expressed as X _{1, (2i + 1) −4} + X _{1, (2i + 1)} + X _{1, (2i + 1) +5} + X _{2, (2i + 1) −3} + X _{2, (2i + 1)} + P _{(2i + 1) −3} + P _{(2i + 1) −1} + P _{(2i + 1)} = 0 is satisfied.

  As described above, the relationship between the parity check polynomial and the check matrix H has been described by taking the case of the coding rate 1/2, 2/3 as an example, but the parity check polynomial and the check are similarly applied regardless of the coding rate. The relationship of the matrix H is established. In particular, LDPC-CC (convolutional code) parity check matrix H is described in detail in Non-Patent Document 17 and Non-Patent Document 18.

(Other Embodiment 13)
Here, a difference between the seventh embodiment, the eighth embodiment, the other embodiment 5, the other embodiment 6, the other embodiment 8, and the non-patent document 16 will be described.

Non-Patent Document 16 describes a method of designing an LDPC-CC having a time-varying period of 4 from an LDPC-BC (Low-Density Parity-Check Block Code) when the coding rate is 1/2.

  Hereinafter, the LDPC-CC design method of Non-Patent Document 16 will be briefly described with reference to the drawings.

  FIG. 58 is a diagram provided for explaining the design method described in Non-Patent Document 16. A method of designing an LDPC-CC with a time varying period of 4 from an LDPC-BC with a coding rate of 1/2 will be described with reference to FIG. In Non-Patent Document 16, an LDPC-CC parity check matrix is generated by steps 1) to 3) described below.

Step 1)
Set LDPC-BC as the base of LDPC-CC. According to Non-Patent Document 16, when creating an LDPC-CC with a coding rate of ½ and a time-varying period of m, an LDPC-BC of m rows × 2 m columns is required.

  A parity check matrix 5801 in FIG. 58A illustrates an example of a configuration of a parity check matrix of an LDPC-BC serving as a base of an LDPC-CC having a time varying period of 4. As described above, in the case of the time varying period 4, the LDPC-BC parity check matrix of 4 rows × 8 columns is the base parity check matrix.

Step 2)
Then, a predetermined process is performed on parity check matrix 5801 to create parity check matrix 5802 (see FIG. 58B). In addition, since it describes in the nonpatent literature 16 about a specific process, description is abbreviate | omitted.

Step 3)
Then, as shown in FIG. 58C, “ 11 ” is added to parity check matrix 5802 to create parity check matrix 5803.

  In this way, in Non-Patent Document 16, an LDPC-CC parity check matrix with a time varying period of 4 is created from 4 rows × 8 columns of LDPC-BC by steps 1) to 3).

A parity check polynomial corresponding to the check matrix 5803 obtained in this way is expressed by Equation (152).

  In the formula (152), a1, a2,..., Ap are integers of 1 or more (provided that a1 ≠ a2 ≠ ... ≠ ap). B1, b2,... Bq are integers of 1 or more (where b1 ≠ b2 ≠ ... ≠ bq).

As can be seen from FIG. 58C, in the LDPC-CC with the time varying period 4, there are four different parity check polynomials based on the equation (152). Therefore, when designing an LDPC-CC with a time varying period of 4, an LDPC-BC parity check matrix of 4 rows × 8 columns is used as a base, and “ 11 ” is added as shown in FIG. 58C in step 3). , 4 different parity check polynomials constituting the base LDPC-BC parity check matrix, ai ≦ 4 (i = 1, 2,..., P) and bj ≦ 4 (j = 1, 2,..., Q).

  That is, when designing an LDPC-CC with a time varying period of 4 according to Non-Patent Document 16, the maximum constraint length is 4 + 1 = 5.

  Similarly, when an LDPC-CC having a time-varying period m is designed by the design method of Non-Patent Document 16, all the m different parity check polynomials constituting the base LDPC-BC check matrix are used. In equation (152), ai ≦ m (i = 1, 2,..., P) and bj ≦ m (j = 1, 2,..., Q) are established.

That is, when designing an LDPC-CC having a time-varying period m according to Non-Patent Document 16, an LDPC-BC parity check matrix of m rows × 2 m columns is used as a base, and in step 3), “ 1
1 ”is added, so the maximum constraint length is m + 1.

  Similarly, when an LDPC-CC having a time-varying period of 2 is designed by the design method of Non-Patent Document 16, in all two different parity check polynomials constituting the base LDPC-BC check matrix, ai ≦ 2 (i = 1, 2,..., p) and bj ≦ 2 (j = 1, 2,..., q) are established.

  That is, when designing an LDPC-CC having a time-varying period of 2 according to Non-Patent Document 16, the maximum constraint length is 2 + 1 = 3. Thus, when designing an LDPC-CC with a time-varying period m by the design method of Non-Patent Document 16, the maximum constraint length is m + 1. Therefore, the constraint length is long in order to improve the reception quality (error correction capability), for example, the constraint length is 100 or more (100, ..., 500, ..., 1000, ..., 2000, ..., When designing an LDPC-CC of 10,000,..., 20000,...), When designing an LDPC-CC according to Non-Patent Document 16, a time-varying period of a value similar to the constraint length is required.

  As described in the seventh embodiment and the like, if the time-varying period is too large, it is difficult to puncture periodically, and for example, it is necessary to puncture at random, and reception quality may be deteriorated. is there. Therefore, when the time-varying LDPC-CC is designed using the design method of Non-Patent Document 16, it is possible to simultaneously apply puncturing to support multiple coding rates and improve reception quality (error correction capability). May be difficult.

  If an LDPC-CC having a time-varying period 2 to about 10 that is applicable to puncture described in the seventh embodiment is designed using the design method of Non-Patent Document 16, the time-varying period 2 The maximum constraint length is 3 (= 2 + 1), and in equation (152), ai ≦ 2 (i = 1, 2,..., P) and bj ≦ 2 (j = 1, 2,. q). Similarly, in the time-varying cycle 10, the maximum constraint length 11 (= 10 + 1) is obtained. That is, in the equation (152), ai ≦ 10 (i = 1, 2,..., P) and bj ≦ 10 ( j = 1, 2,..., q).

  Thus, when the design method disclosed in Non-Patent Document 16 is used, the shorter the time-varying period, the shorter the maximum constraint length. In general, in LDPC-CC, the longer the constraint length, the wider the range in which the reliability is propagated, so that the reception characteristics are improved. However, in Non-Patent Document 16, when the time-varying period is short, the constraint length is also shortened at the same time, so that it is difficult to obtain good reception quality (correction capability).

  That is, in Non-Patent Document 16, when the constraint length of the parity check polynomial is increased in order to obtain good reception quality, the time-varying period is also increased at the same time, so that it is difficult to puncture periodically. In Non-Patent Document 16, when the time-varying period is shortened, the constraint length is also shortened, and it is difficult to obtain good reception quality.

  On the other hand, as described in Embodiment 7 or the like, in LDPC-CC, by adding the following requirements, it is possible to achieve both compatibility with a plurality of coding rates by puncturing and improvement in reception quality.

[Requirements]
・ Set the time-varying cycle to about 2 to 10.
-In the time-varying period m, the constraint length is set to m + 2 or more. In other words, when using m different parity check polynomials based on Equation (152), the maximum value A _{max of} ai (i = 1, 2,..., P) is used for all the m parity check polynomials. A _max ≧ m + 1 holds between m and m, and B _max ≧ m + between the maximum values B _max and m of bi (i = 1, 2,..., Q).
1 is established. In order to obtain good reception quality, it is desirable that either A _max or B _max is 100 or more.
Set the row weight to 7-12.

  On the other hand, when an LDPC-CC having a time-varying period of 2 that can most easily search for a puncture pattern is designed by the design method of Non-Patent Document 16, the maximum constraint length is 3, and two different equations (152) In the above, ai ≦ 2 (i = 1, 2,..., P) and bj ≦ 2 (j = 1, 2,..., Q). Therefore, when the LDPC-CC having the time varying period 2 is designed using the design method of Non-Patent Document 16, the row weight is 6 at the maximum.

  Therefore, among the requirements of the LDPC-CC of time-varying period 2 for achieving both the improvement of the reception quality described in the seventh embodiment and the like and the correspondence of a plurality of coding rates by puncturing, “row weight is set to 7 to 12 “Is set” is a characteristic requirement of the present invention.

(Other Embodiment 14)
Here, the time-invariant LDPC-CC and the LDPC-CC loop 6 of the time-varying period 2 will be described in detail.

  (1) First, the time-invariant LDPC-CC with the coding rate n / (n + 1) will be described.

Data (information) X1 polynomial is X1 (D), data (information) X2 polynomial is X2 (D), data (information) X3 polynomial is X3 (D),..., Data (information) Xn polynomial Let Xn (D) be a polynomial of parity P and P (D), and consider the following parity check polynomial.

In the formula (153), a _1,1 , a ₁ , ₂ ,..., A _{1, r1} are integers (where a _1,1 ≠ a _1,2 ≠... ≠ a _{1, r1} ). . Further, a _2,1 , a _2,2 ,..., A _{2, r2} are integers (provided that a _2,1 ≠ a _2,2 ≠... ≠ a _{2, r2} ). Further, a _{i, 1} , a _{i, 2} ,..., A _{i, ri} (i = 3,..., N−1) are integers (where a _{i, 1} ≠ a _{i, 2} ≠ ... ≠ a _{i, ri} ). A _{n, 1} , a _{n, 2} ,..., A _{n, rn} are integers (where a _{n, 1} ≠ an _{, 2} ≠... ≠ an _{, rn} ). Further, e ₁ , e ₂ ,..., E _w are integers (where e ₁ ≠ e ₂ ≠... ≠ e _w ).

[Theorem 1]
In time-invariant LDPC-CC based on the parity check polynomial of equation (153), any one of X1 (D), X2 (D), X3 (D),..., Xn (D), P (D) When there are three or more terms, at least one loop 6 exists.

[Example]
As regards X1 (D), consider a case where the parity check polynomial has a term (D ⁵ + D ³ +1) X1 (D). Then, a sub-matrix generated by extracting only the portion related to X1 (D) in the parity check matrix H is represented as shown in FIG. 59, and a loop 6 exists as indicated by a dotted line 5901.

[Proof]
If there are three or more terms for X1 (D), if it can be proved that at least one loop 6 exists, X2 (D), X3 (D),..., Xn (D), P (D )), It can be proved that the same is true by considering it as X1 (D). Accordingly, attention is paid to X1 (D).

  For the equation (153), a sub-matrix generated by extracting only the part related to X1 (D) in the parity check matrix H in which two terms exist in X1 (D) is expressed as shown in FIG. There is no loop.

Next, an expression (154) in which three terms exist in X1 (D) with respect to the expression (153) is considered.

ここで、α、βは自然数とする。 At this time, generality is not lost even if a _1,1 > a _1,2 > a _1,3 . Therefore, Expression (154) is expressed as follows.

Here, α and β are natural numbers.

At this time, a term relating to X1 (D) in equation (155), that is, (D ^{a1,3 + α + β} + D ^{a1,3 + β} + D ^a1,3 ) X1 (D) is considered. In the parity check matrix H, a sub-matrix generated by extracting only the portion related to X1 (D) is represented as shown in FIG. Therefore, regardless of the values of α and β, the loop 6 formed by the element 6101 always occurs as shown in FIG.

  When there are four or more terms related to X1 (D), when three terms are selected from the four or more terms, a loop 6 is formed by the selected three elements (see FIG. 61). Therefore, when there are four or more terms related to X1 (D), the loop 6 exists.

  Therefore, if there are three or more terms related to X1 (D) in the parity check polynomial, loop 6 exists. The proof can be similarly applied to X2 (D), X3 (D),..., Xn (D), P (D). Therefore, Theorem 1 was proved. (End of proof)

  (2) Next, important matters regarding the LDPC-CC with the time varying period 2 will be described.

In the LDPC-CC of time varying period 2, the polynomial of data (information) X1 is X1 (D), the polynomial of data (information) X2 is X2 (D), the polynomial of data (information) X3 is X3 (D), The data (information) Xn polynomial is Xn (D), and the parity P polynomial is P (D). Then, the parity check polynomial of equation (156) is considered as “check equation # 1”.

In the formula (156), a _1,1 , a _1,2 ,..., A _{1, r1} are integers (where a _1,1 ≠ a _1,2 ≠... ≠ a _{1, r1} ). . Further, a _2,1 , a _2,2 ,..., A _{2, r2} are integers (provided that a _2,1 ≠ a _2,2 ≠... ≠ a _{2, r2} ). Further, a _{i, 1} , a _{i, 2} ,..., A _{i, ri} (i = 3,..., N−1) are integers (where a _{i, 1} ≠ a _{i, 2} ≠ ... ≠ a _{i, ri} ). A _{n, 1} , a _{n, 2} ,..., A _{n, rn} are integers (where a _{n, 1} ≠ an _{, 2} ≠... ≠ an _{, rn} ). Further, e ₁ , e ₂ ,..., E _w are integers (where e ₁ ≠ e ₂ ≠... ≠ e _w ).

Similarly, the parity check polynomial of equation (157) is considered as “check equation # 2”.

In the formula (157), b _1,1 , b _1,2 ,..., B _{1, s1} are integers (where b _1,1 ≠ b _1,2, ≠... ≠ b _{1, s1} ). . In addition, b _2,1 , b _2,2 ,..., B _{2, s2} are integers (where b _2,1 ≠ b _2,2 ≠... ≠ b _{2, s2} ). Also,
b _{i, 1} , b _{i, 2} ,..., b _{i, si} (i = 3,..., n−1) are integers (where b _{i, 1} ≠ b _{i, 2} ≠... ≠ b _{i , Si} ). In addition, b _{n, 1} , b _{n, 2} ,..., B _{n, sn} are integers (where b _{n, 1} ≠ b _{n, 2} ≠ ... b _{n, sn} ). Further, f ₁ , f ₂ ,..., F _v are integers (where f ₁ ≠ f ₂ ≠... ≠ f _v ).

  Then, consider an LDPC-CC with a time-varying period 2 given by “check equation # 1” and “check equation # 2”.

[Theorem 2]
A time-varying period 2 LDPC-CC based on the parity check polynomial of Equation (156) and the parity check polynomial of Equation (157), and the parity check polynomial of Equation (156):
“There exists y where (a _{y, i} , a _{y, j} , a _{y, k} ) are all odd numbers, or all are even numbers (where i ≠ j ≠ k), or (e _i , e _j , e _k ) is all odd or all even, or there is _{z where} (b _{z, i} , b _{z, j} , b _{z, k} ) are all odd or all even (where i ≠ j ≠ k), or (f _i , f _j , d _k ) are all odd numbers or all are even numbers ”
When this condition is satisfied, at least one loop 6 exists.

[Example]
Consider a case where (D ⁶ + D ² +1) X1 (D) term exists in the parity check polynomial with respect to “check equation # 1” X1 (D). Then, in the parity check matrix H, a sub-matrix generated by extracting only the portion related to X1 (D) is represented as shown in FIG. 62, and a loop 6 exists as indicated by a dotted line 6203.

[Proof]
For X1 (D), if (a _{1, i} , a _{1, j} , a _{1, k} ) are all odd numbers or all are even numbers (where i ≠ j ≠ k), the loop 6 may exist. If it can be proved, X2 (D), X3 (D),..., Xn (D), P (D) can be proved to be the same by considering X1 (D) instead of X1 (D). . Accordingly, attention is paid to X1 (D).

In addition, it is proved in the same way that the parity check polynomial of equation (156), that is, “check equation # 1” is established, so that in the parity check polynomial of equation (157), that is, “check equation # 2”. We can prove that Therefore, the parity check polynomial of equation (156), that is, “check equation # 1” is considered.

In the term related to X1 (D) in the formula (156), when a _{1, i} (i = 1, 2,..., R1) has two even numbers or two odd numbers, a portion related to X1 (D) FIG. 63 shows a sub-matrix generated by extracting only the sub-matrix. In FIG. 63, a sub-matrix 6301 is a sub-matrix corresponding to X1 (D) of “check equation # 1,” and a sub-matrix 6302 is a sub-matrix corresponding to X1 (D) of “check equation # 2.” As can be seen from the submatrix 6301 in FIG. 63, even numbers are 2 in a _{1, i} (i = 1, 2,..., R1) of the parity check polynomial (“check expression # 1”) of the expression (156). If only one or two odd numbers exist, a loop does not occur.

ここで、ｐ、ｑは自然数である。 Next, for expression (156), there are three terms for X1 (D) and (a _{1, i} , a _{1, j} , a _{1, k} ) are all odd or all are even. In this case, considering the equation (158), it can be expressed as the equation (159). Note that generality is not lost even if a _1,1 > a _1,2 > a _1,3 .

Here, p and q are natural numbers.

At this time, a term relating to X1 (D) in equation (159), that is, (D ^{a1,3 + 2p + 2q} + D ^{a1,3 + 2q} + D ^a1,3 ) X1 (D) is considered. In the parity check matrix H, a sub-matrix generated by extracting only the portion related to X1 (D) is represented as shown in FIG. In the case of time-varying period 2, all of the sub-matrixes 6401 in FIG. 64 are “checking expression # 1” in Expression (159), and thus the state is the same as that in FIG. 61 described in the proof of Theorem 1.

  Therefore, regardless of the values of p and q, only “inspection formula # 1” forms the loop 6 by the element 6101 as shown in FIG.

When there are four or more terms related to X1 (D), three terms are selected from the four or more terms, and in the three selected terms, (a _{1, i} , a _{1, j} , a _{1 , K} ) are all odd numbers, or all are even numbers, loop 6 is formed by element 6101 as shown in FIG.

