Open AccessArticle

On Optimal and Quantum Code Construction from Cyclic Codes over $F$ _qPQ with Applications

Shakir Ali

Amal S. Alali

Pushpendra Sharma

^1,*

Kok Bin Wong

³,

Elif Segah Öztas

⁴ and

Mohammad Jeelani

⁵

Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India

Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur 50603, Malaysia

⁴

Department of Mathematics, Kamil Ozdag Science Faculty, Karamanoglu Mehmetbey University, Karaman 70100, Turkey

⁵

Department of Computer Application, Faculty of Science, Integral University, Lucknow 226026, India

Author to whom correspondence should be addressed.

Entropy 2023, 25(8), 1161; https://doi.org/10.3390/e25081161

Submission received: 2 July 2023 / Revised: 30 July 2023 / Accepted: 31 July 2023 / Published: 2 August 2023

(This article belongs to the Special Issue Advances in Information and Coding Theory II)

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Abstract

The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of

F_{q} P Q

, where

P = \frac{F_{q} [v]}{⟨ v^{3} - α_{2}^{2} v ⟩}

and

Q = \frac{F_{q} [u, v]}{⟨ u^{2} - α_{1}^{2}, v^{3} - α_{2}^{2} v ⟩}

are nonchain finite rings and

α_{i}

is in

F_{q} / {0}

for

i \in {1, 2}

, where

q = p^{m}

with

m \geq 1

is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.

Keywords:

cyclic code; dual code; mixed alphabet code; QEC code

MSC:

94B05; 94B15; 94B60

1. Introduction

The most significant families of cyclic codes were first introduced and studied by Prange [1] and Sloane-Thompson [2]. These codes are extensively used because of their robust algebraic structure and simplicity of usage. In recent years, there has been a rapid expansion of research on cyclic codes over finite rings, following the notable work of Hammons et al. [3]. The literature extensively delves into the exploration of cyclic codes and their constructions across various finite rings, such as applications in constructing minimal codes in [4] and projective two-weight codes in [5]. Recently, Pereira and Mancini have given a general method to construct EAQEC codes from cyclic codes in [6]. A particular area of interest in recent years has been the study of codes over mixed alphabets. This research direction was initiated by Brouwer et al. [7] in 1998, where they began investigating linear codes over mixed alphabets. Specifically, the authors focused on describing

Z_{2}

-submodules over

Z_{r^{2}} Z_{s^{3}}

for mixed alphabet codes. Borges et al. [8] made significant contributions to this field by discovering

Z_{2} Z_{4}

-additive codes and their associated

Z_{2} Z_{4}

-linear codes. Notably, extensive studies have been conducted on additive codes, with significant research contributions in [9,10,11,12]. Moreover, the additive codes, additive cyclic codes, and the additive quasi-cyclic codes over different mixed alphabets have also been intensely studied, for example,

Z_{2} Z_{2} [u]

-additive codes [13],

Z_{p} Z_{p^{k}}

-additive codes [14],

Z_{2} Z_{2} [u]

-cyclic and constacyclic [15],

Z_{2} (Z_{2} + u Z_{2})

-additive cyclic codes [16]. Borges et al. [8] explored double cyclic codes over

Z_{2}

. Recently, Gao et al. have studied hulls of double cyclic codes over

Z_{2}

, and obtained some good quantumcodes from hulls in [17]. Gao et al. [18] generalized double cyclic codes over

Z_{4}

. The triple cyclic codes over

Z_{2}

were introduced by Mostafanasab [19] and extended this double cyclic code structure. Recently,

Z_{2} Z_{2} Z_{4}

-additive cyclic codes and

Z_{2} Z_{4} Z_{8}

-cyclic codes were separately explored by Wu et al. [20], and Aydogdu-Gursoy [21], respectively. Moreover, the P and Q used in this paper are finite nonchain rings actually. As we know, there are many papers on quantum codes over finite nonchain rings [22,23,24]. Researchers primarily concentrated on investigating the structural properties of mixed alphabet codes in all of the works, including generator matrices, parity check matrices, generator polynomials, minimal generating sets, generator polynomials for dual codes, etc. In 2020, Dinh et al. [25] delivered quantum and LCD code construction over mixed alphabets. In 2022, Ashraf et al. [26] obtained quantum codes over mixed alphabets.

