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Algorithmic complexity in coding theory and the minimum distance problem

Published: 04 May 1997 Publication History
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References

[1]
M. AJTAI, Generating hard instances of lattice problems, in Proc. 28-th Annum ACM Syrup. Theory of Computing, pp. 99-108, Philadelphia, PA, May 1996.
[2]
N. ALON, Packings with large minimum kissing numbers, preprint, 1996.
[3]
N. ALON, J. BRUCK, J. NAOR, M. NAOR, and R. ROTH, Construction of asymptotically good low-rate errorcorrecting codes through pseudo-random graphs, IEEE Trans. Inform. Theory, vol. 38, pp. 509-516, 1992.
[4]
S. ARORA, Probabilistic checking of proofs and hardness of approximation problems, Ph.D. thesis, University of California, Berkeley, CA., 1994.
[5]
S. ARORA, L. BABAI, J. STERN, and Z. SWEEDYK, The hardness of approximate optima in lattices, codes, and systems of linear equations, in Proc. 34-th Annum Syrup. Found. Computer Science, pp. 724-733, Palo Alto, CA, 1993.
[6]
S. ARORA and C. LUND, Hardness of approximations, in Approximation Algorithms for NP-Hard Problems, D.S. Hochbaum (Ed.), Boston: PWS Publishing Co., pp. 399-446, 1997.
[7]
L.R. BAHL, J. COCKE, F. JELINEK, and J. R. AVIV, Optimal decoding of linear codes for minimizing symbol error rate, IEEE Trans. Inform. Theory, vol. 20, pp. 284- 287, 1974.
[8]
R. BALASUBRAMANIAN, M. FELLOWS, and V. RAMAN, An improved fixed-parameter algorithm for vertex cover, to appear, 1997.
[9]
A. BARG, Some new NP-complete coding problems, Probl. Peredachi Informatsii, voi. 30, pp. 23-28, 1994, (in Russian).
[10]
A. BARG, Complexity issues in coding, survey chapter to appear in Handbook of Coding Theory, R.A. Brualdi, C. Huffman, and V. Pless (Ed.), Amsterdam: Elsevier.
[11]
E.R. BERLV. KAMP, Algebraic Coding Theory, New York: McGraw-Hill, 1968.
[12]
E.R. BP. RLEKAMP, R.J. McELIECF., and H.C.A.vAN TILBORG, On the inherent intractability of certain coding problems, IEEE Trans. Inform. Theory, vol. 24, pp. 384-386, 1978.
[13]
C. BERROU, A. GLAVlEUX, and P. THITIMAJSHIMA, Near Shannon limit error-correcting coding and decoding: turbo codes, in Proc. IEEE Int. Conf. on Communications, pp. 1064-1070, Geneva, Switzerland, 1993.
[14]
M. BLAUM, J. BRUCE, and A. VARDY, On MDS codes and alternants over certain rings, in Proc. 900-th Meeting, American Math. Soc., Chicago, IL., March 1995.
[15]
M. BLAUM, J. BRUCK, and A. VARDY, MDS array codes with independent parity symbols, IEEE Trans. Inform. Theory, vol. 42, pp. 529-542, 1996.
[16]
J. BRUCK and M. NAOR, The hardness of decoding linear codes with preprocessing, IEEE Trans. Inform. Theory, vol. 36, pp. 381-385, 1990.
[17]
G.C. CLARK and J.B. CAIN, Error-Correction Coding for Digital Communications, New York: Plenum Press, 1981.
[18]
R.G. DOWNS7 and M.R. FELLOWS, Fixed parameter tractability and completeness I: basic theory, SIAM J. Comput., vol. 24, pp. 873-921, 1995.
[19]
R.G. DowNEY and M.R. FELLOWS, Fixed parameter tractability and completeness II: completeness for WIll, Theoret. Computer Sci. A vol. 141 pp. 109-131, 1995.
[20]
R.G.DOWNEY, M.R. FELLOWS, A.VARDY, and G.WHI- TTLE, On the parametrized complexity of certain fundamental problems in coding theory, preprint in preparation, 1997.
[21]
P. DIACONIS and R.L. GRAHAM, The Radon transform on Z~, Pacific J. Math., vol. 118, pp. 176-185, 1985.
[22]
I.I. DUMER, On complexity of maximum-likelihood decoding of the best concatenated codes, in Proc. 8-th All- Union Conf. on Coding Theory and Information Theory, Moscow-Kuibishev, pp. 66-69, 1981, (in Russian).