From the above, regarding X1 (D), when (a _{1, i} , a _{1, j} , a _{1, k} ) are all odd numbers or all are even numbers (however, i ≠ j ≠ k), the loop 6 exists. Will do. The same applies to X2 (D), X3 (D),..., Xn (D), P (D).

  The same can be said for “inspection formula # 2” with respect to “inspection formula # 1”, and theorem 2 is proved. (End of proof)

[Theorem 3]
A time-varying period 2 LDPC-CC based on the parity check polynomial of equation (156) and the parity check polynomial of equation (157), and the parity check polynomials X1 (D), X2 (D) of equation (156), In any of X3 (D),..., Xn (D), P (D), when there are five or more terms, or the parity check polynomial X1 (D), X2 ( In any one of D), X3 (D),..., Xn (D), P (D), when there are five or more terms, at least one loop 6 exists.

[Proof]
In any of X1 (D), X2 (D), X3 (D),..., Xn (D), P (D), theorem 2 must be satisfied if there are five or more terms. . Therefore, Theorem 3 was proved. (End of proof)

  From the above, the importance of other Embodiment 9 becomes clear.

(Other Embodiment 15)
First, an LDPC-CC having a time varying period of 4 with good characteristics will be described. In the following, a case where the coding rate is 1/2 will be described as an example.

Formulas (160-1) to (160-4) are considered as parity check polynomials for LDPC-CC with a time-varying period of 4. At this time, X (D) is a polynomial expression of data (information), and P (D) is a polynomial expression of parity. Here, in the equations (160-1) to (160-4), the parity check polynomial is such that four terms exist in each of X (D) and P (D). This is because it is preferable to obtain four terms.

In Expression (160-1), a1, a2, a3, and a4 are integers (where a1 ≠ a2 ≠ a3 ≠ a4). B1, b2, b3, and b4 are integers (where b1 ≠ b2 ≠ b3 ≠ b4). The parity check polynomial of equation (160-1) is referred to as “check equation # 1”, and the sub-matrix based on the parity check polynomial of equation (160-1) is referred to as a first sub-matrix H ₁ .

In Formula (160-2), A1, A2, A3, and A4 are integers (however, A1 ≠ A2 ≠ A3 ≠ A4). B1, B2, B3, and B4 are integers (B1 ≠ B2 ≠ B3 ≠ B4). Referred to parity check polynomial of equation (160-2) and "check equation # 2", the sub-matrix based on a parity check polynomial of equation (160-2), the second sub-matrix _{H 2.}

In Expression (160-3), α1, α2, α3, and α4 are integers (where α1 ≠ α2 ≠ α3 ≠ α4). Β1, β2, β3, and β4 are integers (where β1 ≠ β2 ≠ β3 ≠ β4). The parity check polynomial of equation (160-3) is referred to as “check equation # 3”, and the sub-matrix based on the parity check polynomial of equation (160-3) is referred to as a third sub-matrix H ₃ .

In the equation (160-4), E1, E2, E3, and E4 are integers (however, E1 ≠ E2 ≠ E3 ≠ E4). F1, F2, F3 and F4 are integers (where F1 ≠ F2 ≠
F3 ≠ F4). The parity check polynomial of equation (160-4) is referred to as “check equation # 4”, and the sub-matrix based on the parity check polynomial of equation (160-4) is referred to as a fourth sub-matrix H ₄ .

Then, with respect to the LDPC-CC of the time varying period 4 in which the check matrix is generated as shown in FIG. 17 from the first sub-matrix H ₁ , the second sub-matrix H ₂ , the third sub-matrix H ₃ , and the fourth sub-matrix H ₄ Think.

  At this time, in the formulas (160-1) to (160-4), combinations of orders of X (D) and P (D) (a1, a2, a3, a4), (b1, b2, b3, b4), (A1, A2, A3, A4), (B1, B2, B3, B4), (α1, α2, α3, α4), (β1, β2, β3, β4), (E1, E2, E3, E4), When the remainder obtained by dividing each value of (F1, F2, F3, F4) by 4 is k, the remainder is divided into four coefficient sets (for example, (a1, a2, a3, a4)) expressed as described above. 0, 1, 2, 3 are included one by one, and all four coefficient sets are satisfied.

  For example, if each order (a1, a2, a3, a4) of X (D) of “check formula # 1” is (a1, a2, a3, a4) = (8, 7, 6, 5), each order The remainder k obtained by dividing (a1, a2, a3, a4) by 4 becomes (0, 3, 2, 1), and the remainder (k) 0, 1, 2, 3 is one by one in the four coefficient sets. To be included. Similarly, if each order (b1, b2, b3, b4) of P (D) of “inspection formula # 1” is (b1, b2, b3, b4) = (4, 3, 2, 1), The remainder k obtained by dividing the order (b1, b2, b3, b4) by 4 becomes (0, 3, 2, 1), and 0, 1, 2, 3 are obtained as the remainder (k) in the four coefficient sets. It will be included one by one. For the four coefficient sets of X (D) and P (D) of other inspection formulas (“check formula # 2”, “check formula # 3”, “check formula # 4”), the above “remainder” is also related. It is assumed that the condition is satisfied.

  By doing in this way, it becomes possible to form a regular LDPC code in which the column weights of the parity check matrix H composed of equations (160-1) to (160-4) are 4 in all columns. . Here, the regular LDPC code is an LDPC code defined by a parity check matrix in which each column weight is constant, and has characteristics that characteristics are stable and an error floor is difficult to occur. In particular, when the column weight is 4, the characteristics are good. Therefore, by generating the LDPC-CC as described above, it is possible to obtain the LDPC-CC with good reception performance.

Table 6 is an example of LDPC-CC (LDPC-CC # 1 to # 3) having a time-varying period of 4 and a coding rate of ½, in which the condition regarding the “remainder” is satisfied. In Table 6, the LDPC-CC with time varying period 4 is defined by four parity check polynomials of “check polynomial # 1”, “check polynomial # 2”, “check polynomial # 3”, and “check polynomial # 4”. The

  In the above description, the case where the coding rate is 1/2 has been described as an example. However, when the coding rate is (n−1) / n, information X1 (D), X2 (D),... Xn− In each of the four coefficient sets in 1 (D), if the above-mentioned condition relating to the “remainder” is satisfied, it becomes a regular LDPC code, and good reception quality can be obtained.

  Even in the case of time-varying period 2, it was confirmed that a code with good characteristics can be searched by applying the condition relating to the “remainder”. Hereinafter, an LDPC-CC having a time-varying period 2 with good characteristics will be described. In the following, a case where the coding rate is 1/2 will be described as an example.

As an LDPC-CC parity check polynomial with a time-varying period of 2, equations (161-1) and (161-2) are considered. At this time, X (D) is a polynomial expression of data (information), and P (D) is a polynomial expression of parity. Here, in equations (161-1) and (161-2), the parity check polynomial is such that there are four terms in each of X (D) and P (D). This is because it is preferable to obtain four terms.

In Expression (161-1), a1, a2, a3, and a4 are integers (where a1 ≠ a2 ≠ a3 ≠ a4). B1, b2, b3, and b4 are integers (where b1 ≠ b2 ≠ b3 ≠ b4). Referred to parity check polynomial of equation (161-1) and "check equation # 1", the sub-matrix based on a parity check polynomial of equation (161-1), the first sub-matrix _{H 1.}

In the equation (161-2), A1, A2, A3, and A4 are integers (however, A1 ≠ A2 ≠ A3 ≠ A4). B1, B2, B3, and B4 are integers (B1 ≠ B2 ≠ B3 ≠ B4). Referred to parity check polynomial of equation (161-2) and "check equation # 2", the sub-matrix based on a parity check polynomial of equation (161-2), the second sub-matrix _{H 2.}

Then, consider the LDPC-CC of varying period 2 when producing the first sub-matrix _{H 1} and second sub-matrix _{H 2.}

  At this time, in formulas (161-1) and (161-2), combinations of orders of X (D) and P (D) (a1, a2, a3, a4), (b1, b2, b3, b4), When the remainder obtained by dividing each value of (A1, A2, A3, A4), (B1, B2, B3, B4) by 4 is k, four coefficient sets (e.g., (a1, a2, a3, and a4)) are each made to contain one remainder, 0, 1, 2, and 3, and all the above four coefficient sets are satisfied.

  For example, if each order (a1, a2, a3, a4) of X (D) of “check formula # 1” is (a1, a2, a3, a4) = (8, 7, 6, 5), each order The remainder k obtained by dividing (a1, a2, a3, a4) by 4 becomes (0, 3, 2, 1), and the remainder (k) 0, 1, 2, 3 is one by one in the four coefficient sets. To be included. Similarly, if each order (b1, b2, b3, b4) of P (D) of “inspection formula # 1” is (b1, b2, b3, b4) = (4, 3, 2, 1), The remainder k obtained by dividing the order (b1, b2, b3, b4) by 4 becomes (0, 3, 2, 1), and 0, 1, 2, 3 are obtained as the remainder (k) in the four coefficient sets. It will be included one by one. It is assumed that the condition regarding the “remainder” is also satisfied for each of the four coefficient sets of X (D) and P (D) of “inspection formula # 2.”

  By doing in this way, it becomes possible to form a regular LDPC code in which the column weights of the parity check matrix H composed of the equations (161-1) and (161-2) are 4 in all columns. . Here, the regular LDPC code is an LDPC code defined by a parity check matrix in which each column weight is constant, and has characteristics that characteristics are stable and an error floor is difficult to occur. In particular, when the row weight is 8, since the characteristics are good, by generating the LDPC-CC as described above, it is possible to obtain the LDPC-CC that can further improve the reception performance. Become.

Table 7 shows an example of LDPC-CC (LDPC-CC # 1, # 2) having a time-varying period of 2 and a coding rate of ½, in which the condition regarding the “remainder” is satisfied. In Table 7, the LDPC-CC with time-varying period 2 is defined by two parity check polynomials of “check polynomial # 1” and “check polynomial # 2”.

  In the above description (LDPC-CC with time-varying period 2), the case where the coding rate is 1/2 has been described as an example, but when the coding rate is (n-1) / n, the information X1 (D), In each of the four coefficient sets in X2 (D),..., Xn-1 (D), if the above-mentioned condition relating to “remainder” is satisfied, it becomes a regular LDPC code, and good reception quality can be obtained. it can.

  In addition, even in the case of the time-varying period 3, it was confirmed that a code having good characteristics can be searched by applying the following condition regarding “remainder”. Hereinafter, an LDPC-CC having a time varying period of 3 with good characteristics will be described. In the following, a case where the coding rate is 1/2 will be described as an example.

Formulas (162-1) to (162-3) are considered as parity check polynomials for LDPC-CC with a time-varying period of 3. At this time, X (D) is a polynomial expression of data (information), and P (D) is a polynomial expression of parity. Here, in equations (162-1) to (162-3), the parity check polynomial is such that three terms exist in each of X (D) and P (D).

In Expression (162-1), a1, a2, and a3 are integers (where a1 ≠ a2 ≠ a3). B1, b2, and b3 are integers (where b1 ≠ b2 ≠ b3). Referred to parity check polynomial of equation (162-1) and "check equation # 1", the sub-matrix based on a parity check polynomial of equation (162-1), the first sub-matrix _{H 1.}

In the formula (162-2), A1, A2, and A3 are integers (however, A1 ≠ A2 ≠ A3). B1, B2, and B3 are integers (B1 ≠ B2 ≠ B3). Referred to parity check polynomial of equation (162-2) and "check equation # 2", the sub-matrix based on a parity check polynomial of equation (162-2), the second sub-matrix _{H 2.}

In the equation (162-3), α1, α2, and α3 are integers (where α1 ≠ α2 ≠ α3). Β1, β2, and β3 are integers (where β1 ≠ β2 ≠ β3). The parity check polynomial of equation (162-3) is referred to as “check equation # 3”, and a sub-matrix based on the parity check polynomial of equation (162-3) is referred to as a third sub-matrix H ₃ .

Then, consider LDPC-CC with a time varying period of 3 generated from the first sub-matrix H ₁ , second sub-matrix H ₂ , and third sub-matrix H ₃ .

  At this time, in the formulas (162-1) to (162-3), combinations of orders of X (D) and P (D) (a1, a2, a3), (b1, b2, b3), (A1, A2) , A3), (B1, B2, B3), (α1, α2, α3), (β1, β2, β3) divided by 3, and the remainder is k, the three expressed as above The coefficient set (for example, (a1, a2, a3)) includes one remainder, 0, 1 and 2, and is established by all three coefficient sets.

  For example, if the orders (a1, a2, a3) of X (D) of “inspection formula # 1” are (a1, a2, a3) = (6, 5, 4), the orders (a1, a2, a3) ) Divided by 3, the remainder k is (0, 2, 1), and the remainder (k) 0, 1, 2 is included in each of the three coefficient sets. Similarly, if the orders (b1, b2, b3) of P (D) of “inspection formula # 1” are (b1, b2, b3) = (3, 2, 1), the orders (b1, b2, The remainder k obtained by dividing b3) by 4 is (0, 2, 1), and the three coefficient sets include 0, 1, and 2 as the remainder (k). It is assumed that the above “remainder” condition is also satisfied for each of the three coefficient sets of X (D) and P (D) of “Checking Formula # 2” and “Checking Formula # 3”.

  By generating LDPC-CC in this way, a regular LDPC-CC code can be generated. Further, when BP decoding is performed, the reliability in “check equation # 2” and the reliability in “check equation # 3” are accurately propagated to “check equation # 1”, and “check equation # 1”. And the reliability in “inspection equation # 3” are accurately propagated to “inspection equation # 2”, and the reliability in “inspection equation # 1” and the reliability in “inspection equation # 2” are Properly propagates to “inspection formula # 3”. For this reason, LDPC-CC with better reception quality can be obtained. This is because, when considered in units of columns, the positions where “1” exists are arranged so as to accurately propagate the reliability as described above.

  Hereinafter, the above-described reliability propagation will be described with reference to the drawings. FIG. 65A shows the configuration of an LDPC-CC parity check polynomial and check matrix H of time-varying period 3.

  “Check expression # 1” is the parity check polynomial of Expression (162-1), in which (a1, a2, a3) = (2,1,0), (b1, b2, b3) = (2,1,0) ) And the remainder obtained by dividing each coefficient by 3 is (a1% 3, a2% 3, a3% 3) = (2,1,0), (b1% 3, b2% 3, b3% 3) ) = (2, 1, 0). “Z% 3” represents the remainder obtained by dividing Z by 3.

  “Check expression # 2” is the parity check polynomial of Expression (162-2), where (A1, A2, A3) = (5, 1, 0), (B1, B2, B3) = (5, 1, 0) ), And the remainder of dividing each coefficient by 3 is (A1% 3, A2% 3, A3% 3) = (2,1,0), (B1% 3, B2% 3, B3% 3) ) = (2, 1, 0).

  “Checking equation # 3” is the parity check polynomial of equation (162-3), where (α1, α2, α3) = (4, 2, 0), (β1, β2, β3) = (4, 2, 0 ), And the remainder of dividing each coefficient by 3 is (α1% 3, α2% 3, α3% 3) = (1,2,0), (β1% 3, β2% 3, β3% 3 ) = (1, 2, 0).

  Therefore, the example of the LDPC-CC with the time-varying period 3 shown in FIG. 65A is the above-described condition relating to the “remainder”, that is, (a1% 3, a2% 3, a3% 3), (b1% 3, b2% 3, b3% 3), (A1% 3, A2% 3, A3% 3), (B1% 3, B2% 3, B3% 3), (α1% 3, α2% 3, α3% 3), ( β1% 3, β2% 3, β3% 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0 1), (2, 1, 0) is satisfied.

  Returning to FIG. 65A again, reliability propagation will be described. By column operation of column 6506 in BP decoding, “1” in region 6501 of “check equation # 1” becomes “1” in region 6504 of “check matrix # 2” and “1” in region 6505 of “check matrix # 3”. The reliability is propagated from “1”. As described above, “1” in the area 6501 of the “check equation # 1” is a coefficient such that the remainder after division by 3 is 0 (a3% 3 = 0 (a3 = 0) or b3% 3 = 0 (b3 = 0)). In addition, “1” in the area 6504 of “check matrix # 2” is a coefficient whose remainder after division by 3 is 1 (A2% 3 = 1 (A2 = 1) or B2% 3 = 1 (B2 = 1)). In addition, “1” in the area 6505 of the “inspection formula # 3” is a coefficient with a remainder obtained by dividing by 3 (α2% 3 = 2 (α2 = 2) or β2% 3 = 2 (β2 = 2)).

  As described above, “1” in the region 6501 in which the remainder of the coefficient of “check expression # 1” is 0 is 1 in the coefficient of “check expression # 2” in the column calculation of the column 6506 in BP decoding. The reliability is propagated from “1” in the region 6504 and “1” in the region 6505 in which the remainder is 2 in the coefficient of the “check equation # 3”.

  Similarly, “1” in the region 6502 in which the remainder is 1 in the coefficient of “check equation # 1” is a region in which the remainder is 2 in the coefficient of “check equation # 2” in the column calculation of the column 6509 in BP decoding. The reliability is propagated from “1” in 6507 and “1” in the region 6508 in which the remainder is 0 in the coefficient of “check equation # 3”.

  Similarly, “1” in the region 6503 in which the remainder is 2 in the coefficient of “check equation # 1” is a region in which the remainder is 0 in the coefficient of “check equation # 2” in the column calculation of the column 6512 in BP decoding. The reliability is propagated from “1” of 6510 “1” and “1” of the region 6511 in which the remainder is 1 in the coefficient of “check equation # 3”.