Motivated by the above study, in this paper, we describe cyclic codes, new quantum error-correcting (QEC) codes and several optimal codes over mixed alphabets. Firstly, we provide linear and cyclic codes over

F_{q} P Q

, where

F_{q}

are the finite fields with q elements,

P = F_{q} [v] / 〈 v^{3} - α_{2}^{2} v 〉

and

Q = F_{q} [u, v] / 〈 u^{2} - α_{1}^{2}, v^{3} - α_{2}^{2} v 〉

, where

α_{1}

and

α_{2}

are the nonzero elements of

F_{q}

. Section 2 presents some basic definitions, the construction of cyclic codes over

F_{q} P Q

, and some important structural properties over

F_{q} P Q

. Section 3 describes Gray images and linear codes over P and Q. Further, we define a Gray map with the help of a matrix. In Section 4, like Section 3, we define the Gray map and linear codes over

F_{q} P Q

. Section 5 discusses the structural properties of cyclic codes over P, Q,

F_{q} P Q

and describes quantum error-correcting (QEC) codes and their construction over

F_{q} P Q

. Finally, in Section 6, we discuss some applications of cyclic codes over mixed alphabets and provide the conclusion of our results.

2. Preliminaries

Let m be a positive integer, p be an odd prime, and q be an odd prime power such that

q = p^{m}

. Next, let

F_{q}

be a finite field with q elements having characteristic p. Our construction depends on

F_{q} P Q

, where P and Q are the commutative, nonchain, semi-local ring. We begin with some key remarks and basic definitions as follows:

Remark 1.

Let R be a local ring. Then, the following conditions are equivalent:

(i).: R has a unique maximal left ideal.
(ii).: R has a unique maximal right ideal.
(iii).: The sum of any two nonunit elements of R is also a nonunit as well as $0 \neq 1$ .
(iv).: If x is an arbitrary element of R, then x or $1 - x$ is unit as well as $0 \neq 1$ .

Remark 2.

In the case of a commutative ring, R contains a unique maximal ideal if and only if it is local.

Definition 1.

Let

x, y \in F_{q}^{n}

, the Hamming distance between two vectors

x = x_{1} \dots x_{n}

and

y = y_{1} \dots y_{n}

be defined to be the number of places at which they differ and be denoted by

d (x, y)

Definition 2.

The Hamming weight of a vector

x = x_{1} x_{2} \dots x_{n}

is defined to be the number of nonzero coordinates

x_{i}

x

and is denoted by

w_{H} (x)

Definition 3.

Each element of code

E

is referred to as a codeword, and a code of length n over R is said to be linear if it is an R-submodule of

R^{n} .

Definition 4.

A code

E

is said to be self-orthogonal if

E \subseteq E^{⊥}

, self-dual if

E = E^{⊥}

, and dual containing if

E^{⊥} \subseteq E

Definition 5

([27]). A code

E

is a cyclic code of length n over R if it is linear and every cyclic shift of each codeword is also in

E

, i.e.,

σ (c) = (E_{n - 1}, E_{0}, E_{1}, \dots, E_{n - 2}) \in E

, whenever

c = (E_{n - 1}, E_{0}, \dots, E_{n - 2}) \in E

. The operator σ is known as cyclic shift.