[23]
I.I. DuMER, Concatenated codes and their generalizations, to appear in Handbook of Coding Theory, R.A. Brualdi, W.C. Huffman, V. Pless (Ed.), Amsterdam: Elsevier.
[24]
S. EVEN and Y. YACOBI, Cryptography and NP-completeness, Lect. Notes Comp. Science, vol. 85, pp. 195- 207, Springer-Verlag, 1982.
[25]
J. FANG, G.D. COHEN, PH. GODLEWSKI, and G.BAT- TAIL, On the inherent intractability of soft decision decoding of linear codes, Lect. Notes Comp. Science, vol. 311, pp. 141-149, Springer-Verlag, 1988.
[26]
R.M. FANO, A heuristic discussion of probabilistic decoding, IEEE Trans. Inform. Theory, vol. 9, pp. 64-73, 1963.
[27]
J. FEIGENBAUM, The use of coding theory in computational complexity, in Proc. Syrup. Appl. Math, A.R. Calderbank (Ed.), Providence, RI: AMS Press, pp. 203-229, 1995.
[28]
J. FEIGENBAUM, G.D. FORNEY, B. MARCUS, R.J. Mc ELIECE, and A. VARDY, Special issue on "Codes and Complexity," IEEE Trans. Inform. Theory, vol. 42, November 1996.
[29]
M.R. FELLOWS and A. VARDY, The hardness of theta series in lattices, preprint in preparation, 1997.
[30]
G.-L. FENG and K.K. TZENG, A new procedure for decoding cyclic and BCH codes up to actual minimum distance, IEEE Trans. Inform. Theory, vol. 40, pp. 1364- 1374, 1994.
[31]
G.D. FORNEY, JR., Concatenated Codes, Cambridge, MA: M.I.T. Press, 1966.
[32]
G.D. FORNEV, JR., The Viterbi algorithm, Proc. IEEE, vol. 61, pp. 268-278, 1973.
[33]
G.D. FORNEY, JR., Convolutional codes III: sequential decoding, Inform. Control, vol. 25, pp. 267-297, 1974.
[34]
G.D. FORN~.Y, JR., Coset codes II: Binary lattices and related codes, IEEE Trans. Inform. Theory, vol. 34, pp. 1152-1187, 1988.
[35]
G.D. FOaNEY, Ja., The forward-backward algorithm, in Proc. 34-th Allerton Conference on Comm., Control, and Computing, Monticello, IL., pp. 432-446, October 1996.
[36]
G.D. FORNF. Y, JR. and A. VARD~, Generalized minimum distance decoding of Euclidean-space codes and lattices, IEEE Trans. Inform. Theory, vol. 42, pp. 1992- 2026, 1996.
[37]
B.J. FREY and F.R. KSCHmCHANG, Probability propagation and iterative decoding, in Proc. 34-th Allerton Conference on Comm., Control, and Computing, Monticello, IL., pp. 482-493, October 1996.
[38]
R.G. GALLAGER, Low Density Parity-Check Codes, Cambridge: M.I.T. Press, 1962.
[39]
M.R. GARSY and D.S. JOHNSON, Computers and Intractability: A Guide to the Theory of NP-Completeness, San Francisco: Freeman, 1979.
[40]
C.R.P. HARTMAN and K.K. TZP. N6, Decoding beyond the BCH bound using multiple sets of syndrome sequences, IEEE Trans. inform. Theory, vol. 20, pp. 292- 295, 1974.
[41]
D.S.HOCHBAUM, (Editor), Approximation Algorithms for NP-Hard Problems, Boston: PWS Publishing Co., 1997.
[42]
T. HOHOLDT and R. PELLIKAAN, On the decoding of algebraic-geometric codes, IEEE Trans. Inform. Theory, vol. 41, pp. 1589-1614, 1995.
[43]
G.B. HOaN and F.R. KSCHmCHANG, On the intractability of permuting a block code to minimize trellis complexity, IEEE Trans. Inform. Theory, vol. 42, pp. 2042- 2048, 1996.
[44]
D.S. JOHNSON, The NP-completeness column: An ongoing guide, J. Algorithms, vol. 3, pp. 182-195, 1982.
[45]
D.S. JOHNSON, The NP-completeness column: An ongoing guide, J. Algorithms, vol. 7, pp. 584-601, 1986.