  A supplementary explanation of the reliability propagation will be given with reference to FIG. 65B. FIG. 65B shows the relationship of reliability propagation between the terms relating to X (D) of “checking formula # 1” to “checking formula # 3” in FIG. 65A. 65A, “examination formula # 1” to “examination formula # 3” are expressed in terms of X (D) in formulas (162-1) to (162-3) as follows: (a1, a2, a3) = (2, 1, 0), (A1, A2, A3) = (5, 1, 0), (α1, α2, α3) = (4, 2, 0).

  In FIG. 65B, terms (a3, A3, α3) enclosed by squares indicate coefficients with a remainder of 0 divided by 3. Further, the terms (a2, A2, α1) surrounded by circles indicate a coefficient whose remainder is 1 after dividing by 3. In addition, the terms (a1, A1, α2) surrounded by rhombuses indicate coefficients with a remainder of 2 divided by 3.

  As can be seen from FIG. 65B, the reliability of a1 of “check equation # 1” is propagated from A3 of “check equation # 2” and α1 of “check equation # 3”, which have different remainders after division by 3. The reliability of a2 of “checking formula # 1” is propagated from A1 of “checking formula # 2” and α3 of “checking formula # 3”, which have different remainders after division by 3. The reliability of “a3” of “check equation # 1” is propagated from A2 of “check equation # 2” and α2 of “check equation # 3”, which have different remainders after division by 3. FIG. 65B shows the relationship of reliability propagation between the terms related to X (D) in “checking formula # 1” to “checking formula # 3”, but the same applies to the terms related to P (D). I can say that.

  As described above, the reliability is propagated to the “check equation # 1” from the coefficients of which the remainder obtained by dividing by 3 is 0, 1, 2 among the coefficients of the “check equation # 2”. That is, the reliability is propagated to the “checking formula # 1” from the coefficients of the “checking formula # 2”, all of which are different from each other when they are divided by 3. Therefore, all the reliability levels having low correlation are propagated to “checking formula # 1”.

  Similarly, the reliability is propagated to the “check formula # 2” from the coefficients of which the remainder obtained by dividing by 3 among the coefficients of the “check formula # 1” is 0, 1 and 2. That is, the reliability is propagated to the “checking formula # 2” from the coefficients of the “checking formula # 1”, all of which have different remainders after division by 3. Also, the reliability is propagated to “check formula # 2” from the coefficients of which the remainder obtained by dividing by 3 is 0, 1, and 2 among the coefficients of “check formula # 3”. That is, the reliability is propagated to the “check formula # 2” from the coefficients of the “check formula # 3”, all of which have different remainders after division by 3.

  Similarly, the reliability is propagated to “check formula # 3” from the coefficients of “check formula # 1” whose remainders after division by 3 are 0, 1, and 2. That is, the reliability is propagated to the “checking formula # 3” from the coefficients of the “checking formula # 1”, all of which have different remainders after division by 3. Also, the reliability is propagated to “check formula # 3” from the coefficients of which the remainder obtained by dividing by 3 becomes 0, 1 and 2 among the coefficients of “check formula # 2”. That is, the reliability is propagated to the “checking equation # 3” from the coefficients of the “checking equation # 2”, all of which have different remainders after division by 3.

  As described above, by ensuring that the orders of the parity check polynomials of the equations (162-1) to (162-3) satisfy the above-described condition regarding the “remainder”, the reliability is always ensured in all column operations. Since it is propagated, the reliability can be propagated efficiently in all the check expressions, and the error correction capability can be further increased.

  As described above, the LDPC-CC with the time varying period 3 has been described by taking the case of the coding rate 1/2 as an example, but the coding rate is not limited to 1/2. In the case of coding rate (n-1) / n (n is an integer of 2 or more), each of the three coefficients in information X1 (D), X2 (D),... Xn-1 (D) If the condition regarding the “remainder” is satisfied in the set, it becomes a regular LDPC code, and good reception quality can be obtained.

  Hereinafter, the case of coding rate (n−1) / n (n is an integer of 2 or more) will be described.

Formulas (163-1) to (163-3) are considered as LDPC-CC parity check polynomials with a time-varying period of 3. At this time, X1 (D), X2 (D),... Xn-1 (D) is a polynomial expression of data (information) X1, X2,... Xn-1, and P (D) is a parity expression. It is a polynomial expression. Here, in the formulas (163-1) to (163-3), there are three terms in each of X1 (D), X2 (D),... Xn-1 (D), P (D). Parity check polynomial.

In the formula (163-1), a _{i, 1} , a _{i, 2} , a _{i, 3} (i = 1, 2,..., N−1) are integers (where a _{i, 1} ≠ a _{i, 2} ≠ a _{i, 3} ). B1, b2, and b3 are integers (where b1 ≠ b2 ≠ b3). The parity check polynomial of equation (163-1) is called “check equation # 1”, and the sub-matrix based on the parity check polynomial of equation (163-1) is referred to as a first sub-matrix H ₁ .

In Formula (163-2), A _{i, 1} , A _{i, 2} , A _{i, 3} (i = 1, 2,..., N−1 are integers (where A _{i, 1} ≠ A _{i , 2} ≠ A _{i, 3} ), and B1, B2, and B3 are integers (where B1 ≠ B2 ≠ B3), and the parity check polynomial of equation (163-2) is “check equation # 2”. Yobi, a sub-matrix based on a parity check polynomial of equation (163-2), the second sub-matrix _{H 2.}

In Formula (163-3), α _{i, 1} , α _{i, 2} , α _{i, 3} (i = 1, 2,..., N−1 are integers (where α _{i, 1} ≠ α _{i , 2} ≠ α _{i, 3} ) and β1, β2, and β3 are integers (where β1 ≠ β2 ≠ β3), and the parity check polynomial of equation (163-3) is “check equation # 3”. A sub-matrix based on the parity check polynomial of Expression (163-3) is referred to as a third sub-matrix H ₃ .

Then, consider LDPC-CC with a time varying period of 3 generated from the first sub-matrix H ₁ , second sub-matrix H ₂ , and third sub-matrix H ₃ .

At this time, in formulas (163-1) to (163-3), combinations of orders of X1 (D), X2 (D),... Xn-1 (D) and P (D) (a _1,1 , _A1,2 , _a1,3 ), ( _a2,1 , _a2,2 , _a2,3 ), ... (an _-1,1 , an _-1,2 , _{an- 1,3), (b1, b2,} b3), (A 1,1, A 1,2, A 1,3), (A 2,1, A 2,2, A 2,3), ··· (A _n-1,1 , A _n-1,2 , A _n-1,3 ), (B1, B2, B3), (α _1,1 , α _1,2 , α _1,3 ), (α _2,1 , α _2,2 , α _2,3 ),... (Α _n−1,1 , α _n−1 , ₂ , α _n−1,3 ), (β1, β2, β3) When the remainder obtained by dividing the value by 3 is k, three coefficient sets expressed as described above (for example, _{(A 1,1, a 1,2, a} 1,3) in), less so 0,1,2 are included one by one, and, so as to hold in all three coefficient sets as described above.

That is, (a _1,1 % 3, a _1,2 % 3, a _1,3 % 3), (a _2,1 % 3, a _2,2 % 3, a _2,3 % 3), ...・ ( _An-1,1 % 3, _{ann -1,2} % 3, an _-1,1,3 % 3), (b1% 3, b2% 3, b3% 3), ( _A1,1 % 3, A1, ₂ % 3, _A1,3 % 3), ( _A2,1 % 3, _A2,2 % 3, _A2,3 % 3), ... ( _{An-1,1 % 3, A n-1,2%} 3, A n-1,3% 3), (B1% 3, B2% 3, B3% 3), (α 1,1% 3, α 1,2% 3 , Α _1,3 % 3), (α _2,1 % 3, α _2,2 % 3, α _2,3 % 3), ... (α _n-1,1 % 3, α _{n-1, 2} % 3, α _n-1,3 % 3), (β1% 3, β2% 3, β3% 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, ), (2,0,1), so that the one of (2,1,0).

  By generating LDPC-CC in this way, a regular LDPC-CC code can be generated. Further, when BP decoding is performed, the reliability in “check equation # 2” and the reliability in “check equation # 3” are accurately propagated to “check equation # 1”, and “check equation # 1”. And the reliability in “inspection equation # 3” are accurately propagated to “inspection equation # 2”, and the reliability in “inspection equation # 1” and the reliability in “inspection equation # 2” are Properly propagates to “inspection formula # 3”. For this reason, LDPC-CC with better reception quality can be obtained as in the case of the coding rate of 1/2.

Table 8 shows an example of an LDPC-CC (LDPC-CC # 1, # 2, # 3, # 4, # 5) having a time-varying period of 3 and a coding rate of ½, in which the condition regarding the “remainder” is satisfied. ). In Table 8, LDPC-CC with time-varying period 3 is represented by three parity check polynomials of “check (multinomial) equation # 1”, “check (multinomial) equation # 2”, and “check (multinomial) equation # 3”. Defined.

  Similarly to the time-varying period 3, the following conditions regarding “remainder” are applied to LDPC-CC whose time-varying period is a multiple of 3 (for example, the time-varying period is 6, 9, 12,...). Then, it was confirmed that a code with good characteristics can be searched. Hereinafter, the LDPC-CC having a multiple of the time-varying period 3 with good characteristics will be described. In the following, a case of LDPC-CC with a coding rate of 1/2 and a time varying period of 6 will be described as an example.

Formulas (164-1) to (164-6) are considered as parity check polynomials for LDPC-CC with a time-varying period of 6.

At this time, X (D) is a polynomial expression of data (information), and P (D) is a polynomial expression of parity. In an LDPC-CC with a time-varying period 6, if the parity Pi and the information Xi at time i are i% 6 = k (k = 0, 1, 2, 3, 4, 5), the equation (164− (k + 1)) ) Parity check polynomial. For example, if i = 1, i% 6 = 1 (k = 1), and thus Expression (165) is established.

  Here, in equations (164-1) to (164-6), the parity check polynomial is such that three terms exist in each of X (D) and P (D).

In Expression (164-1), a1,1, a1,2, and a1,3 are integers (where a1,1 ≠ a1,2 ≠ a1,3). Further, b1,1, b1,2, b1,3 are integers (where b1,1 ≠ b1,2 ≠ b1,3). Referred to parity check polynomial of equation (164-1) and "check equation # 1", the sub-matrix based on a parity check polynomial of equation (164-1), the first sub-matrix _{H 1.}

In the formula (164-2), a2, 1, a2, 2, a2, 3 are integers (where a2, 1 ≠ a2, 2 ≠ a2, 3). In addition, b2,1, b2,2, b2,3 are integers (where b2,1 ≠ b2,2 ≠ b2,3). Referred to parity check polynomial of equation (164-2) and "check equation # 2", the sub-matrix based on a parity check polynomial of equation (164-2), the second sub-matrix _{H 2.}

In the formula (164-3), a3, 1, a3, 2, and a3, 3 are integers (where a3, 1 ≠ a3, 2 ≠ a3, 3). B3, 1, b3, 2, b3, 3 are integers (where b3, 1 ≠ b3, 2 ≠ b3, 3). The parity check polynomial of equation (164-3) is referred to as “check equation # 3”, and the sub-matrix based on the parity check polynomial of equation (164-3) is referred to as a third sub-matrix H ₃ .

In the formula (164-4), a4, 1, a4, 2, and a4, 3 are integers (where a4, 1 ≠ a4, 2 ≠ a4, 3). Also, b4, 1, b4, 2, b4, 3 are integers (where b4, 1 ≠ b4, 2 ≠ b4, 3). Referred to parity check polynomial of equation (164-4) and "check equation # 4", a sub-matrix based on a parity check polynomial of equation (164-4), and a fourth sub-matrix _{H 4.}

In the formula (164-5), a5, 1, a5, 2, and a5, 3 are integers (where a5, 1 ≠ a5, 2 ≠ a5, 3). Also, b5, 1, b5, 2, and b5, 3 are integers (where b5, 1 ≠ b5, 2 ≠ b5, 3). Referred to parity check polynomial of equation (164-5) and "check equation # 5", a sub-matrix based on a parity check polynomial of equation (164-5), and the fifth sub-matrix _{H 5.}

In the formula (164-6), a6, 1, a6, 2, a6, 3 are integers (where a6, 1 ≠ a6, 2 ≠ a6, 3). B6, 1, b6, 2, b6, 3 are integers (where b6, 1 ≠ b6, 2 ≠ b6, 3). Referred to parity check polynomial of equation (164-6) and "check equation # 6", the sub-matrix based on a parity check polynomial of equation (164-6), and the sixth sub-matrix _{H 6.}

The first sub-matrix _{H 1,} second sub-matrix _{H 2,} third sub-matrix _{H 3,} fourth sub-matrix _{H 4,} fifth sub-matrix _{H 5,} varying period 6 when generating the sixth sub-matrix _{H 6} Think about LDPC-CC.

  At this time, in the formulas (164-1) to (164-6), combinations of the orders of X (D) and P (D) (a1, 1, a1, 2, a1, 3), (b1, 1, b1) , 2, b1, 3), (a2, 1, a2, 2, a2, 3), (b2, 1, b2, 2, b2, 3), (a3, 1, a3, 2, a3, 3), (B3, 1, b3, 2, b3, 3), (a4, 1, a4, 2, a4, 3), (b4, 1, b4, 2, b4, 3), (a5, 1, a5, 2 , A5, 3), (b5, 1, b5, 2, b5, 3), (a6, 1, a6, 2, a6, 3), (b6, 1, b6, 2, b6, 3) When the remainder k is divided by 3, the remainder is 0, 1 and 2 in the three coefficient sets (for example, (a1, 1, a1, 2, a1, 3)) expressed as described above. To be included one by one And, so as to hold in all three coefficient sets as described above. That is, (a1, 1% 3, a1, 2% 3, a1, 3% 3), (b1, 1% 3, b1, 2% 3, b1, 3% 3), (a2, 1% 3, a2 , 2% 3, a2, 3% 3), (b2, 1% 3, b2, 2% 3, b2, 3% 3), (a3, 1% 3, a3, 2% 3, a3, 3% 3 ), (B3, 1% 3, b3, 2% 3, b3, 3% 3), (a4, 1% 3, a4, 2% 3, a4, 3% 3), (b4, 1% 3, b4) , 2% 3, b4, 3% 3), (a5, 1% 3, a5, 2% 3, a5, 3% 3), (b5, 1% 3, b5, 2% 3, b5, 3% 3) ), (A6, 1% 3, a6, 2% 3, a6, 3% 3), (b6, 1% 3, b6, 2% 3, b6, 3% 3) is (0, 1, 2) , (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0) It made.

  By generating LDPC-CC in this way, when a Tanner graph is drawn with respect to “inspection formula # 1”, if there is an edge, “examination formula # 2 or inspection formula # 5 ”And the reliability in“ Checking Formula # 3 or Checking Formula # 6 ”are accurately propagated. Further, when an edge is present when a Tanner graph is drawn for “inspection formula # 2”, the reliability in “inspection formula # 1 or inspection formula # 4” is accurately determined, “inspection formula # 3, Or, the reliability in the inspection formula # 6 ”is accurately propagated. Further, when an edge is present when a Tanner graph is drawn with respect to “inspection formula # 3”, the reliability in “inspection formula # 1 or inspection formula # 4” is accurately determined, “inspection formula # 2, Or, the reliability in the inspection formula # 5 ”is accurately transmitted. When the Tanner graph is drawn for “inspection formula # 4”, if there is an edge, the reliability in “inspection formula # 2 or inspection formula # 5” is accurately determined, “inspection formula # 3, or The reliability in the inspection formula # 6 "is accurately transmitted. Further, when the Tanner graph is drawn, if there is an edge, the reliability in “inspection formula # 1 or inspection formula # 4” is accurately compared with “inspection formula # 5”, “inspection formula # 3, Or, the reliability in the inspection formula # 6 ”is accurately propagated. Further, when an edge is present when a Tanner graph is drawn for “inspection formula # 6”, the reliability in “inspection formula # 1 or inspection formula # 4” is accurately determined, “inspection formula # 2, Or, the reliability in the inspection formula # 5 ”is accurately transmitted. For this reason, as in the case where the time varying period is 3, the LDPC-CC having the time varying period 6 retains better error correction capability.

About this, reliability propagation is demonstrated using FIG. 65C. FIG. 65C shows the relationship of reliability propagation between the terms related to X (D) of “checking formula # 1” to “checking formula # 6”. In FIG. 65C, a square indicates a coefficient with a remainder of 0 divided by 3 in ax, y (x = 1, 2, 3, 4, 5, 6; y = 1, 2, 3). In addition, the circles indicate coefficients with a remainder of 1 after dividing by 3 in ax, y (x = 1, 2, 3, 4, 5, 6; y = 1, 2, 3). The rhombus indicates a coefficient with a remainder of 2 obtained by dividing 3 in ax, y (x = 1, 2, 3, 4, 5, 6; y = 1, 2, 3). As can be seen from FIG. 65C, when the Tanner graph is drawn, if there is an edge, a1,1 of “inspection equation # 1” is different from “inspection equation # 2 or # 5” and “ The reliability is propagated from the inspection formula # 3 or # 6 ". Similarly, when the Tanner graph is drawn, if there is an edge, a1 and a2 of “inspection formula # 1” are different in “examination formula # 2 or # 5” and “inspection formula # 3” obtained by dividing remainders by 3. Or the reliability is propagated from # 6 ". Similarly, when the Tanner graph is drawn, if there is an edge, a1, 3 of “inspection formula # 1” is different from “inspection formula # 2 or # 5” and “inspection formula # 3” with different remainders divided by 3 Or the reliability is propagated from # 6 ". FIG. 65C shows the relationship of reliability propagation between the terms related to X (D) of “checking formula # 1” to “checking formula # 6”, but the same applies to the terms related to P (D). I can say that.