By using the properties of cyclic code over the finite commutative nonchain ring, we can define cyclic code over

F_{q} P Q

. Clearly, the ring P can be expressed as

P = F_{q} + v F_{q} + v^{2} F_{q}

, where

v^{3} = α_{2}^{2} v

, the set

{1, v, v^{2}}

is a set of basis elements of P, and we denote the basis elements of P as follows:

Δ_{1} = 1, Δ_{2} = v, Δ_{3} = v^{2},

and every element of ring P is of the form

p_{r} = a_{1} + v a_{2} + v^{2} a_{3}

, for

a_{1}, a_{2}, a_{3} \in F_{q}

. Orthogonal idempotents of this ring is given as follows:

E_{1} = (α_{2}^{2} - v^{2}),

E_{2} = \frac{(α_{2} v + α_{2}^{2} v^{2})}{2},

and

E_{3} = \frac{(- α_{2} v + α_{2}^{2} v^{2})}{2} .

It is straightforward to see that

E_{i}^{2} = E_{i}, E_{i} E_{j} = 0

for

i \neq j

, and

E_{1} + E_{2} + E_{3} = 1

where

i, j = 1, 2, 3

. By using orthogonal idempotents

E_{1}, E_{2}

and

E_{3}

, we can write the arbitrary element

p_{r}

of the ring P as

p_{r} = E_{1} p_{r_{1}} + E_{2} p_{r_{2}} + E_{3} p_{r_{3}}

, where

p_{r_{1}}, p_{r_{2}}, p_{r_{3}} \in F_{q}

. Similarly, the ring Q can be expressed as

Q = F_{q} + u F_{q} + v F_{q} + u v F_{q} + v^{2} F_{q} + u v^{2} F_{q}

, where

u^{2} = α_{1}^{2}

v^{3} = α_{2}^{2} v

, the set

{1, u, v, u v, v^{2}, u v^{2}}

is a set of basis elements of Q and we denote the basis elements of Q as follows

δ_{1} = 1, δ_{2} = u

and

δ_{3} = v, δ_{4} = u v, δ_{5} = v^{2}, δ_{6} = u v^{2}

and every element of ring Q is of the form

q_{r} = b_{1} + u b_{2} + v b_{3} + u v b_{4} + v^{2} b_{5} + u v^{2} b_{6}

, for

b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6} \in F_{q}

. Orthogonal idempotents of the ring Q are given as follows:

ζ_{1} = \frac{(α_{1} + u) (α_{2}^{2} - v^{2})}{2 α_{1} α_{2}^{2}},

ζ_{2} = \frac{(α_{1} - u) (α_{2}^{2} - v^{2})}{2 α_{1} α_{2}^{2}},

ζ_{3} = \frac{(α_{1} + u) (α_{2} v + v^{2})}{4 α_{1} α_{2}^{2}},

ζ_{4} = \frac{(α_{1} - u) (α_{2} v + v^{2})}{4 α_{1} α_{2}^{2}},

ζ_{5} = \frac{(α_{1} + u) (- α_{2} v + v^{2})}{4 α_{1} α_{2}^{2}}

and

ζ_{6} = \frac{(α_{1} - u) (- α_{2} v + v^{2})}{4 α_{1} α_{2}^{2}} .

It is easy to see that

ζ_{i}^{2} = ζ_{i}, ζ_{i} ζ_{j} = 0

and

ζ_{1} + ζ_{2} + ζ_{3} + ζ_{4} + ζ_{5} + ζ_{6} = 1

where

i, j = 1, 2, 3, 4, 5, 6

and

i \neq j .

By using orthogonal idempotents

ζ_{1}, ζ_{2}, ζ_{3}, ζ_{4}, ζ_{5}

and

ζ_{6}

, we can write the arbitrary element

q_{r}

of the ring Q as

q_{r} = ζ_{1} q_{r_{1}} + ζ_{2} q_{r_{2}} + ζ_{3} q_{r_{3}} + ζ_{4} q_{r_{4}} + ζ_{5} q_{r_{5}} + ζ_{6} q_{r_{6}}