[46]
J. JUSTESEN, A class of constructive asymptotically good algebraic codes, IEEE Trans. Inform. Theory, vol. 18, pp. 652-656, 1972.
[47]
R. KARP and R. LIPTON, Some connections between nonuniform and uniform complexity classes, in Proc. 12-th Annual Syrup. Theory of Computing, pp. 302- 309, 1980.
[48]
L. KHACHtYAN, On the complexity of approximating extremal determinants in matrices, J. Complexity, vol. 11, pp. 138-153, 1995.
[49]
F.R. KSCHmCHANG and V. SOROKINE, Oll the trellis structure of block codes, IEEE Trans. Inform. Theory, vol. 41, pp. 1924-1937, 1995.
[50]
B.D. KUDRYASHOV and T.G. ZAKHAaOVA, Block codes from convolutional codes, Problemy Peredachi Informats/i, vol. 25, pp. 98-102, 1989, (in Russian).
[51]
A. LAFOURCADF. and A. VARDY, Asymptotically good codes have infinite trellis complexity, IEEE Transl Inform. Theory, vol. 41, pp. 555-559, 1995.
[52]
A. LAFOURCADE and A. V^RDY, Lower bounds on trellis complexity of block codes, IEEE Trans. Inform. Theory, vol. 41, pp. 1938-1954, 1995.
[53]
L.A. LE:WN, Average case completeness, SIAM J. Cornput., vol. 15, pp. 285-286, 1986.
[54]
S. LIN and E.J. WELDON, Long BCH codes are bad, Inform. and Control, vol. 11, pp. 445-451, 1967.
[55]
A. LOBSTEiN and G.D. COHrN, Sur la complexit~ d'un probl~me de codage, Theoretical Informatics Appl., vol. 21, pp. 25-32, 1987.
[56]
B. LdPr. Z JIM~.NrZ, Plane models of Drinfeld modulax curves, Ph.D. thesis, University of Complutense, Madrid, March 1996.
[57]
D.J.C. MACKAY and R.M. NEAL, Near Shannon limit performance of low-density parity-check codes, Elect. Lett., to appear, 1997.
[58]
D.J.C. MACKAY and R.M. NS:AL, Good error-correcting codes based on very sparse matrices, IEEE Trans. Inform. Theory, submitted for publication, 1996.
[59]
F.J. MACWmLIAMS and N.J.A. SLOANS, The Theory of Error Correcting Codes, Amsterdam: North-Holland, 1977.
[60]
YU.I. MANm and S.C. VL/~DUTS, Linear codes and modular curves, J. Soviet Math., vol. 30, pp. 2611-2643, 1985, (in Russian).
[61]
J.L. MASSrY, Shift-register synthesis and BCH decoding, IEEE Trans. Inform. Theory, vol. 15, pp. 122-127, 1969.
[62]
J.L. MASSE:Y, Foundation and methods of channel encoding, Proc. Int. Cone Information Theory and Systems, NTG-Fachberichte, Berlin, 1978.
[63]
R.J. M cELmcP, On the BCJR trellis for linear block codes, IEEE Trans. Inform. Theory, vol. 42, pp. 1072- 1092, 1996.
[64]
R.J. MCELIECE, E.R. RODEMICH, and J.-F. CHENG, The turbo decision algorithm, in Proc. 33-rd Allerton Conference on Comm., Control, and Computing, Monticello, IL., pp. 366-379, October 1995.
[65]
D.J. MUDER, Minimal trellises for block codes, IEEE Trans. Inform. Theory, vol. 34, pp. 1049-1053, 1988.
[66]
T. MUIR, Treatise on the Theory of Determinants, New York: Dover, 1960.
[67]
S.C. NTAFOS and S.L. HAKIMI, On the complexity of some coding problems, IEEE Trans. Inform. Theory, vol. 27, pp. 794-796, 1981.
[68]
J.K. OMURA, On the Viterbi decoding algorithm, IEEE Trans. Inform. Theory, vol. 15, pp. 177-179, 1969.
[69]
J. PEARL, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, San Mateo, CA: Kaufmann, 1988.
[70]
R. PELLIKAAN, Asymptotically good sequences of curves and codes, in Proc. 34-th A11erton Conference on Comm., Control, and Computing, Monticello, IL., pp. 276-285, October 1996.
[71]
I.S. REED and G. SOLOMON, Polynomial codes over certain finite fields, SIAM J. Appl. Math., vol. 8, pp. 300- 304, 1960.