  As described above, the reliability is propagated to each node in the Tanner graph of “check formula # 1” from the coefficient nodes other than “check formula # 1”. Therefore, all the reliability levels having low correlation are propagated to “checking formula # 1”, so that it is considered that the error correction capability is improved.

  In FIG. 65C, attention is paid to “inspection formula # 1”, but a “tanner graph” can be similarly drawn for “inspection formula # 2” to “inspection formula # 6”. The reliability is propagated to each node from coefficient nodes other than “check expression #K”. Therefore, all of the reliability levels with low correlation are propagated to “checking formula #K”, which is considered to improve the error correction capability. (K = 2, 3, 4, 5, 6)

  In this way, by making the respective orders of the parity check polynomials of the equations (164-1) to (164-6) satisfy the above-mentioned condition regarding the “remainder”, the reliability is efficiently obtained in all the check equations. Can be propagated, and the possibility that the error correction capability can be further increased is increased.

  As described above, the LDPC-CC with the time varying period 6 has been described by taking the case of the coding rate 1/2 as an example, but the coding rate is not limited to 1/2. In the case of coding rate (n-1) / n (n is an integer of 2 or more), each of the three coefficients in information X1 (D), X2 (D),... Xn-1 (D) If the condition regarding the “remainder” is satisfied in the set, the possibility of obtaining good reception quality is increased.

  Hereinafter, the case of coding rate (n−1) / n (n is an integer of 2 or more) will be described.

Equations (166-1) to (166-6) are considered as LDPC-CC parity check polynomials with a time-varying period of 6.

  At this time, X1 (D), X2 (D),... Xn-1 (D) is a polynomial expression of data (information) X1, X2,... Xn-1, and P (D) is a parity expression. It is a polynomial expression. Here, in the formulas (166-1) to (166-6), there are three terms in each of X1 (D), X2 (D),... Xn-1 (D), P (D). Parity check polynomial. When the coding rate is ½, and when considered in the same manner as in the time varying period 3, the time varying period 6, code represented by the parity check polynomials of the equations (166-1) to (166-6) In an LDPC-CC with a conversion rate (n-1) / n (n is an integer of 2 or more), if the following condition (<condition # 1>) is satisfied, there is a possibility that higher error correction capability can be obtained. Rise.

However, in an LDPC-CC with a time varying period of 6 and a coding rate (n−1) / n (n is an integer of 2 or more), the parity at time i is Pi and the information is X _{i, 1} , X _{i, 2} , ..., represented by Xi _{, n-1} . At this time, if i% 6 = k (k = 0, 1, 2, 3, 4, 5), the parity check polynomial of Expression (166- (k + 1)) is established. For example, if i = 8, i% 6 = 2 (k = 2), and therefore Expression (167) is established.

<Condition # 1>
In formulas (166-1) to (166-6), the combinations of the orders of X1 (D), X2 (D),... Xn-1 (D) and P (D) satisfy the following conditions.
(A _{# 1,1,1} % 3, a _{# 1,1,2} % 3, a _{# 1,1,3} % 3), (a _{# 1,2,1} % 3, a _{# 1,2,2} % 3, a _{# 1, 2, 3} % 3), ..., (a _{# 1, k, 1} % 3, a
_{# 1, k, 2} % 3, a _{# 1, k, 3} % 3), ..., (a _{# 1, n-1, 1} % 3, a _{# 1, n-1, 2} % 3, a _{# 1, n-1,3} % 3), (b _{# 1,1} % 3, b _{# 1,2} % 3, b _{# 1,3} % 3) are (0,1,2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0). (K = 1, 2, 3,..., N-1)
And,
_{(A # 2,1,1% 3, a} # 2,1,2% 3, a # 2,1,3% 3), (a # 2,2,1% 3, a # 2,2,2 % 3, a _{# 2,2,3} % 3), ..., (a _{# 2, k, 1} % 3, a _{# 2, k, 2} % 3, a _{# 2, k, 3} % 3), ..., (a _{# 2, n-1, 1} % 3, a _{# 2, n-1, 2} % 3, a _{# 2, n-1, 3} % 3), (b _{# 2,1} % 3) , B _{# 2,2} % 3, b _{# 2,3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0) , (2, 0, 1), or (2, 1, 0). (K = 1, 2, 3,..., N-1)
And,
(A _{# 3,1,1} % 3, a _{# 3,1,2} % 3, a _{# 3,1,3} % 3), (a _{# 3,2,1} % 3, a _{# 3,2,2} % 3, a _{# 3} , _{2, 3} % 3), ..., (a _{# 3, k, 1} % 3, a _{# 3, k, 2} % 3, a _{# 3, k, 3} % 3), ..., (a _{# 3, n-1, 1} % 3, a _{# 3, n-1, 2} % 3, a _{# 3, n-1, 3} % 3), (b _{# 3} , ₁ % 3) , B _{# 3, 2} % 3, b _{# 3, 3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0) , (2, 0, 1), or (2, 1, 0). (K = 1, 2, 3,..., N-1)
And,
(A _{# 4,1,1} % 3, a _{# 4,1,2} % 3, a _{# 4,1,3} % 3), (a _{# 4,2,1} % 3, a _{# 4,2,2} % 3, a _{# 4,2,3} % 3), ..., (a _{# 4, k, 1} % 3, a _{# 4, k, 2} % 3, a _{# 4, k, 3} % 3), ..., (a _{# 4, n-1, 1} % 3, a _{# 4, n-1, 2} % 3, a _{# 4, n-1, 3} % 3), (b _{# 4} , ₁ % 3) , B _{# 4,2} % 3, b _{# 4,3} % 3) are (0,1,2), (0,2,1), (1,0,2), (1,2,0) , (2, 0, 1), or (2, 1, 0). (K = 1, 2, 3,..., N-1)
And,
(A _{# 5,1,1} % 3, a _{# 5,1,2} % 3, a _{# 5,1,3} % 3), (a _{# 5,2,1} % 3, a _{# 5,2,2} % 3, a _{# 5, 2} , 3% 3), ..., (a _{# 5, k, 1} % 3, a _{# 5, k, 2} % 3, a _{# 5, k, 3} % 3), ..., (a _{# 5, n-1, 1} % 3, a _{# 5, n-1, 2} % 3, a _{# 5, n-1,3} % 3), (b _{# 5} , ₁ % 3) , B _{# 5, 2} % 3, b _{# 5, 3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0). , (2, 0, 1), or (2, 1, 0). (K = 1, 2, 3,..., N-1)
And,
(A _{# 6,1,1} % 3, a _{# 6,1,2} % 3, a _{# 6,1,3} % 3), (a _{# 6,2,1} % 3, a _{# 6,2,2} % 3, a _{# 6, 2} , 3% 3), ..., (a _{# 6, k, 1} % 3, a _{# 6, k, 2} % 3, a _{# 6, k, 3} % 3), ..., (a _{# 6, n-1, 1} % 3, a _{# 6, n-1, 2} % 3, a _{# 6, n-1, 3} % 3), (b _{# 6} , ₁ % 3) , B _{# 6, 2} % 3, b _{# 6, 3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0) , (2, 0, 1), or (2, 1, 0). (K = 1, 2, 3,..., N-1)

In the above description, a code having a high error correction capability in the LDPC-CC with the time varying period 6 has been described. However, as with the LDPC-CC design method with the time varying periods 3 and 6, the time varying period 3g (g = 1). 2, 3, 4,...) LDPC-CC (that is, LDPC-CC whose time-varying period is a multiple of 3), a code having high error correction capability can be generated. Hereinafter, a method for configuring the code will be described in detail.

An LDPC-CC parity check polynomial with a time-varying period of 3 g (g = 1, 2, 3, 4,...) And coding rate (n−1) / n (n is an integer of 2 or more) Consider (168-1) to (168-3g).

  At this time, X1 (D), X2 (D),... Xn-1 (D) is a polynomial expression of data (information) X1, X2,... Xn-1, and P (D) is a parity expression. It is a polynomial expression. Here, in the equations (168-1) to (168-3g), there are three terms in each of X1 (D), X2 (D),... Xn-1 (D), P (D). Parity check polynomial.

  When considered in the same manner as the LDPC-CC with the time varying period 3 and the LDPC-CC with the time varying period 6, the time varying period 3g represented by the parity check polynomials of the equations (168-1) to (168-3g), the coding rate In the LDPC-CC of (n-1) / n (n is an integer of 2 or more), when the following condition (<condition # 2>) is satisfied, the possibility that higher error correction capability can be obtained increases.

However, in an LDPC-CC with a time varying period of 3 g and a coding rate (n−1) / n (n is an integer of 2 or more), the parity at time i is Pi and the information is X _{i, 1} , X _{i, 2} , ..., represented by Xi _{, n-1} . At this time, if i% 3g = k (k = 0, 1, 2,..., 3g−1), the parity check polynomial of Expression (168− (k + 1)) is established. For example, if i = 2, i% 3g = 2 (k = 2) is satisfied, and therefore equation (169) is established.

Moreover, in Formula (168-1)-Formula (168-3g), a _{# k, p, 1} , a _{# k, p, 2} , a _{# k, p, 3} is an integer (however, a _{# k, p , 1} ≠ a _{#k, p, 2} ≠ a _{#k, p, 3} ) (k = 1, 2, 3,..., 3g: p = 1, 2, 3,..., N− 1). Also, b _{# k, 1} , b _{# k, 2} , and b _{# k, 3} are integers (where b _{# k, 1} ≠ b _{# k, 2} ≠ b _{# k, 3} ). The parity check polynomial (k = 1, 2, 3,..., 3g) of the equation (168-k) is called “check equation #k”, and the sub-matrix based on the parity check polynomial of the equation (168-k) is K-th sub-matrix H _k . Consider an LDPC-CC with a time varying period of 3 g generated from the first sub-matrix H ₁ , the second sub-matrix H ₂ , the third sub-matrix H ₃ ,..., The third g sub-matrix H _3g .

<Condition # 2>
In the formulas (168-1) to (168-3g), combinations of orders of X1 (D), X2 (D),... Xn-1 (D) and P (D) satisfy the following conditions.
(A _{# 1,1,1} % 3, a _{# 1,1,2} % 3, a _{# 1,1,3} % 3), (a _{# 1,2,1} % 3, a _{# 1,2,2} % 3, a _{# 1,2,3} % 3), ..., (a _{# 1, p, 1} % 3, a _{# 1, p, 2} % 3, a _{# 1, p, 3} % 3), ..., (a _{# 1, n-1,1} % 3, a _{# 1, n-1,2} % 3, a _{# 1, n-1,3} % 3), (b _{# 1,1} % 3 , B _{# 1,2} % 3, b _{# 1,3} % 3) are (0,1,2), (0,2,1), (1,0,2), (1,2,0) , (2, 0, 1), or (2, 1, 0). (P = 1, 2, 3,..., N-1)
And,
_{(A # 2,1,1% 3, a} # 2,1,2% 3, a # 2,1,3% 3), (a # 2,2,1% 3, a # 2,2,2 % 3, a _{# 2,2,3} % 3), ..., (a _{# 2, p, 1} % 3, a _{# 2, p, 2} % 3, a _{# 2, p, 3} % 3), ..., (a _{# 2, n-1, 1} % 3, a _{# 2, n-1, 2} % 3, a _{# 2, n-1, 3} % 3), (b _{# 2,1} % 3) , B _{# 2,2} % 3, b _{# 2,3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0) , (2, 0, 1), or (2, 1, 0). (P = 1, 2, 3,..., N-1)
And,
(A _{# 3,1,1} % 3, a _{# 3,1,2} % 3, a _{# 3,1,3} % 3), (a _{# 3,2,1} % 3, a _{# 3,2,2} % 3, a _{# 3} , _{2, 3} % 3), ..., (a _{# 3, p, 1} % 3, a _{# 3, p, 2} % 3, a _{# 3, p, 3} % 3), ..., (a _{# 3, n-1, 1} % 3, a _{# 3, n-1, 2} % 3, a _{# 3, n-1, 3} % 3), (b _{# 3} , ₁ % 3) , B _{# 3, 2} % 3, b _{# 3, 3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0) , (2, 0, 1), or (2, 1, 0). (P = 1, 2, 3,..., N-1)
And,
・
・
・
And,
(A _{# k, 1,1} % 3, a _{# k, 1,2} % 3, a _{# k, 1,3} % 3), (a _{# k, 2,1} % 3, a _{# k, 2,2} % 3, a _{# k, 2,3} % 3), ..., (a _{# k, p, 1} % 3, a _{# k, p, 2} % 3, a _{# k, p, 3} % 3), ..., (a _{# k, n-1, 1} % 3, a _{# k, n-1, 2} % 3, a _{# k, n-1, 3} % 3), (b _{# k, 1} % 3) , B _{# k, 2} % 3, b _{# k, 3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0) , (2, 0, 1), or (2, 1, 0). (P = 1, 2, 3,..., N-1) (hence, k = 1, 2, 3,..., 3g)
And,
・
・
・
And,
_{(A # 3g-2,1,1% 3} , a # 3g-2,1,2% 3, a # 3g-2,1,3% 3), (a # 3g-2,2,1% 3 , A _{# 3g-2,2,2} % 3, a _{# 3g-2,2,3} % 3), ..., (a _{# 3g-2, p, 1} % 3, a _{# 3g-2, p , 2} % 3, a _{# 3g-2, p, 3} % 3), ..., (a _{# 3g-2, n-1, 1} % 3, a _{# 3g-2, n-1, 2} % 3 , A _{# 3g-2, n-1,3} % 3), (b _{# 3g-2,1} % 3, b _{# 3g-2,2} % 3, b _{# 3g-2,3} % 3) 0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0) . (P = 1, 2, 3,..., N-1)
And,
(A _{# 3g-1,1,1} % 3, a _{# 3g-1,1,2} % 3, a _{# 3g-1,1,3} % 3), (a _{# 3g-1,1,1} % 3) , A _{# 3g-1,2,2} % 3, a _{# 3g-1,2,3} % 3), ..., (a _{# 3g-1, p, 1} % 3, a _{# 3g-1, p , 2} % 3, a _{# 3g-1, p, 3} % 3), ..., (a _{# 3g-1, n-1, 1} % 3, a _{# 3g-1, n-1, 2} % 3 , A _{# 3g-1, n-1,3} % 3), (b _{# 3g-1,1} % 3, b _{# 3g-1,2} % 3, b _{# 3g-1,3} % 3) 0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0) . (P = 1, 2, 3,..., N-1)
And,
(A _{# 3g, 1,1} % 3, a _{# 3g, 1,2} % 3, a _{# 3g, 1,3} % 3), (a _{# 3g, 2,1} % 3, a _{# 3g, 2,2} % 3, a _{# 3g, 2, 3} % 3), ..., (a _{# 3g, p, 1} % 3, a _{# 3g, p, 2} % 3, a _{# 3g, p, 3} % 3), ..., (a _{# 3g, n-1, 1} % 3, a _{# 3g, n-1, 2} % 3, a _{# 3g, n-1, 3} % 3), (b _{# 3g, 1} % 3 , B _{# 3g, 2} % 3, b _{# 3g, 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0) , (2, 0, 1), or (2, 1, 0). (P = 1, 2, 3,..., N-1)

However, as described in other than the present embodiment, in consideration of the point that the encoding is easily performed, in the equations (168-1) to (168-3g),
It is preferable that one “0” exists among three (b _{# k, 1} % 3, b _{# k, 2} % 3, b _{# k, 3} % 3) (where k = 1, 2,... 3g).

In addition, in order to correlate the parity bit and the data bit at the same time and easily search for a code having a high correction capability,
There is one “0” among the three (a _{# k, 1,1} % 3, a _{# k, 1,2} % 3, a _{# k, 1,3} % 3),
There is one “0” among the three (a _{# k, 2,1} % 3, a _{# k, 2,2} % 3, a _{# k, 2,3} % 3),
・
・
・
There is one “0” among the three (a _{# k, p, 1} % 3, a _{# k, p, 2} % 3, a _{# k, p, 3} % 3),
・
・
・
There should be one “0” among the three (a _{# k, n−1, 1} % 3, a _{# k, n−1, 2} % 3, a _{# k, n−1, 3} % 3). (However, k = 1, 2,... 3 g).

Next, consider an LDPC-CC with a time-varying period of 3 g (g = 2, 3, 4, 5,...) In consideration of easy encoding. At this time, assuming that the coding rate is (n-1) / n (n is an integer of 2 or more), the parity check polynomial of LDPC-CC can be expressed as follows.

At this time, X1 (D), X2 (D),... Xn-1 (D) is a polynomial expression of data (information) X1, X2,... Xn-1, and P (D) is a parity expression. It is a polynomial expression. Here, in the formulas (170-1) to (170-3g), there are three terms in each of X1 (D), X2 (D),... Xn-1 (D), P (D). Parity check polynomial. However, in an LDPC-CC with a time varying period of 3 g and a coding rate (n−1) / n (n is an integer of 2 or more), the parity at time i is Pi and the information is X _{i, 1} , X _{i, 2} , ..., represented by Xi _{, n-1} . At this time, if i% 3g = k (k = 0, 1, 2,..., 3g−1), the parity check polynomial of Expression (170− (k + 1)) is established. For example, if i = 2, i% 3g = 2 (k = 2), and therefore Equation (171) is established.

  At this time, satisfying <Condition # 3> and <Condition # 4> increases the possibility that a code having higher error correction capability can be created.