, where

q_{r_{1}}, q_{r_{2}}, q_{r_{3}}, q_{r_{4}}, q_{r_{5}}, q_{r_{6}} \in F_{q}

. Now, we define two ring homomorphisms

η

and

ζ

η : Q ⟶ F_{q}

such that

η (q_{r}) = η (ζ_{1} q_{r_{1}} + ζ_{2} q_{r_{2}} + ζ_{3} q_{r_{3}} + ζ_{4} q_{r_{4}} + ζ_{5} q_{r_{5}} + ζ_{6} q_{r_{6}}) = q_{r_{1}}

and

Γ : Q ⟶ P

such that

Γ (ζ_{1} q_{r_{1}} + ζ_{2} q_{r_{2}} + ζ_{3} q_{r_{3}} + ζ_{4} q_{r_{4}} + ζ_{5} q_{r_{5}} + ζ_{6} q_{r_{6}}) = E_{1} q_{r_{1}} + E_{2} q_{r_{2}} + E_{3} q_{r_{3}}

. For arbitrary

q_{r} \in Q

and

(x, y, z) \in F_{q} P Q

, we define Q-scalar multiplication on

F_{q} P Q

by:

⧫ : Q \times F_{q} P Q ⟶ F_{q} P Q

such that

q_{r} ⧫ (x, y, z) = (η (q_{r}) (x), Γ (q_{r}) (y), q_{r} z) .

This multiplication is well defined and we can extend this multiplication over

F_{q}^{α} \times P^{β} \times Q^{γ}

⧫ : Q \times (F_{q}^{α} \times P^{β} \times Q^{γ}) ⟶ F_{q}^{α} \times P^{β} \times Q^{γ}

such that

q_{r} ⧫ l = (η (q_{r}) x_{0}, η (q_{r}) x_{1}, \dots, η (q_{r}) x_{α - 1}, Γ (q_{r}) y_{0}, Γ (q_{r}) y_{1}, \dots, Γ (q_{r}) y_{β - 1}, q_{r} z_{0}, q_{r} z_{1},

\dots, q_{r} z_{γ - 1})

, where

q_{r} \in Q

and

l = (x_{0}, x_{1}, \dots, x_{α - 1}, y_{0}, \dots, y_{β - 1}, z_{0}, z_{1}, \dots, z_{γ - 1}) \in F_{q}^{α} \times P^{β} \times Q^{γ} .

In view of this scalar multiplication,

F_{q}^{α} \times P^{β} \times Q^{γ}

forms a Q-module.

Definition 6.

A nonempty subset

E

F_{q}^{α} \times P^{β} \times Q^{γ}

is a

F_{q} P Q

-linear code of length

(α + β + γ)

E

is a Q-submodule of

F_{q}^{α} \times P^{β} \times Q^{γ}

Let

l = (x_{0}, x_{1}, \dots, x_{α - 1}, y_{0}, \dots, y_{β - 1}, z_{0}, z_{1}, \dots, z_{γ - 1})

and

l^{'} = (x_{0}^{'}, x_{1}^{'}, \dots, x_{α - 1}^{'}, y_{0}^{'}, \dots,

y_{β - 1}^{'}, z_{0}^{'}, z_{1}^{'}, \dots, z_{γ - 1}^{'})

, where

l, l^{'} \in F_{q}^{α} \times P^{β} \times Q^{γ}

. After this, we also define inner product of l and

l^{'}

l \cdot l^{'} = \sum_{i = 0}^{α - 1} x_{i} x_{i}^{'} + \sum_{j = 0}^{β - 1} y_{j} y_{j}^{'} + \sum_{k = 0}^{γ - 1} z_{k} z_{k}^{'} \in Q .

Here, the dual of

E

, i.e.,

E^{⊥} = {l^{'} \in F_{q}^{α} \times P^{β} \times Q^{γ} ∣ l \cdot l^{'} = 0, \forall l \in E}

Definition 7.