[72]
R.M. RoTH and A. LEMPV~L, A construction of non- Reed-Solomon type MDS codes, IEEE 7kans. Inform. Theory, vol. 35, pp. 655-657, 1989.
[73]
C.E. SHANNON, A mathematical theory of communication, Bell Syst. Tech. J., voi. 27, pp. 379-423 and pp. 623-656, 1948.
[74]
B.-Z. SHEN, A Justesen construction of binary concatenated codes that asymptotically meet the Zyablov bound for low rate, IEEE Trans. Inform. Theory, vol. 39, pp. 239-242, 1993.
[75]
V. SHOUP, New algorithms for finding irreducible polynomials over finite fields, Math. Computation, vol. 54, pp. 435-447, 1990.
[76]
M. SIPSER and D.A. SPIELMAN, Expander codes, IEEE Trans. Inform. Theory, vol. 42, pp. 1710-1722, 1996.
[77]
D.A. SPmLMAN, Linear-time encodable and decodable codes, IEBE Trans. Inform. Theory, vol. 42, pp. 1723- 1731, 1996.
[78]
J. STERN, Approximating the number of error locations within a constant ratio is NP-complete, Lect. Notes Comp. Science, vol. 673, pp. 325-331, Springer, 1993.
[79]
M. SUDAN, Efficient checking of polynomials and proofs and the hardness of approximation problems, Ph.D. thesis, University of California, Berkeley, CA., 1992.
[80]
M. SUDAN, Decoding of Reed-Solomon codes beyond the error-correction bound, J. Complexity, vol. 1, 1997, to appear.
[81]
R.M. TANNER, A recursive approach to low-complexity codes, IEEE Trans. Inform. Theory, vol. 27, pp. 533- 547, 1981.
[82]
A. TRACHTENBERG and A. VARDY, Which codes have cycle-free Tanner graphs?, preprint, 1997.
[83]
M.A. TSrASMAN and S.G. VL~DUTS, Algebraic Geometry Codes, Dodrecht: Kluwer Academic, 1991.
[84]
M.A. TSFASMAN, S.G. VLXDUTS, and T. ZINK, Modular curves, Shimura curves, and Goppa codes better than the Varshamov-Gilbert bound, Math. Nachrichten, vol. 104, pp. 13-28, 1982.
[85]
P. VAN EMDE BOAS, Another NP-complete partition problem and the complexity of computing short vectors in a lattice, Tech. Report 81-04, Dept. of Mathematics, Univ. of Amsterdam, 1980.
[86]
A. VARDY, Trellis structure of block codes, to appear in Handbook of Coding Theory, R. Brualdi, W.C. Huffman, and V. Pless (Ed.), Amsterdam: Elsevier.
[87]
A. VARDY and F.R. KSCHmCHANG, Proof of a conjecture of McEliece regarding the expansion index of the minimal trellis, IEEB Trans. Inform. Theory, vol. 42, pp. 2027-2033, 1996.
[88]
A. VARDY, J. SNYDERS, and Y. BE'ERY, Bounds on the dimension of codes and subcodes with prescribed contraction index, Linear Algebra Appl., vol. 142, pp. 237- 261, 1990.
[89]
S.G. VL/kDUTS, G.L. KATSMAN, and M.A. TSFASMAN, Modular curves and codes with polynomial complexity of construction, Problemy Peredachi Int'ormatsii, vol. 20, pp. 47-55, 1984, (in Russian).
[90]
D.J.A. WELSH, Combinatorial problems in matroid theory, pp. 291-307 in Combinatorial Mathematics and its Applications, D.J.A. Welsh (Ed.), London: Academic Press, 1971.
[91]
N. WIBERG, Codes and decoding on general graphs, Ph.D. thesis, University of LinkSping, Sweden, 1996.
[92]
N. WmERG, H.-A. LOELIGER, and R. KOTTER, Codes and iterative decoding on general graphs, Euro. Trans. Telecommun., vol. 6, pp. 513-526, 1995.
[93]
J.K. WOLF, Efficient maximum-likelihood decoding of linear block codes using a trellis, IEEE Trans. Inform. Theory, vol. 24, pp. 76-80, 1978.
[94]
V.V. ZYABLOV, An estimate of the complexity of constructing binary linear concatenated codes, Problemy Peredachi Informatsii, vol. 7, pp. 5-13, 1971.

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cover image ACM Conferences
STOC '97: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
May 1997
752 pages
ISBN:0897918886
DOI:10.1145/258533
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