<Condition # 3>
In the formulas (170-1) to (170-3g), combinations of orders of X1 (D), X2 (D),... Xn-1 (D) satisfy the following conditions.
(A _{# 1,1,1} % 3, a _{# 1,1,2} % 3, a _{# 1,1,3} % 3), (a _{# 1,2,1} % 3, a _{# 1,2,2} % 3, a _{# 1,2,3} % 3), ..., (a _{# 1, p, 1} % 3, a _{# 1, p, 2} % 3, a _{# 1, p, 3} % 3), ... (a _{# 1, n-1, 1} % 3, a _{# 1, n-1, 2} % 3, a _{# 1, n-1, 3} % 3) is (0, 1, 2) , (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0). (P = 1, 2, 3,..., N-1)
And,
_{(A # 2,1,1% 3, a} # 2,1,2% 3, a # 2,1,3% 3), (a # 2,2,1% 3, a # 2,2,2 % 3, a _{# 2,2,3} % 3), ..., (a _{# 2, p, 1} % 3, a _{# 2, p, 2} % 3, a _{# 2, p, 3} % 3), ... (a _{# 2, n-1, 1} % 3, a _{# 2, n-1, 2} % 3, a _{# 2, n-1, 3} % 3) is (0, 1, 2) , (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0). (P = 1, 2, 3,..., N-1)
And,
(A _{# 3,1,1} % 3, a _{# 3,1,2} % 3, a _{# 3,1,3} % 3), (a _{# 3,2,1} % 3, a _{# 3,2,2} % 3, a _{# 3} , _{2, 3} % 3), ..., (a _{# 3, p, 1} % 3, a _{# 3, p, 2} % 3, a _{# 3, p, 3} % 3), ... (a _{# 3, n-1, 1} % 3, a _{# 3, n-1, 2} % 3, a _{# 3, n-1, 3} % 3) is (0, 1, 2) , (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0). (P = 1, 2, 3,..., N-1)
And,
・
・
・
And,
(A _{# k, 1,1} % 3, a _{# k, 1,2} % 3, a _{# k, 1,3} % 3), (a _{# k, 2,1} % 3, a _{# k, 2,2} % 3, a _{# k, 2,3} % 3), ..., (a _{# k, p, 1} % 3, a _{# k, p, 2} % 3, a _{# k, p, 3} % 3), ... (a _{# k, n-1, 1} % 3, a _{# k, n-1, 2} % 3, a _{# k, n-1, 3} % 3) is (0, 1, 2) , (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0). (P = 1, 2, 3,..., N-1) (hence, k = 1, 2, 3,..., 3g)
And,
・
・
・
And,
_{(A # 3g-2,1,1% 3} , a # 3g-2,1,2% 3, a # 3g-2,1,3% 3), (a # 3g-2,2,1% 3 , A _{# 3g-2,2,2} % 3, a _{# 3g-2,2,3} % 3), ..., (a _{# 3g-2, p, 1} % 3, a _{# 3g-2, p , 2} % 3, a _{# 3g-2, p, 3} % 3), ..., (a _{# 3g-2, n-1, 1} % 3, a _{# 3g-2, n-1, 2} % 3 , A _{# 3g-2, n-1,3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), ( 2, 0, 1) or (2, 1, 0). (P = 1, 2, 3,..., N-1)
And,
(A _{# 3g-1,1,1} % 3, a _{# 3g-1,1,2} % 3, a _{# 3g-1,1,3} % 3), (a _{# 3g-1,1,1} % 3) , A _{# 3g-1,2,2} % 3, a _{# 3g-1,2,3} % 3), ..., (a _{# 3g-1, p, 1} % 3, a _{# 3g-1, p , 2} % 3, a _{# 3g-1, p, 3} % 3), ..., (a _{# 3g-1, n-1, 1} % 3, a _{# 3g-1, n-1, 2} % 3 , A _{# 3g-1, n-1, 3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), ( 2, 0, 1) or (2, 1, 0). (P = 1, 2, 3,..., N-1)
And,
(A _{# 3g, 1,1} % 3, a _{# 3g, 1,2} % 3, a _{# 3g, 1,3} % 3), (a _{# 3g, 2,1} % 3, a _{# 3g, 2,2} % 3, a _{# 3g, 2, 3} % 3), ..., (a _{# 3g, p, 1} % 3, a _{# 3g, p, 2} % 3, a _{# 3g, p, 3} % 3), ..., (a _{# 3g, n-1, 1} % 3, a _{# 3g, n-1, 2} % 3, a _{# 3g, n-1, 3} % 3) is (0, 1, 2) , (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0). (P = 1, 2, 3,..., N-1)

In addition, in the formulas (170-1) to (170-3g), the combination of the orders of P (D) satisfies the following condition.
(B _{# 1,1} % 3, b _{# 1,2} % 3), (b _{# 2,1} % 3, b _{# 2,2} % 3), (b _{# 3,1} % 3, b _{# 3,2} % 3), ..., (b _{# k, 1} % 3, b _{# k, 2} % 3), ..., (b _{# 3g-2,1} % 3, b _{# 3g-2,2} % 3 ), (B _{# 3g-1,1} % 3, b _{# 3g-1,2} % 3), (b _{# 3g, 1} % 3, b _{# 3g, 2} % 3) are (1,2), ( 2 or 1) (k = 1, 2, 3,..., 3g).

  <Condition # 3> for Expressions (170-1) to (170-3g) has the same relationship as <Condition # 2> for Expressions (168-1) to (168-3g). If the following condition (<condition # 4>) is added to the expressions (170-1) to (170-3g) in addition to <condition # 3>, an LDPC-CC having higher error correction capability is created. The possibility of being able to do increases.

<Condition # 4>
The following conditions are satisfied in the order of P (D) in the equations (170-1) to (170-3g).
(B _{# 1,1} % 3g, b _{# 1,2} % 3g), (b _{# 2,1} % 3g, b _{# 2,2} % 3g), (b _{# 3, 1} % 3g, b _{# 3,2} % 3g), ..., (b _{# k, 1} % 3g, b _{# k, 2} % 3g), ..., (b _{# 3g-2, 1} % 3g, b _{# 3g-2, 2} % 3g ), (B _{# 3g-1,1} % 3g, b _{# 3g-1,2} % 3g), (b _{# 3g, 1} % 3g, b _{# 3g, 2} % 3g) To 3g-1 (0, 1, 2, 3, 4, ..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6, ..., 3g- All values other than 3) exist.

  By the way, in the parity check matrix, if there is regularity at a position where “1” exists, but there is randomness, there is a high possibility that a good error correction capability can be obtained. Time-varying period 3g (g = 2, 3, 4, 5,...) Having parity check polynomials of equations (170-1) to (170-3g), and a coding rate of (n−1) / n ( In LDPC-CC (where n is an integer of 2 or more), if a code is created with <condition # 4> in addition to <condition # 3>, there is regularity at the position where “1” exists in the parity check matrix. However, since randomness can be given, the possibility that a good error correction capability can be obtained increases.

Next, in a time-varying period of 3 g (g = 2, 3, 4, 5,...) That can be easily encoded and has a relationship between parity bits and data bits at the same time. LDPC-CC
think about. At this time, assuming that the coding rate is (n-1) / n (n is an integer of 2 or more), the parity check polynomial of LDPC-CC can be expressed as follows.

At this time, X1 (D), X2 (D),... Xn-1 (D) is a polynomial expression of data (information) X1, X2,... Xn-1, and P (D) is a parity expression. It is a polynomial expression. And in the formulas (172-1) to (172-3g), there are three terms in each of X1 (D), X2 (D),... Xn-1 (D), P (D). A parity check polynomial is used, and a term of D ⁰ exists in X1 (D), X2 (D),... Xn−1 (D), P (D). (K = 1, 2, 3, ..., 3g)

However, in an LDPC-CC with a time varying period of 3 g and a coding rate (n−1) / n (n is an integer of 2 or more), the parity at time i is Pi and the information is X _{i, 1} , X _{i, 2} , ..., represented by Xi _{, n-1} . At this time, if i% 3g = k (k = 0, 1, 2,..., 3g−1), the parity check polynomial of Expression (172− (k + 1)) is established. For example, if i = 2, i% 3g = 2 (k = 2) is established, and therefore the equation (173) is established.

  At this time, if the following conditions (<condition # 5> and <condition # 6>) are satisfied, there is a high possibility that a code having a higher error correction capability can be created.

<Condition # 5>
In the formulas (172-1) to (172-3g), combinations of orders of X1 (D), X2 (D),... Xn-1 (D) satisfy the following conditions.
(A _{# 1,1,1} % 3, a _{# 1,1,2} % 3), (a _{# 1,2,1} % 3, a _{# 1,2,2} % 3), ..., (a _{# 1, p, 1} % 3, a _{# 1, p, 2} % 3), ..., (a _{# 1, n-1,1} % 3, a _{# 1, n-1,2} % 3) , (1,2), (2,1). (P = 1, 2, 3,..., N-1)
And,
_{(A # 2,1,1% 3, a} # 2,1,2% 3), (a # 2,2,1% 3, a # 2,2,2% 3), ···, (a _{# 2, p, 1} % 3, a _{# 2, p, 2} % 3), ..., (a _{# 2, n-1,1} % 3, a _{# 2, n-1,2} % 3) , (1,2), (2,1). (P = 1, 2, 3,..., N-1)
And,
_{(A # 3,1,1% 3, a} # 3,1,2% 3), (a # 3,2,1% 3, a # 3,2,2% 3), ···, (a _{# 3, p, 1} % 3, a _{# 3, p, 2} % 3), ..., (a _{# 3, n-1, 1} % 3, a _{# 3, n-1, 2} % 3) , (1,2), (2,1). (P = 1, 2, 3,..., N-1)
And,
・
・
・
And,
(A _{# k, 1,1} % 3, a _{# k, 1,2} % 3), (a _{# k, 2,1} % 3, a _{# k, 2,2} % 3), (a _{# K, p, 1} % 3, a _{# k, p, 2} % 3), ... (a _{# k, n-1,1} % 3, a _{# k, n-1,2} % 3) , (1,2), (2,1). (P = 1, 2, 3,..., N-1) (hence, k = 1, 2, 3,..., 3g)
And,
・
・
・
And,
_{(A # 3g-2,1,1% 3} , a # 3g-2,1,2% 3), (a # 3g-2,2,1% 3, a # 3g-2,2,2% 3 ), ..., (a _{# 3g-2, p, 1} % 3, a _{# 3g-2, p, 2} % 3), ..., (a _{# 3g-2, n-1, 1} % 3) , A _{# 3g-2, n-1,2} % 3) is either (1,2) or (2,1). (P = 1, 2, 3,..., N-1)
And,
(A _{# 3g-1,1,1} % 3, a _{# 3g-1,1,2} % 3), (a _{# 3g-1,2,1} % 3, a _{# 3g-1,2,2} % 3 ), ..., (a _{# 3g-1, p, 1} % 3, a _{# 3g-1, p, 2} % 3), ..., (a _{# 3g-1, n-1, 1} % 3) , A _{# 3g-1, n-1,2} % 3) is either (1,2) or (2,1). (P = 1, 2, 3,..., N−
1)
And,
(A _{# 3g, 1,1} % 3, a _{# 3g, 1,2} % 3), (a _{# 3g, 2,1} % 3, a _{# 3g, 2,2} % 3), (a _{# 3g, p, 1} % 3, a _{# 3g, p, 2} % 3), ... (a _{# 3g, n-1,1} % 3, a _{# 3g, n-1,2} % 3) , (1,2), (2,1). (P = 1, 2, 3,..., N-1)

In addition, in the formulas (172-1) to (172-3g), the combination of the orders of P (D) satisfies the following condition.
(B _{# 1,1} % 3, b _{# 1,2} % 3), (b _{# 2,1} % 3, b _{# 2,2} % 3), (b _{# 3,1} % 3, b _{# 3,2} % 3), ..., (b _{# k, 1} % 3, b _{# k, 2} % 3), ..., (b _{# 3g-2,1} % 3, b _{# 3g-2,2} % 3 ), (B _{# 3g-1,1} % 3, b _{# 3g-1,2} % 3), (b _{# 3g, 1} % 3, b _{# 3g, 2} % 3) are (1,2), ( 2 or 1) (k = 1, 2, 3,..., 3g).

  <Condition # 5> for Expressions (172-1) to (172-3g) has the same relationship as <Condition # 2> for Expressions (168-1) to (168-3g). An LDPC-CC having high error correction capability can be created by adding the following condition (<condition # 6>) to the expressions (172-1) to (172-3g) in addition to <condition # 5>. The possibility increases.

<Condition # 6>
The following conditions are satisfied in the order of X1 (D) in the formulas (172-1) to (172-3g).
_{(A # 1,1,1% 3g, a} # 1,1,2% 3g), (a # 2,1,1% 3g, a # 2,1,2% 3g), ···, (a _{# P, 1,1} % 3g, a _{# p, 1,2} % 3g), ..., 6g values of (a _{# 3g, 1,1} % 3g, a _{# 3g, 1,2} % 3g) Is an integer from 0 to 3g-1 (0, 1, 2, 3, 4,..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6,... -All values other than 3g-3) exist. (P = 1, 2, 3, ..., 3g)
And,
The following conditions are satisfied in the order of X2 (D) in Expressions (172-1) to (172-3g).
(A _{# 1,2,1} % 3g, a _{# 1,2,2} % 3g), (a _{# 2,2,1} % 3g, a _{# 2,2,2} % 3g), (a _{# P, 2,1} % 3g, a _{# p, 2,2} % 3g), ..., 6g values of (a _{# 3g, 2,1} % 3g, a _{# 3g, 2,2} % 3g) Is an integer from 0 to 3g-1 (0, 1, 2, 3, 4,..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6,... -All values other than 3g-3) exist. (P = 1, 2, 3, ..., 3g)
And,
The following conditions are satisfied in the order of X3 (D) in the formulas (172-1) to (172-3g).
_{(A # 1,3,1% 3g, a} # 1,3,2% 3g), (a # 2,3,1% 3g, a # 2,3,2% 3g), ···, (a _{# P, 3,1} % 3g, a _{# p, 3,2} % 3g), ..., 6g values of (a _{# 3g, 3,1} % 3g, a _{# 3g, 3,2} % 3g) Is an integer from 0 to 3g-1 (0, 1, 2, 3, 4,..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6,... -All values other than 3g-3) exist. (P = 1, 2, 3, ..., 3g)
And,
・
・
・
And,
The following conditions are satisfied in the order of Xk (D) in Expressions (172-1) to (172-3g).
(A _{# 1, k, 1} % 3g, a _{# 1, k, 2} % 3g), (a _{# 2, k, 1} % 3g, a _{# 2, k, 2} % 3g), (a _{# P, k, 1} % 3g, a _{# p, k, 2} % 3g), ..., 6g values of (a _{# 3g, k, 1} % 3g, a _{# 3g, k, 2} % 3g) Is an integer from 0 to 3g-1 (0, 1, 2, 3, 4,..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6,... -All values other than 3g-3) exist. (P = 1, 2, 3, ..., 3g)
(K = 1, 2, 3,..., N-1)
And,
・
・
・
And,
The following conditions are satisfied in the order of Xn-1 (D) in the equations (172-1) to (172-3g).
(A _{# 1, n-1, 1} % 3g, a _{# 1, n-1, 1} % 3g), (a _{# 2, n-1, 1} % 3g, a _{# 2, n-1, 2} % 3g ), ..., (a _{# p, n-1,1} % 3g, a _{# p, n-1,2} % 3g), ..., (a _{# 3g, n-1,1} % 3g, a _{# 3g, n-1, 2} % 3g) is an integer from 0 to 3g-1 (0, 1, 2, 3, 4, ..., 3g-2, 3g-1) Of these, all values other than multiples of 3 (that is, 0, 3, 6,..., 3g−3) exist. (P = 1, 2, 3, ..., 3g)
And,
The following conditions are satisfied in the order of P (D) in Expressions (172-1) to (172-3g). (B _{# 1,1} % 3g, b _{# 1,2} % 3g), (b _{# 2,1} % 3g, b _{# 2,2} % 3g), (b _{# 3, 1} % 3g, b _{# 3,2} % 3g), ..., (b _{# k, 1} % 3g, b _{# k, 2} % 3g), ..., (b _{# 3g-2, 1} % 3g, b _{# 3g-2, 2} % 3g ), (B _{# 3g-1,1} % 3g, b _{# 3g-1,2} % 3g), (b _{# 3g, 1} % 3g, b _{# 3g, 2} % 3g) To 3g-1 (0, 1, 2, 3, 4, ..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6, ..., 3g- All values other than 3) exist. (K = 1, 2, 3, ..., 3g)

  By the way, in the parity check matrix, if there is regularity at a position where “1” exists, but there is randomness, there is a high possibility that a good error correction capability can be obtained. Time-varying period 3g (g = 2, 3, 4, 5,...) Having parity check polynomials of equations (172-1) to (172-3g), and a coding rate of (n−1) / n ( In LDPC-CC (where n is an integer of 2 or more), when a code is created by adding the condition of <condition # 6> in addition to <condition # 5>, the regularity is at the position where “1” exists in the parity check matrix. Since the randomness can be given while having the error rate, the possibility that a better error correction capability can be obtained increases.

  Also, instead of <Condition # 6>, <Condition # 6 ′> is used, that is, even if <Condition # 6 ′> is added in addition to <Condition # 5>, a higher error correction is possible. The possibility that an LDPC-CC having the capability can be created is increased.