A linear code

E

is a

F_{q} P Q

-cyclic code of length

(α + β + γ)

if every cyclic shift of

E

is also in

E

, i.e.,

σ (c) = (x_{α - 1}, x_{0}, x_{1}, \dots, x_{α - 2}, p_{β - 1}, p_{0}, p_{1}, \dots, p_{β - 2}, q_{γ - 1}, q_{0}, q_{1}, \dots, q_{γ - 2}) \in E, \forall c \in E

, where

c = (x_{0}, x_{1}, \dots, x_{α - 1}, p_{0}, \dots, p_{β - 1}, q_{0}, q_{1}, \dots, q_{γ - 1})

and

σ (c)

is a cyclic shift of

E

Proposition 1.

Suppose

E

is a

F_{q} P Q

-cyclic code of length (

α + β + γ

). Then, the dual of

E

is a also

F_{q} P Q

-cyclic code of length (

α + β + γ

Proof

Suppose

E

is a

F_{q} P Q

-cyclic code of length

(α + β + γ)

, and next, let us consider that

l^{'} \in E^{⊥}

such that

l^{'} = (x_{0}^{'}, x_{1}^{'}, \dots, x_{α - 1}^{'}, p_{0}^{'}, p_{1}^{'}, \dots, p_{β - 1}^{'}, q_{0}^{'}, q_{1}^{'}, \dots, q_{γ - 1}^{'})

, and also, we take

l c m (α, β, γ) = t

and

l = (x_{0}, x_{1}, \dots, x_{α - 1}, p_{0}, p_{1}, \dots, p_{β - 1}, q_{0}, q_{1}, \dots, q_{γ - 1}) \in E

. Then, we will show that

σ (l^{'}) = (x_{α - 1}^{'}, x_{0}^{'}, x_{1}^{'}, \dots, x_{α - 2}^{'}, p_{β - 1}^{'}, p_{0}^{'}, p_{1}^{'}, \dots, p_{β - 2}^{'}, q_{γ - 1}^{'}, q_{0}^{'}, q_{1}^{'}, \dots, q_{γ - 2}^{'}) \in E^{⊥} .

From above described inner product, we have

\begin{matrix} l . σ (l^{'}) & = & (x_{0} x_{α - 1}^{'} + x_{1} x_{0}^{'} + \dots + x_{α - 1} x_{α - 2}^{'}) + (p_{0} p_{β - 1}^{'} + p_{1} p_{0}^{'} + \dots + p_{β - 1} p_{β - 2}^{'}) + \\ (q_{0} q_{γ - 1}^{'} + q_{1} q_{0}^{'} + \dots + q_{γ - 1} q_{γ - 2}^{'}) . \end{matrix}

Since

E

is a

F_{q} P Q

-cyclic code and

l c m (α, β, γ) = t

σ^{t - 1} (l) = (x_{1}, \dots, x_{α - 1}, x_{0}, p_{1}, \dots, p_{β - 1}, p_{0}, q_{1}, \dots, q_{γ - 1}, q_{0}) .

Now, we take the inner product of

σ^{t - 1} (l)

and

l^{'}

, we have

σ^{t - 1} (l) \cdot l^{'} = 0,

where

\begin{matrix} σ^{t - 1} (l) . l^{'} & = & (x_{1} x_{0}^{'} + x_{2} x_{1}^{'} + \dots + x_{0} x_{α - 1}^{'}) + (p_{1} p_{0}^{'} + p_{2} p_{1}^{'} + \dots + p_{0} p_{β - 1}^{'}) + \\ (q_{1} q_{0}^{'} + q_{2} q_{1}^{'} + \dots + q_{0} q_{γ - 1}^{'}) \end{matrix}

On comparing the coefficients of both sides, we have

\begin{matrix} x_{1} x_{0}^{'} + x_{2} x_{1}^{'} + \dots + x_{0} x_{α - 1}^{'} & = & 0 \\ p_{1} p_{0}^{'} + p_{2} p_{1}^{'} + \dots + p_{0} p_{β - 1}^{'} & = & 0 \\ q_{1} q_{0}^{'} + q_{2} q_{1}^{'} + \dots + q_{0} q_{γ - 1}^{'} & = & 0 \end{matrix}

we obtain

l \cdot σ (l^{'}) = 0 .