<Condition # 6 '>
The following conditions are satisfied in the order of X1 (D) in the formulas (172-1) to (172-3g).
(A _{# 1,1,1} % 3g, a _{# 1,1,2} % 3g), (a _{# 2,1,1} % 3g, a _{# 2,
1, 2} % 3g), ..., (a _{# p, 1,1} % 3g, a _{# p, 1,2} % 3g), ..., (a _{# 3g, 1,1} % 3g, a _{# 3g, 1, 2} % 3g) is an integer from 0 to 3g-1 (0, 1, 2, 3, 4, ..., 3g-2, 3g-1). All values other than multiples of 0 (ie, 0, 3, 6,..., 3g−3) exist. (P = 1, 2, 3, ..., 3g)
Or
The following conditions are satisfied in the order of X2 (D) in Expressions (172-1) to (172-3g).
(A _{# 1,2,1} % 3g, a _{# 1,2,2} % 3g), (a _{# 2,2,1} % 3g, a _{# 2,2,2} % 3g), (a _{# P, 2,1} % 3g, a _{# p, 2,2} % 3g), ..., 6g values of (a _{# 3g, 2,1} % 3g, a _{# 3g, 2,2} % 3g) Is an integer from 0 to 3g-1 (0, 1, 2, 3, 4,..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6,... -All values other than 3g-3) exist. (P = 1, 2, 3, ..., 3g)
Or
The following conditions are satisfied in the order of X3 (D) in the formulas (172-1) to (172-3g). _{(A # 1,3,1% 3g, a} # 1,3,2% 3g), (a # 2,3,1% 3g, a # 2,3,2% 3g), ···, (a _{# P, 3,1} % 3g, a _{# p, 3,2} % 3g), ..., 6g values of (a _{# 3g, 3,1} % 3g, a _{# 3g, 3,2} % 3g) Is an integer from 0 to 3g-1 (0, 1, 2, 3, 4,..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6,... -All values other than 3g-3) exist. (P = 1, 2, 3, ..., 3g)
Or
・
・
・
Or
The following conditions are satisfied in the order of Xk (D) in Expressions (172-1) to (172-3g).
(A _{# 1, k, 1} % 3g, a _{# 1, k, 2} % 3g), (a _{# 2, k, 1} % 3g, a _{# 2, k, 2} % 3g), (a _{# P, k, 1} % 3g, a _{# p, k, 2} % 3g), ..., 6g values of (a _{# 3g, k, 1} % 3g, a _{# 3g, k, 2} % 3g) Is an integer from 0 to 3g-1 (0, 1, 2, 3, 4,..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6,... -All values other than 3g-3) exist. (P = 1, 2, 3, ..., 3g)
(K = 1, 2, 3,..., N-1)
Or
・
・
・
Or
The following conditions are satisfied in the order of Xn-1 (D) in the equations (172-1) to (172-3g).
(A _{# 1, n-1, 1} % 3g, a _{# 1, n-1, 1} % 3g), (a _{# 2, n-1, 1} % 3g, a _{# 2, n-1, 2} % 3g ), ..., (a _{# p, n-1,1} % 3g, a _{# p, n-1,2} % 3g), ..., (a _{# 3g, n-1,1} % 3g, a _{# 3g, n-1, 2} % 3g) is an integer from 0 to 3g-1 (0, 1, 2, 3, 4, ..., 3g-2, 3g-1) Of these, all values other than multiples of 3 (that is, 0, 3, 6,..., 3g−3) exist. (P = 1, 2, 3, ..., 3g)
Or
The following conditions are satisfied in the order of P (D) in Expressions (172-1) to (172-3g). (B _{# 1,1} % 3g, b _{# 1,2} % 3g), (b _{# 2,1} % 3g, b _{# 2,2} % 3g), (b _{# 3, 1} % 3g, b _{# 3,2} % 3g), ..., (b _{# k, 1} % 3g, b _{# k, 2} % 3g), ..., (b _{# 3g-2, 1} % 3g, b _{# 3g-2, 2} % 3g ), (B _{# 3g-1,1} % 3g, b _{# 3g-1,2} % 3g), (b _{# 3g, 1} % 3g, b _{# 3g, 2} % 3g) To 3g-1 (0, 1, 2, 3, 4, ..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6, ..., 3g- All values other than 3) exist. (K = 1, 2, 3, ..., 3g)

  The LDPC-CC having a time varying period of 3 g and a coding rate (n−1) / n (n is an integer of 2 or more) has been described. Hereinafter, the condition of the degree of the parity check polynomial of the LDPC-CC having a time varying period of 3 g and a coding rate of ½ (n = 2) will be described.

As a parity check polynomial of an LDPC-CC having a time-varying period of 3 g (g = 1, 2, 3, 4,...) And a coding rate of 1/2 (n = 2) 174-3g).

  At this time, X is a polynomial expression of data (information) X, and P (D) is a polynomial expression of parity. Here, in equations (174-1) to (174-3g), the parity check polynomial is such that three terms exist in each of X (D) and P (D).

  When considered in the same manner as the LDPC-CC with the time varying period 3 and the LDPC-CC with the time varying period 6, the time varying period 3g represented by the parity check polynomials of the equations (174-1) to (174-3g), the coding rate In the case of 1/2 (n = 2) LDPC-CC, if the following condition (<condition # 2-1>) is satisfied, the possibility that higher error correction capability can be obtained increases.

However, in an LDPC-CC with a time varying period of 3 g and a coding rate of ½ (n = 2), the parity at time i is represented by Pi and the information is represented by X _{i, 1} . At this time, if i% 3g = k (k = 0, 1, 2,..., 3g−1), the parity check polynomial of Expression (174− (k + 1)) is established. For example, if i = 2, i% 3g = 2 (k = 2), and therefore Expression (175) is established.

In addition, in the formulas (174-1) to (174-3g), a _{# k, 1,1} , a _{# k, 1,2} , a _{# k, 1,3} are integers (however, a _{# k, 1 , 1} ≠ a _{# k, 1,2} ≠ a _{# k, 1,3} ) (k = 1, 2, 3,..., 3g). Also, b _{# k, 1} , b _{# k, 2} , and b _{# k, 3} are integers (where b _{# k, 1} ≠ b _{# k, 2} ≠ b _{# k, 3} ). The parity check polynomial (k = 1, 2, 3,..., 3g) in Expression (174-k) is called “check expression #k”, and the sub-matrix based on the parity check polynomial in Expression (174-k) is K-th sub-matrix H _k . Consider an LDPC-CC with a time varying period of 3 g generated from the first sub-matrix H ₁ , the second sub-matrix H ₂ , the third sub-matrix H ₃ ,..., The third g sub-matrix H _3g .

<Condition # 2-1>
In formulas (174-1) to (174-3g), the combination of the orders of X (D) and P (D) satisfies the following condition.
(A _{# 1,1,1} % 3, a _{# 1,1,2} % 3, a _{# 1,1,3} % 3), (b _{# 1,1} % 3, b _{# 1,2} % 3, b _{# 1,3} % 3) is (0,1,2), (0,2,1), (1,0,2), (1,2,0), (2,0,1), ( 2, 1, 0).
And,
_{(A # 2,1,1% 3, a} # 2,1,2% 3, a # 2,1,3% 3), (b # 2,1% 3, b # 2,2% 3, b _{# 2, 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), ( 2, 1, 0).
And,
(A _{# 3,1,1} % 3, a _{# 3,1,2} % 3, a _{# 3,1,3} % 3) (b _{# 3,1} % 3, b _{# 3,2} % 3, b _{# 3,3} % 3) is (0,1,2), (0,2,1), (1,0,2), (1,2,0), (2,0,1), (2 1, 0).
And,
・
・
・
And,
(A _{# k, 1,1} % 3, a _{# k, 1,2} % 3, a _{# k, 1,3} % 3), (b _{# k, 1} % 3, b _{# k, 2} % 3, b #K _{, 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), ( 2, 1, 0). (Thus, k = 1, 2, 3,..., 3g)
And,
・
・
・
And,
_{(A # 3g-2,1,1% 3} , a # 3g-2,1,2% 3, a # 3g-2,1,3% 3), (b # 3g-2,1% 3, b _{# 3g-2,2} % 3, b _{# 3g-2,3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), or (2, 1, 0).
And,
(A _{# 3g-1,1,1} % 3, a _{# 3g-1,1,2} % 3, a _{# 3g-1,1,3} % 3), (b _{# 3g-1,1} % 3, b _{# 3g-1,2} % 3, b _{# 3g-1,3} % 3) is (0,
1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).
And,
(A _{# 3g, 1,1} % 3, a _{# 3g, 1,2} % 3, a _{# 3g, 1,3} % 3), (b _{# 3g, 1} % 3, b _{# 3g, 2} % 3, b _{# 3g, 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), ( 2, 1, 0).

However, in consideration of the point that the encoding is easily performed as described in other than this embodiment, in the equations (174-1) to (174-3g),
It is preferable that one “0” exists among three (b _{# k, 1} % 3, b _{# k, 2} % 3, b _{# k, 3} % 3) (where k = 1, 2,... 3g).

In addition, in order to correlate the parity bit and the data bit at the same time and easily search for a code having a high correction capability,
There is one “0” among the three (a _{# k, 1,1} % 3, a _{# k, 1,2} % 3, a _{# k, 1,3} % 3) (where k = 1) 2, ... 3g).

Next, consider an LDPC-CC with a time-varying period of 3 g (g = 2, 3, 4, 5,...) In consideration of easy encoding. At this time, when the coding rate is 1/2 (n = 2), the parity check polynomial of LDPC-CC can be expressed as follows.

At this time, X (D) is a polynomial expression of data (information) X, and P (D) is a polynomial expression of parity. Here, in the equations (176-1) to (176-3g), the parity check polynomial is such that three terms exist in each of X and P (D). However, in an LDPC-CC with a time varying period of 3 g and a coding rate of ½ (n = 2), the parity at time i is represented by Pi and the information is represented by X _{i, 1} . At this time, if i% 3g = k (k = 0, 1, 2,..., 3g−1), the parity check polynomial of Expression (176− (k + 1)) is established. For example, if i = 2, i% 3g = 2 (k = 2), and therefore equation (177) is established.

  At this time, satisfying <Condition # 3-1> and <Condition # 4-1> increases the possibility that a code having higher error correction capability can be created.

<Condition # 3-1>
In the formulas (176-1) to (176-3g), the combination of the orders of X (D) satisfies the following condition.
(A _{# 1,1,1} % 3, a _{# 1,1,2} % 3, a _{# 1,1,3} % 3) are (0, 1, 2), (0, 2, 1), ( 1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).
And,
_{(A # 2,1,1% 3, a} # 2,1,2% 3, a # 2,1,3% 3) is (0,1,2), (0,2,1), ( 1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).
And,
(A _{# 3,1,1} % 3, a _{# 3,1,2} % 3, a _{# 3,1,3} % 3) is (0,1,2), (0,2,1), ( 1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).
And,
・
・
・
And,
(A _{# k, 1,1} % 3, a _{# k, 1,2} % 3, a _{# k, 1,3} % 3) is (0,1,2), (0,2,1), ( 1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0). (Thus, k = 1, 2, 3,..., 3g)
And,
・
・
・
And,
_{(A # 3g-2,1,1% 3} , a # 3g-2,1,2% 3, a # 3g-2,1,3% 3) is (0,1,2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).
And,
(A _{# 3g-1,1,1} % 3, a _{# 3g-1,1,2} % 3, a _{# 3g-1,1,3} % 3) are (0,1,2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).
And,
(A _{# 3g, 1,1} % 3, a _{# 3g, 1,2} % 3, a _{# 3g, 1,3} % 3) is (0,1,2), (0,2,1), ( 1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).

In addition, in the formulas (176-1) to (176-3g), the combination of the orders of P (D) satisfies the following condition.
(B _{# 1,1} % 3, b _{# 1,2} % 3), (b _{# 2,1} % 3, b _{# 2,2} % 3), (b _{# 3,1} % 3, b _{# 3,2} % 3), ..., (b _{# k, 1} % 3, b _{# k, 2} % 3), ...,
(B _{# 3g-2,1} % 3, b _{# 3g-2,2} % 3), (b _{# 3g-1,1} % 3, b _{# 3g-1,2} % 3), (b _{# 3g, 1} % 3, b _{# 3g, 2} % 3) is either (1, 2) or (2, 1) (k = 1, 2, 3,..., 3g).

  <Condition # 3-1> for Expressions (176-1) to (176-3g) has the same relationship as <Condition # 2-1> for Expressions (174-1) to (174-3g). When the following condition (<condition # 4-1>) is added to the expressions (176-1) to (176-3g) in addition to <condition # 3-1>, LDPC having higher error correction capability -The possibility of creating a CC increases.

<Condition # 4-1>
The following conditions are satisfied in the order of P (D) in the equations (176-1) to (176-3g).
(B _{# 1,1} % 3g, b _{# 1,2} % 3g), (b _{# 2,1} % 3g, b _{# 2,2} % 3g), (b _{# 3, 1} % 3g, b _{# 3,2} % 3g), ..., (b _{# k, 1} % 3g, b _{# k, 2} % 3g), ..., (b _{# 3g-2, 1} % 3g, b _{# 3g-2, 2} % 3g ), (B _{# 3g-1,1} % 3g, b _{# 3g-1,2} % 3g), (b _{# 3g, 1} % 3g, b _{# 3g, 2} % 3g) To 3g-1 (0, 1, 2, 3, 4, ..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6, ..., 3g- All values other than 3) exist.

  By the way, in the parity check matrix, if there is regularity at a position where “1” exists, but there is randomness, there is a high possibility that a good error correction capability can be obtained. Time-varying period 3g (g = 2, 3, 4, 5,...) Having a parity check polynomial of equations (176-1) to (176-3g), and coding rate 1/2 (n = 2) In LDPC-CC, when a code is created with the condition of <condition # 4-1> in addition to <condition # 3-1>, the randomness is maintained while the regularity is present at the position where “1” exists in the parity check matrix. Therefore, the possibility that a better error correction capability can be obtained increases.

Next, in a time-varying period of 3 g (g = 2, 3, 4, 5,...) That can be easily encoded and has a relationship between parity bits and data bits at the same time. Consider LDPC-CC. At this time, when the coding rate is 1/2 (n = 2), the parity check polynomial of LDPC-CC can be expressed as follows.

At this time, X (D) is a polynomial expression of data (information) X, and P (D) is a polynomial expression of parity. In equations (178-1) to (178-3g), the parity check polynomial is such that three terms exist in each of X (D) and P (D), and X (D) and P (D) Will have a term of D ⁰ . (K = 1, 2, 3, ..., 3g)

However, in an LDPC-CC with a time varying period of 3 g and a coding rate of ½ (n = 2), the parity at time i is represented by Pi and the information is represented by X _{i, 1} . At this time, if i% 3g = k (k = 0, 1, 2,..., 3g−1), the parity check polynomial of Expression (178− (k + 1)) is established. For example, if i = 2, i% 3g = 2 (k = 2), and therefore Equation (179) is established.

  At this time, if the following conditions (<Condition # 5-1> and <Condition # 6-1>) are satisfied, the possibility that a code having higher error correction capability can be created increases.

<Condition # 5-1>
In the expressions (178-1) to (178-3g), the combination of the orders of X (D) satisfies the following condition.
(A _{# 1,1,1} % 3, a _{# 1,1,2} % 3) is either (1,2) or (2,1).
And,
_{(A # 2,1,1% 3, a} # 2,1,2% 3) is a one of the (1,2), (2,1).
And,
(A _# 3, _{1, 1} % 3, a _{# 3, 1, 2} % 3) is either (1, 2) or (2, 1).
And,
・
・
・
And,
(A _{# k, 1,1} % 3, a _{# k, 1,2} % 3) is either (1,2) or (2,1). (Thus, k = 1, 2, 3,..., 3g)
And,
・
・
・
And,
_{(A # 3g-2,1,1% 3} , a # 3g-2,1,2% 3) is a one of the (1,2), (2,1).
And,
(A _{# 3g-1,1,1} % 3, a _{# 3g-1,1,2} % 3) is either (1,2) or (2,1).
And,
(A _{# 3g, 1,1} % 3, a _{# 3g, 1,2} % 3) is either (1,2) or (2,1).

In addition, in the formulas (178-1) to (178-3g), the order combination of P (D) satisfies the following condition.
(B _{# 1,1} % 3, b _{# 1,2} % 3), (b _{# 2,1} % 3, b _{# 2,2} % 3), (b _{# 3,1} % 3, b _{# 3,2} % 3), ..., (b _{# k, 1} % 3, b _{# k, 2} % 3), ..., (b _{# 3g-2,1} % 3, b _{# 3g-2,2} % 3 ), (B _{# 3g-1,1} % 3, b _{# 3g-1,2} % 3), (b _{# 3g, 1} % 3, b _{# 3g, 2} % 3) are (1,2), ( 2 or 1) (k = 1, 2, 3,..., 3g).

  <Condition # 5-1> for Expressions (178-1) to (178-3g) has the same relationship as <Condition # 2-1> for Expressions (174-1) to (174-3g). When the following condition (<condition # 6-1>) is added to the expressions (178-1) to (178-3g) in addition to <condition # 5-1>, LDPC having higher error correction capability -The possibility of creating a CC increases.

<Condition # 6-1>
The following conditions are satisfied in the order of X (D) in formulas (178-1) to (178-3g). _{(A # 1,1,1% 3g, a} # 1,1,2% 3g), (a # 2,1,1% 3g, a # 2,1,2% 3g), ···, (a _{# P, 1,1} % 3g, a _{# p, 1,2} % 3g), ..., 6g values of (a _{# 3g, 1,1} % 3g, a _{# 3g, 1,2} % 3g) Is an integer from 0 to 3g-1 (0, 1, 2, 3, 4,..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6,... -All values other than 3g-3) exist. (P = 1, 2, 3, ..., 3g)
And,
The following conditions are satisfied in the order of P (D) in Expressions (178-1) to (178-3g). (B _{# 1,1} % 3g, b _{# 1,2} % 3g), (b _{# 2,1} % 3g, b _{# 2,2} % 3g), (b _{# 3, 1} % 3g, b _{# 3,2} % 3g), ..., (b _{# k, 1} % 3g, b _{# k, 2} % 3g), ..., (b _{# 3g-2, 1} % 3g, b _{# 3g-2, 2} % 3g ), (B _{# 3g-1,1} % 3g, b _{# 3g-1,2} % 3g), (b _{# 3g, 1} % 3g, b _{# 3g, 2} % 3g) To an integer from 3g-1 (0, 1, 2, 3, 4, ..., 3g-2,
3g-1), all values other than multiples of 3 (that is, 0, 3, 6, ..., 3g-3) exist. (K = 1, 2, 3, ..., 3g)

  By the way, in a parity check matrix, if there is regularity at a position where “1” exists, there is a high possibility that a good error correction capability can be obtained. In an LDPC-CC with a time-varying period of 3 g (g = 2, 3, 4, 5,...) Having a parity check polynomial of equations (178-1) to (178-3g) and a coding rate of 1/2, When a code is created by adding the condition of <condition # 6-1> in addition to <condition # 5-1>, a randomness is given to the check matrix with regularity at the position where “1” exists. Therefore, the possibility that a better error correction capability can be obtained increases.