Thus,

σ (l^{'}) \in E^{⊥} .

This shows that

E^{⊥}

is a

F_{q} P Q

-cyclic code of length

(α + β + γ)

. □

Let

Q_{α + β + γ} = \frac{F_{q} [x]}{〈 x^{α} - 1 〉} \times \frac{P [x]}{〈 x^{β} - 1 〉} \times \frac{Q [x]}{〈 x^{γ} - 1 〉}

and

f = (a_{0}, a_{1}, \dots, a_{α - 1}, b_{0}, b_{1}, \dots, b_{β - 1}, c_{0}, c_{1}, \dots, c_{γ - 1}) \in F_{q}^{α} \times P^{β} \times Q^{γ}

. Let f be an arbitrary element of

Q_{α + β + γ}

, and then f can be identified as

\begin{matrix} f (x) & = & (a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{α - 1} x^{α - 1}, b_{0} + b_{1} x + b_{2} x^{2} + \dots + b_{β - 1} x^{β - 1}, c_{0} + c_{1} x + c_{2} x^{2} \\ + \dots + c_{γ - 1} x^{γ - 1}) \\ = & (a (x), b (x), c (x)) . \end{matrix}

This gives bijective mapping between

F_{q}^{α} \times P^{β} \times Q^{γ}

and

Q_{α + β + γ}

. Next, let us consider that

g (x) = q_{0} + q_{1} x + q_{2} x^{2} + \dots + q_{n} x^{n} \in Q [x]

and

(a (x), b (x), c (x)) \in Q_{α + β + γ}

. With the help of previously defined Q-scalar multiplication, we induce the multiplication ⨀ in

Q_{α + β + γ}

g (x) ⨀ (a (x), b (x), c (x)) = (η (g (x)) a (x), Γ (g (x)) b (x), g (x) c (x),

where

η (g (x)) = η (q_{0}) + η (q_{1}) x + \dots + η (q_{n}) x^{n}

and

Γ (g (x)) = Γ (q_{0}) + Γ (q_{1}) x + \dots + Γ (q_{n}) x^{n}

. It is simple to demonstrate that

Q_{α + β + γ}

makes an

Q [x]

-submodule with respect to multiplication ⨀.

Proposition 2

([25]). A code

E

is a

F_{q} P Q

-cyclic code of length

(α + β + γ)

if and only if

E

is a

Q [x]

-submodule of

Q_{α + β + γ}

3. Linear Codes and Gray Images over P and Q

In this part, we study the linear codes over P and Q as well as Gray maps. We construct Gray maps with the help of matrices. Gray maps are more intuitive and give better results. We see that P is a semi-local, commutative, and nonchain ring. An element

p_{r}

of P is of the form

p_{r} = a_{1} + v a_{2} + v^{2} a_{3}

such that

v^{3} = α_{2}^{2} v

, where

a_{1}, a_{2}, a_{3} \in F_{q}

In view of Chinese Remainder Theorem, it is clear to observe that

P = E_{1} F_{q} \oplus E_{2} F_{q} \oplus E_{3} F_{q}

, P is a semi-local, commutative, and nonchain ring, and each

p_{r}

has representation

p_{r} = \sum_{i = 1}^{3} Δ_{i} a_{i} = \sum_{i = 1}^{3} E_{i} p_{r_{i}}

, where

a_{i}

p_{r_{i}}

\in F_{q}

, for

i = 1, 2, 3 .