  Also, instead of <Condition # 6-1>, <Condition # 6′-1> is used. That is, in addition to <Condition # 5-1>, <Condition # 6′-1> is added to create a code. Even so, the possibility that an LDPC-CC having higher error correction capability can be created increases.

<Condition # 6′-1>
The following conditions are satisfied in the order of X (D) in formulas (178-1) to (178-3g). _{(A # 1,1,1% 3g, a} # 1,1,2% 3g), (a # 2,1,1% 3g, a # 2,1,2% 3g), ···, (a _{# P, 1,1} % 3g, a _{# p, 1,2} % 3g), ..., 6g values of (a _{# 3g, 1,1} % 3g, a _{# 3g, 1,2} % 3g) Is an integer from 0 to 3g-1 (0, 1, 2, 3, 4,..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6,... -All values other than 3g-3) exist. (P = 1, 2, 3, ..., 3g)
Or
The following conditions are satisfied in the order of P (D) in Expressions (178-1) to (178-3g). (B _{# 1,1} % 3g, b _{# 1,2} % 3g), (b _{# 2,1} % 3g, b _{# 2,2} % 3g), (b _{# 3, 1} % 3g, b _{# 3,2} % 3g), ..., (b _{# k, 1} % 3g, b _{# k, 2} % 3g), ..., (b _{# 3g-2, 1} % 3g, b _{# 3g-2, 2} % 3g ), (B _{# 3g-1,1} % 3g, b _{# 3g-1,2} % 3g), (b _{# 3g, 1} % 3g, b _{# 3g, 2} % 3g) To 3g-1 (0, 1, 2, 3, 4, ..., 3g-2, 3g-1), a multiple of 3 (that is, 0, 3, 6, ..., 3g- All values other than 3) exist. (K = 1, 2, 3, ..., 3g)

As an example, Table 9 lists LDPC-CC having a good error correction capability and a coding rate of 1/2 and a time varying period of 6.

(Other Embodiment 16)
In the other embodiment 9, the time-varying period 2 LDPC-CC that gives good reception quality has been described. Here, an LDPC-CC having a time-varying period of 2 with good characteristics to which another embodiment 14 is applied will be described. In the following, the case of coding rate (n−1) / n (n is an integer of 2 or more) will be described as an example.

As an LDPC-CC parity check polynomial with a time-varying period of 2, equations (180-1) and (180-2) are considered. At this time, X1 (D), X2 (D),... Xn-1 (D) is a polynomial expression of data (information) X1, X2,... Xn-1, and P (D) is a parity expression. It is a polynomial expression. Here, in Formula (180-1) and Formula (180-2), there are three terms in each of X1 (D), X2 (D),... Xn-1 (D), P (D). The parity check polynomial is as follows.

In the equation (180-1), a _{i, 1} , a _{i, 2} , a _{i, 3} (i = 1, 2,..., N−1) are integers (where a _{i, 1} ≠ a _{i, 2} ≠ a _{i, 3} ). B1, b2, and b3 are integers (where b1 ≠ b2 ≠ b3). The parity check polynomial of equation (180-1) is referred to as “check equation # 1”, and the sub-matrix based on the parity check polynomial of equation (180-1) is referred to as a first sub-matrix H ₁ .

In Formula (180-2), A _{i, 1} , A _{i, 2} , A _{i, 3} (i = 1, 2,..., N−1) are integers (where A _{i, 1} ≠ A _{i, 2} ≠ A _{i, 3} ). B1 and B
2, B3 is an integer (B1 ≠ B2 ≠ B3). Referred to parity check polynomial of equation (180-2) and "check equation # 2", the sub-matrix based on a parity check polynomial of equation (180-2), the second sub-matrix _{H 2.}

Then, consider an LDPC-CC with a time-varying period 2 generated from the first sub-matrix H ₁ and the second sub-matrix H ₂ .

At this time, in Formula (180-1) and Formula (180-2),
“For the coefficients (a _1,1 , a _1,2 , a _1,3 ) and the coefficients (A _1,1 , A _1,2 , A _1,3 ) relating to X1 (D),
・ Of (a _1,1 , a _1,2 , a _1,3 ), two are odd, one is even, and two of (A _1,1 , A _1,2 , A _1,3 ) are Odd number, one is even number. Among (a _1,1 , a ₁ , ₂ , a _1,3 ), one is odd, two is even, and (A _1,1 , A _1,2 , A _1,3 ) Meet one of odd numbers and two even numbers ", and
“Xi (D) (i = 2, 3,..., N−1) coefficients (a _{i, 1} , a _{i, 2} , a _{i, 3} ) and coefficients (A _{i, 1} , A _{i, 2} , A _{i, 3} )
-Of (a _{i, 1} , a _{i, 2} , a _{i, 3} ), two are odd, one is even, and two of (A _{i, 1} , A _{i, 2} , A _{i, 3} ) are Odd number, one is even number. Among (a _{i, 1} , a _{i, 2} , a _{i, 3} ), one is odd number, two is even number, and (A _{i, 1} , A _{i, 2} , A _{i, 3} ) One is odd, two are even "
Meet one of the following, and
“・ (B ₁ , b ₂ , b ₃ ), two are odd, one is even, and two of (B ₁ , B ₂ , B ₃ ) are odd, and one is even (b ₁ , b ₂ , b ₃ ), one is odd, two are even, and one of (B ₁ , B ₂ , B ₃ ) is odd and two is even ”
In this case, since the conditions described in the other fourteenth embodiment are satisfied, the loop 6 does not always occur and becomes a regular LDPC-CC, so that a good error correction capability can be obtained.

(Other Embodiment 17)
In the other fifteenth embodiment, the LDPC-CC having the time varying period 3 has been described. Here, a puncturing method suitable for the LDPC-CC having the time varying period 3 described in other embodiment 15 will be described. Hereinafter, as an example, a case where a code with a coding rate of 1/2 (coding rate of 1/2) is made larger than the coding rate of 1/2 by puncturing will be described.

  Consider an LDPC-CC with a time-varying period 3 defined by the equations (162-1) to (162-3). At this time, even if a1> a2> a3, b1> b2> b3, A1> A2> A3, B1> B2> B3, α1> α2> α3, β1> β2> β3, the generality is not lost. Therefore, the following will be described based on these relationships.

  The maximum degree of the information X (D) of the “check expression # 1” in the equation (162-1) is a1, and the maximum degree of the parity P (D) is b1. Further, the maximum degree of the information X (D) of the “check expression # 2” of the expression (162-2) is A1, and the maximum degree of the parity P (D) is B1. In addition, the maximum degree of the information X (D) of the “check expression # 3” in the expression (162-3) is α1, and the maximum degree of the parity P (D) is β1. Here, the following two conditions are given.

[Condition # 1]
Consider the maximum order among the maximum orders a1, A1, and α1 of the data X (D) in “check formula # 1,” “check formula # 2,” and “check formula # 3”. For example, if a1 is the maximum of these three maximum orders, the bit related to a1 is not punctured, that is, the bit related to a1 is transmitted and the bit other than a1 is punctured (not transmitted) Select. If A1 is the maximum of these three maximum orders, the bits related to A1 are not punctured, and puncture bits are selected from bits other than A1. If α1 is the maximum of these three maximum orders, the bits related to α1 are not punctured, and puncture bits are selected from bits other than α1.

[Condition # 2]
Consider the order that takes the maximum value among the maximum orders b1, B1, and β1 of the parity P (D) in “check formula # 1,” “check formula # 2,” and “check formula # 3”. For example, if b1 is the maximum of these three maximum orders, the bit related to b1 is not punctured, that is, the bit related to b1 is transmitted, and the bit other than b1 is punctured (not transmitted) Select. If B1 is the maximum of these three maximum orders, the bit related to B1 is not punctured, the bit related to B1 is transmitted, and the punctured bit (bit not transmitted) is transmitted from other bits than B1. select. If β1 is the maximum of these three maximum orders, bits related to β1 are not punctured, bits related to β1 are transmitted, and bits other than β1 are punctured (bits that are not transmitted). select.

  Puncturing is performed for the LDPC-CC having the time varying period 3 described in the other embodiment 15 so that either [Condition # 1] or [Condition # 2] is satisfied. Thereby, even when puncturing is performed, good error correction capability can be obtained. As a matter of course, when both [Condition # 1] and [Condition # 2] are satisfied, even better error correction capability can be obtained. Below, it demonstrates in detail using drawing.

FIG. 66 shows a correspondence relationship between the parity check matrix H of LDPC-CC with time-varying period 3, the transmission sequence u, and the parity patterns according to the above [Condition # 1] and [Condition # 2]. FIG. 66 shows a case where the parity check polynomial of time-varying period 3 is composed of the same parity check polynomial as in FIG. 65A. Therefore, the sub-matrix _{H 1,} H 2, _{H 3} in FIG. 66 is similar to the sub-matrix _{H 1,} H 2, _{H 3} of FIG 65A.

When the transmission sequence is u = (X ₁ , P ₁ , X ₂ , P ₂ ,..., X _i , P _i , X _{i + 1} , P _{i + 1} ,...) ^T , the relationship of Hu = 0 is established. . Therefore, the relationship between the transmission sequence and the parity check matrix is expressed as shown in FIG. 66 as described in the seventh embodiment (see FIG. 16).

  In the LDPC-CC of time varying period 3 in FIG. 66, the maximum order (a1, A1, α1) of the data X (D) in “check equation # 1,” “check equation # 2,” and “check equation # 3” = Of (2, 5, 4), focus on the maximum order A1 = 5, which is the maximum value.

  In FIG. 66, “1” at position 6601 indicates the position of the bit corresponding to the maximum order A1. Since “1” at position 6601 is the position of the bit corresponding to the maximum order A1, it is important for obtaining high reliability in the column operation in BP decoding. This is because, when LDPC-CC is considered as a convolutional code, the bit corresponding to the maximum order A1 is involved in the maximum constraint length. In general, with a convolutional code, the longer the constraint length, the higher the reliability. Thus, the maximum order A1 is important because it is related to the maximum constraint length. Therefore, if the bit corresponding to the maximum order A1 is punctured by puncturing and the bit corresponding to the maximum order A1 is not transmitted, the maximum constraint length is shortened.

Therefore, “1” at position 6601 in FIG. 66 remains so that the maximum constraint length is not shortened by puncturing. That is, information Xi, Xi + 3, Xi + 6, Xi + 9 in FIG.
,... Are not selected as puncture bits, are transmitted bits, and information other than information Xi, Xi + 3, Xi + 6, Xi + 9,. Select a bit. Thereby, even when puncturing is performed, a good error correction capability can be obtained.

  Similarly, in the LDPC-CC having a time varying period of 3 in FIG. 66, the maximum order (b1, B1,...) Of parity P (D) in “check equation # 1,” “check equation # 2,” “check equation # 3”. Of β1) = (2, 5, 4), attention is paid to B1 = 5 which is the maximum value.

  In FIG. 66, “1” at position 6602 indicates the position of the bit corresponding to the maximum order B1. As described above, “1” at the position 6602 is a bit position corresponding to the maximum order B1 and is important because it is related to the maximum constraint length.

  Therefore, “1” at position 6602 in FIG. 66 remains so that the maximum constraint length is not shortened by puncturing. That is, the parities Pi, Pi + 3, Pi + 6, Pi + 9,... In FIG. 66 are not selected as puncture bits, but are set as transmission bits. , Pi + 3, Pi + 6, Pi + 9,..., Select bits not to be transmitted. Thereby, even when puncturing is performed, a good error correction capability can be obtained.

  In this way, a bit that is not punctured in information X according to [Condition # 1] (a bit to be transmitted) is determined, and a bit that is not punctured in parity P according to [Condition # 2] (a bit to be transmitted) is defined as [Condition # 1]. Can be determined separately. It should be noted that better error correction capability can be obtained if bits that are not punctured (bits to be transmitted) are determined for information X and parity P in accordance with both [Condition # 1] and [Condition # 2].

FIG. 67 shows another correspondence relationship different from FIG. The parity check matrix H is different between FIG. 67 and FIG. FIG. 67 shows the parity check polynomial of time-varying period 3 and the sub-matrices H ₁ , H ₂ , and H ₃ in FIG.

When the transmission sequence is u = (X ₁ , P ₁ , X ₂ , P ₂ ,..., X _i , P _i , X _{i + 1} , P _{i + 1} ,...) ^T , the relationship of Hu = 0 is established. . Therefore, the relationship between the transmission sequence and the parity check matrix is expressed as shown in FIG. 67 as described in Embodiment 7 (see FIG. 16).

  In the LDPC-CC of time varying period 3 in FIG. 67, the maximum order (a1, A1, α1) of data X (D) in “check equation # 1,” “check equation # 2,” and “check equation # 3” = Of (2, 5, 4), focus on the maximum order A1 = 5, which is the maximum value.

  In FIG. 67, “1” at position 6701 indicates the position of the bit corresponding to the maximum order A1. Since “1” at the position 6701 is the position of the bit corresponding to the maximum degree A1, it is important for obtaining high reliability in the column operation in the BP decoding. This is because, when LDPC-CC is considered as a convolutional code, the bit corresponding to the maximum order A1 is involved in the maximum constraint length.

Therefore, “1” at position 6701 in FIG. 67 remains so that the maximum constraint length is not shortened by puncturing. That is, information Xi, Xi + 3, Xi + 6, Xi + 9,... In FIG. 67 is not selected as a puncture bit, but is set as a transmission bit, and information other than information Xi, Xi + 3, Xi + 6, Xi + 9,. , Xi + 3, Xi + 6, X
Bits not to be transmitted are selected from other than i + 9,. Thereby, even when puncturing is performed, good error correction capability can be obtained.

  Similarly, in the LDPC-CC having a time varying period of 3 in FIG. 67, the maximum order (b1, B1,...) Of parity P (D) in “check equation # 1,” “check equation # 2,” and “check equation # 3”. Of β1) = (5, 2, 7), focus on β1 = 7, which is the maximum value.

  In FIG. 67, “1” at position 6702 indicates the position of the bit corresponding to the maximum order β1. As described above, “1” at position 6702 is the position of the bit corresponding to the maximum order β1, and is important because it is related to the maximum constraint length.

  Therefore, “1” at position 6702 in FIG. 67 remains so that the maximum constraint length is not shortened by puncturing. That is, the parities Pi-1, Pi + 2, Pi + 5, Pi + 8,... In FIG. , Parity Pi-1, Pi + 2, Pi + 5, Pi + 8,..., Select bits not to be transmitted. Thereby, even when puncturing is performed, a good error correction capability can be obtained.

  In this way, a bit that is not punctured in information X according to [Condition # 1] (a bit to be transmitted) is determined, and a bit that is not punctured in parity P according to [Condition # 2] (a bit to be transmitted) is defined as [Condition # 1]. Can be determined separately. It should be noted that a better error correction capability can be obtained if bits that are not punctured (bits to be transmitted) are determined for information X and parity P in accordance with both [Condition # 1] and [Condition # 2].

  The case where the coding rate is ½ has been described above as an example, but the coding rate is not limited to ½, and the coding rate (n−1) / n (n is an integer of 2 or more) The same puncturing can be performed even when the coding rate is made larger than (n−1) / n by puncturing from the coding rate (n−1) / n). The outline is as follows.

Consider an LDPC-CC with a time-varying period 3 defined by equations (163-1) to (163-3). At this time, a _{i, 1} > a _{i, 2} > a _{i, 3} , b1>b2> b3, A _{i, 1} > A _{i, 2} > A _{i, 3} , B1>B2> B3, α _{i, 1} > Even if α _{i, 2} > α _{i, 3} , β1>β2> β3 (i = 1, 2,..., n−1), generality is not lost. Therefore, the following description is based on these relationships.

The maximum degree of information Xi (D) of “check expression # 1” in Expression (163-1) is a _{i, 1} , and the maximum degree of parity P (D) is b1. In addition, the maximum degree of the information Xi (D) of the “check expression # 2” of the expression (163-2) is A _{i, 1} and the maximum degree of the parity P (D) is B1. Further, the maximum degree of the information Xi (D) of the “check expression # 3” in the expression (162-3) is α _{i, 1} and the maximum degree of the parity P (D) is β1. Here, the following two conditions are given.

[Condition # 1]
The maximum order among the maximum orders a _{i, 1} , A _{i, 1} , α _{i, 1} of the data Xi (D) in “check formula # 1”, “check formula # 2”, and “check formula # 3”. think of. For _{example, a i, 1} is transmitted when the maximum of these three maximum _orders, without puncturing the bits associated with _{a i, 1,} that _is, the bit associated with _{a i, 1, a i,} 1 Select puncture bits (bits not to be transmitted) from other bits. _{Also, A i, 1} is, if the maximum of these three maximum _orders, without puncturing the bits associated with _{A i, 1,} and transmits the bits associated with _{A i, 1, A i,} 1 other Select puncture bits (bits not to be transmitted) from other bits. Also, alpha _{i, 1} is, if the maximum of these three maximum orders, without puncturing the bits associated with the alpha _{i, 1,} and transmits the bits associated with the _{α i, 1, α i,} 1 other Select puncture bits (bits not to be transmitted) from other bits.

[Condition # 2]
Consider the maximum order among the maximum orders b1, B1, and β1 of the parity P (D) in “check formula # 1,” “check formula # 2,” and “check formula # 3”. For example, if b1 is the maximum of these three maximum orders, the bit related to b1 is not punctured, that is, the bit related to b1 is transmitted, and the puncture bit (the bit not to be transmitted) is transmitted from other bits than b1. ) Is selected. If B1 is the maximum of these three maximum orders, the bit related to B1 is not punctured, the bit related to B1 is transmitted, and the punctured bit (bit not transmitted) is transmitted from other bits than B1. select. If β1 is the maximum of these three maximum orders, bits related to β1 are not punctured, bits related to β1 are transmitted, and bits other than β1 are punctured (bits that are not transmitted). select.

  Puncturing is performed for an LDPC-CC with a time-varying period of 3 and a coding rate (n-1) / n so that either [Condition # 1] or [Condition # 2] is satisfied. Thereby, even when puncturing is performed, good error correction capability can be obtained. As a matter of course, when both [Condition # 1] and [Condition # 2] are satisfied, even better error correction capability can be obtained. Note that the setting method not to be a candidate for puncture bits (bits not to be transmitted) is as described with reference to FIGS. 66 and 67.

  The present invention is not limited to all the above embodiments, and can be implemented with various modifications. For example, in the above-described embodiment, the case of performing as a wireless communication device has been described.

  It is also possible to perform this communication method as software. For example, a program for executing the communication method may be stored in advance in a ROM (Read Only Memory), and the program may be operated by a CPU (Central Processor Unit).

  Further, a program for executing the communication method is stored in a computer-readable storage medium, the program stored in the storage medium is recorded in a RAM (Random Access Memory) of the computer, and the computer is operated according to the program. You may do it.

  Needless to say, the present invention is useful not only in wireless communication but also in power line communication (PLC), visible light communication, and optical communication.

  The present invention can be widely applied to communication systems using LDPC-CC.

The figure which shows the encoder of a (7,5) convolutional code. The figure which shows the check matrix of a (7,5) convolutional code. The figure which shows the check matrix of a (7,5) convolutional code. The figure which shows an example at the time of adding "1" to the approximate lower triangular matrix of the check matrix of FIG. The figure which shows an example of a structure of the check matrix of LDPC-CC in Embodiment 1. FIG. The figure which shows the check matrix of a (7,5) convolutional code. The figure which shows an example of a structure of the check matrix of LDPC-CC in Embodiment 1. FIG. The figure which shows the check matrix of a (7,5) convolutional code. The figure which shows an example at the time of adding "1" to the upper trapezoid matrix of the check matrix of FIG. The figure which shows an example of a structure of the check matrix of LDPC-CC in Embodiment 2. FIG. The figure which shows an example of a structure of the check matrix at the time of termination | terminus in Embodiment 3 The figure which shows an example of a structure of the check matrix at the time of termination | terminus in Embodiment 3 The figure which shows an example of a structure of the check matrix at the time of termination | terminus in Embodiment 3 The figure explaining the method of producing LDPC-CC from a convolutional code FIG. 15 is a diagram illustrating an example of a configuration of an LDPC-CC parity check matrix according to Embodiment 7 The figure which shows an example of a structure of the check matrix of LDPC-CC of the time-varying period 1 in Embodiment 7. FIG. The figure which shows an example of a structure of the check matrix of LDPC-CC of the time-varying period m in Embodiment 7. FIG. Illustration for explaining the number of puncture patterns The figure which shows the relationship between an encoding sequence and a puncture pattern Diagram showing the number of parity check polynomials that must be checked to select a puncture pattern The figure which shows an example of a structure of the check matrix of LDPC-CC of time-varying period 2 The figure which shows an example of a structure of the check matrix of LDPC-CC of time-varying period 4 The figure explaining the method of producing LDPC-CC from the convolutional code of coding rate 1 / n The figure explaining the method of producing LDPC-CC from the convolutional code of coding rate 1 / n The figure which shows an example of a structure of an encoder. The figure which shows an example of a structure of an encoder. Diagram showing decoder configuration using sum-product decoding algorithm The figure which shows an example of a structure of a transmitter Diagram showing an example of transmission format The figure which shows an example of a structure of a receiver The figure which shows an example of a structure of an encoding part. The figure which shows an example of a structure of the LDPC-CC encoding part using the check matrix which added "1" to the upper trapezoid matrix of the check matrix. The figure which shows an example of a structure of the check matrix of LDPC-CC of time-varying period 2 The figure which shows an example of a structure of the check matrix of LDPC-CC of time-varying period 7 The figure which shows the structure of the LDPC-CC check matrix in other Embodiment 7. FIG. Diagram for explaining a general puncture method The figure which shows a response | compatibility with the transmission codeword sequence v by the general puncture method, and the LDPC-CC check matrix H The figure for demonstrating the puncturing method in Embodiment 7 The figure which shows the response | compatibility with the transmission codeword sequence v by the puncture method in Embodiment 7, and the LDPC-CC check matrix H FIG. 10 is a block diagram showing another main configuration of the transmitting apparatus according to Embodiment 7. The figure which shows an example of the puncture pattern in Embodiment 7 The figure which shows another puncture pattern in Embodiment 7. FIG. The figure which shows another puncture pattern in Embodiment 7. FIG. The figure which shows another puncture pattern in Embodiment 7. FIG. The figure which shows another puncture pattern in Embodiment 7. FIG. Diagram for explaining the decoding processing timing Diagram showing FEC Encoder Diagram showing Structure of LDPC Convolusional Encoder a structure of LDPC-CC encoder The block diagram which shows another principal part structure of the transmitter in other Embodiment 9. FIG. The figure which shows the relation between the maximum degree and the second degree of two polynomials The figure for demonstrating the relationship between the maximum degree of two polynomials, and a 2nd degree. The figure which shows an example of the radio | wireless communications system in other Embodiment 11. FIG. The figure which shows another example of the radio | wireless communications system in other Embodiment 11. FIG. The figure which shows an example of a structure of the check matrix H of LDPC-CC of coding rate 2/3 and time-varying period 2 The figure which shows an example of a structure of the parity check matrix of LDPC-CC of coding rate 2/3 and time-varying period m The figure which shows an example of a structure of the parity check matrix of LDPC-CC of coding rate (n-1) / n and time-varying period m. The figure which shows an example of the 1st sub-matrix and the 2nd sub-matrix in other Embodiment 12. The figure which shows the structural example of the check matrix H of the LDPC-CC of the coding rate 1/2 comprised by the 1st submatrix and the 2nd submatrix, and the time-varying period 2. The figure which shows another example of the 1st sub matrix in other Embodiment 12, and a 2nd sub matrix. The figure which shows the structural example of the check matrix H of LDPC-CC of the coding rate 2/3 comprised by the 1st sub-matrix and the 2nd sub-matrix, and the time-varying period 2 Figure provided to explain the design method described in Non-Patent Document 16 The figure which shows the submatrix used to explain Theorem 1 The figure which shows the submatrix used to explain Theorem 1 The figure which shows the submatrix used to explain Theorem 1 The figure which shows the submatrix used to explain Theorem 2 The figure which shows the submatrix used to explain Theorem 2 The figure which shows the submatrix used to explain Theorem 2 The figure which shows the structure of the parity check polynomial and check matrix H of LDPC-CC of time-varying period 3 The figure which shows the relationship of the reliability propagation | transmission of each item regarding X (D) of "check type | formula # 1"-"check type | formula # 3" of FIG. 65A The figure which shows the relationship of the reliability propagation | transmission of each item regarding X (D) of "check type | formula # 1"-"check type | formula # 6" The figure which shows the correspondence with the parity check matrix H according to the parity check matrix H of the LDPC-CC of time-varying period 3, the transmission sequence u, and [condition # 1] and [condition # 2] The figure which shows another correspondence with the parity check matrix H of LDPC-CC of the time-varying period 3, the transmission sequence u, and the parity pattern according to [Condition # 1] and [Condition # 2] The figure which shows the check matrix of LDPC-CC The figure which shows the structure of a LDPC-CC encoder.

Explanation of symbols

101, 102, 2511-1 to 2511 -M, 2514-1 to 2514 -M, 2911-1 to 2911 -K Shift register 103, 104, 105 Exclusive OR circuit 304, 402, 501, 502, 1001 Check matrix "1" to add to
1105, 1205, 1305, 1306 “1” added to the check matrix at the end
2202 Parity calculation unit 2204 Data storage unit 2206 Parity storage unit 2302 Storage unit 2403 Log likelihood ratio storage unit 2405 Row processing operation unit 2407 Row processing data storage unit 2409 Column processing operation unit 2411 Column processing data storage unit 2413 Control unit 2415 Log Likelihood Ratio Operation Unit 2417 Determination Unit 2500, 3700, 4700 Transmitter 2510, 2910, 4710 LDPC-CC Encoding Unit (LDPC-CC Encoder)
2512-0 to 2512-M, 2513-0 to 2513-M, 2912-1 to 2912-K Weight multiplier 2316 Weight control unit 2515 mod2 adder 2520, 3710, 4720 Puncture unit 2530 Interleave unit 2540 Modulation unit 2700 Receiver 2710 Receiving Unit 2720 Log Likelihood Ratio Generation Unit 2730 Control Information Generation Unit 7540 Deinterleaving Unit 2750 Depuncture Unit 2760 BP Decoding Unit 3711 First Puncture Unit 3712 Second Puncture Unit 3713 Switching Unit

Claims (8)

A coding method for creating a low-density parity check convolutional code (LDPC-CC) having a time varying period of 3 g (g is a positive integer),
Of the parity check polynomials represented by Equation 1-1,
(A _{# 1,1,1} % 3, a _{# 1,1,2} % 3, a _{# 1,1,3} % 3), (a _{# 1,2,1} % 3, a _{# 1,2,2} % 3, a _{# 1,2,3} % 3), ..., (a _{# 1, n-1,1} % 3, a _{# 1, n-1,2} % 3, a _{# 1, n-1 , 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1 or 0),
(B _{# 1,1} % 3, b _{# 1,2} % 3, b _{# 1,3} % 3) are (0,1,2), (0,2,1), (1,0,2) , (1, 2, 0), (2, 0, 1), (2, 1, 0),
A first parity check polynomial;
Among the parity check polynomials expressed by Equation (1-2),
_{(A # 2,1,1% 3, a} # 2,1,2% 3, a # 2,1,3% 3), (a # 2,2,1% 3, a # 2,2,2 % 3, a _{# 2,2,3} % 3), ..., (a _{# 2, n-1,1} % 3, a _{# 2, n-1,2} % 3, a _{# 2, n-1 , 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1 or 0),
(B _{# 2,1} % 3, b _{# 2,2} % 3, b _{# 2,3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2) , (1, 2, 0), (2, 0, 1), (2, 1, 0),
A second parity check polynomial;
Of the parity check polynomials expressed by the equation (1-kk) (kk = 3, 4,..., 3g−1),
(A _{# kk, 1,1} % 3, a _{# kk, 1,2} % 3, a _{# kk, 1,3} % 3), (a _{# kk, 2,1} % 3, a _{# kk, 2,2} % 3, a _{# kk, 2,3} % 3), ..., (a _{# kk, n-1,1} % 3, a _{# kk, n-1,2} % 3, a _{# kk, n-1 , 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1 or 0),
(B _{# kk, 1} % 3, b _{# kk, 2} % 3, b _{# kk, 3} % 3) are (0, 1, 2), (0, 2, 1), (1, 0, 2) , (1, 2, 0), (2, 0, 1), (2, 1, 0),
The kk parity check polynomial;
Of the parity check polynomials expressed by the formula (1-3g),
(A _{# 3g, 1,1} % 3, a _{# 3g, 1,2} % 3, a _{# 3g, 1,3} % 3), (a _{# 3g, 2,1} % 3, a _{# 3g, 2,2} % 3, a _{# 3g, 2, 3} % 3), ..., (a _{# 3g, n-1, 1} % 3, a _{# 3g, n-1, 2} % 3, a _{# 3g, n-1 , 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1 or 0),
(B _{# 3g, 1} % 3, b _{# 3g, 2} % 3, b _{# 3g, 3} % 3) is (0, 1, 2), (0, 2, 1), (1, 0, 2) , (1, 2, 0), (2, 0, 1), (2, 1, 0),
In the LDPC-CC defined based on the third g parity check polynomial,
Providing the first to third g parity check polynomials;
Obtaining an LDPC-CC codeword by linear operation of the first to third g parity check polynomials and input data;
An encoding method comprising:

In formula (1-k), n = 2.
The encoding method according to claim 1. In the first parity check polynomial,
a _{# 1, p, 3} = 0, (a _{# 1,1,1} % 3, a _{# 1,1,2} % 3), (a _{# 1,2,1} % 3, a _{# 1,2 , 2} % 3), ..., (a _{# 1, n-1,1} % 3, a _{# 1, n-1,2} % 3) is either (1,2) or (2,1) Suddenly
b _{# 1,3} = 0, and (b _{# 1,1} % 3, b _{# 1,2} % 3) is either (1,2) or (2,1)
In the second parity check polynomial,
a _# a _{2, p, 3 = 0,} (a # 2,1,1% 3, a # 2,1,2% 3), (a # 2,2,1% 3, a # 2,2 _{, 2} % 3), ..., (a _{# 2, n-1,1} % 3, a _{# 2, n-1,2} % 3) is either (1,2) or (2,1) Suddenly
b _{# 2,3} = 0, and (b _{# 2,1} % 3, b _{# 2,2} % 3) is either (1,2) or (2,1)
In the kk-th parity check polynomial, a _{# kk, p, 3} = 0, and (a _{# kk, 1,1} % 3, a _{# kk, 1,2} % 3), (a _{# kk, 2, 1} % 3, a _{# kk, 2,2} % 3), ..., (a _{# kk, n-1,1} % 3, a _{# kk, n-1,2} % 3) are (1,2 ), (2, 1),
b _{# kk, 3} = 0, and (b _{# kk, 1} % 3, b _{# kk, 2} % 3) is either (1,2) or (2,1)
In the 3g parity check polynomial,
a _{# 3g, p, 3} = 0, (a _{# 3g, 1,1} % 3, a _{# 3g, 1,2} % 3), (a _{# 3g, 2,1} % 3, a _{# 3g, 2 , 2} % 3), ..., (a _{# 3g, n-1,1} % 3, a _{# 3g, n-1,2} % 3) is either (1,2) or (2,1) Become
b _{# 3g, 3} = 0, and (b _{# 3g, 1} % 3, b _{# 3g, 2} % 3) is either (1,2) or (2,1).
The encoding method according to claim 1. In formula (1-k), n = 2.
The encoding method according to claim 3. In the order of X ₁ (D) of the first to third g parity check polynomials, (a _{# 1,1,1} % 3g, a _{# 1,1,2} % 3g), (a _{# 2,1,1} % 3g, a _{# 2,1,2} % 3g), ..., (a _{# 3g, 1,1} % 3g, a _{# 3g, 1,2} % 3g), 0 to 3g All integer values of -1 except for multiples of 3 exist,
And,
In the order of X ₂ (D) of the first to third g parity check polynomials, (a _{# 1,2,1} % 3g, a _{# 1,2,2} % 3g), (a _{# 2,2,1} % 3g, a _{# 2,2,2} % 3g), ..., (a _{# 3g, 2,1} % 3g, a _{# 3g, 2,2} % 3g) values of 0 to 3g All integer values of -1 except for multiples of 3 exist,
And,
In the order of X _p (D) of the first to third g parity check polynomials, (a _{# 1, p, 1} % 3g, a _{# 1, p, 2} % 3g), (a _{# 2, p, 1} % 3g, a _{# 2, p, 2} % 3g), ..., (a _{# 3g, p, 1} % 3g, a _{# 3g, p, 2} % 3g) 0 to 3g for 6g values All integer values of -1 except for multiples of 3 exist,
And,
In the _order of X _n-1 (D) of the first to third g parity check polynomials, (a _{# 1, n-1, 1} % 3g, a _{# 1, n-1, 2} % 3g), (a _{# 2, n-1,1} % 3g, a _{# 2, n-1,2} % 3g), ..., (a _{# 3g, n-1,1} % 3g, a _{# 3g, n-1,2} % Of 3g) includes all values other than multiples of 3, out of integers from 0 to 3g-1.
And,
In the order of P (D) of the first to third g parity check polynomials, (b _{# 1,1} % 3g, b _{# 1,2} % 3g), (b _{# 2,1} % 3g, b _{# 2, 2} % 3g), ..., (b _{# 3g, 1} % 3g, b _{# 3g, 2} % 3g) are values other than multiples of 3 among the integers from 0 to 3g-1. All values of exist,
The encoding method according to claim 3. In formula (1-k), n = 2.
The encoding method according to claim 5. An encoder that generates a low-density parity check convolutional code (LDPC-CC) from a convolutional code,
An encoder comprising a parity calculation unit for obtaining a parity sequence by the encoding method according to claim 1. A decoder for decoding low density parity check convolutional codes (LDPC-CC: Low-Density Parity-Check Convolutional Codes) using reliability propagation (BP: Belief Propagation),
A row processing operation unit that performs a row processing operation using a parity check matrix corresponding to the parity check polynomial used in the encoder according to claim 7;
A column processing operation unit that performs a column processing operation using the check matrix;
A determination unit that estimates codewords using calculation results in the row processing calculation unit and the column processing calculation unit;
A decoder comprising:

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