We define the Gray map

φ_{P} : P ⟶ F_{q}^{3}

(1)

φ_{P} (p_{r}) = φ_{P} (E_{1} p_{r_{1}} + E_{2} p_{r_{2}} + E_{3} p_{r_{3}}) = (p_{r_{1}}, p_{r_{2}}, p_{r_{3}}) A_{1} = e A_{1},

where

A_{1} \in G L_{3} (F_{q})

is a fixed matrix and

G L_{3} (F_{q})

is the linear group of all

3 \times 3

invertible matrices over the field

F_{q}

such that

A_{1} A_{1}^{T} = ϵ I_{3 \times 3},

where

A_{1}^{T}

is the transpose of

A_{1}

and

ϵ \in F_{q} ∖ {0}

. Here, we use

e

for the vector

(p_{r_{1}}, p_{r_{2}}, p_{r_{3}})

. With the orthogonal idempotent, we have

P = E_{1} F_{q} \oplus E_{2} F_{q} \oplus E_{3} F_{q}

. Every element

p_{r} \in P

can be uniquely expressed as

p_{r} = E_{1} p_{r_{1}} + E_{2} p_{r_{2}} + E_{3} p_{r_{3}}

, where

p_{r_{i}} \in F_{q}

and

1 \leq i \leq 3

The above-described map (1) can be extended as

φ_{P} : P^{β} ⟶ F_{q}^{3 β}

component-wise as

(p_{0}, p_{1}, \dots, p_{β - 1}) ⟶ ((p_{r_{0, 1}}, p_{r_{0, 2}}, p_{r_{0, 3}}) A_{1}, (p_{r_{1, 1}}, p_{r_{1, 2}}, p_{r_{1, 3}}) A_{1}, \dots, (p_{r_{β - 1, 1}},

p_{r_{β - 1, 2}}, p_{r_{β - 1, 3}}) A_{1}) = e_{0} A_{1}, e_{1} A_{1}, e_{2} A_{1}, \dots, e_{β - 1} A_{1}

, here we take

p_{r} = (p_{0}, p_{1}, \dots, p_{β - 1})

and

p_{i} = E_{1} p_{r_{i, 1}} + E_{2} p_{r_{i, 2}} + E_{3} p_{r_{i, 3}} \in P

, where

i = 0, 1, 2, \dots, β - 1

.We define

w_{L} (p_{i}) = w_{H} (φ_{P} (p_{i}))

, where

w_{L} (p_{i})

denotes the Lee weight of

p_{i}

and

w_{H}

stands for the Hamming weight over

F_{q}

. Let

B_{β}

be linear code of length

β

over P; we define

\begin{matrix} B_{β_{1}} & = & {p_{r_{1}} \in F_{q}^{β} ∣ \sum_{i = 1}^{3} E_{i} p_{r_{i}} \in B_{β}; p_{r_{2}}, p_{r_{3}} \in F_{q}^{β} f o r 1 \leq i \leq 3}, \\ B_{β_{2}} & = & {p_{r_{2}} \in F_{q}^{β} ∣ \sum_{i = 1}^{3} E_{i} p_{r_{i}} \in B_{β}; p_{r_{1}}, p_{r_{3}} \in F_{q}^{β} f o r 1 \leq i \leq 3}, \\ B_{β_{3}} & = & {p_{r_{3}} \in F_{q}^{β} ∣ \sum_{i = 1}^{3} E_{i} p_{r_{i}} \in B_{β}; p_{r_{1}}, p_{r_{2}} \in F_{q}^{β} f o r 1 \leq i \leq 3} . \end{matrix}

Then,

B_{β_{i}}

is a linear code of length

β

over

F_{q}

, for

i = 1, 2, 3 .

Proposition 3.

The Gray map

φ_{P}

is a linear, bijective and distance preserving map from

(P^{β}, d_{L})

(F_{q}^{3 β}, d_{H})

, where

d_{L} = d_{H}

Proof.

Suppose

p_{r}, p_{r}^{'} \in P^{β}

. Then, we have

\begin{matrix} p_{r} & = & E_{1} p_{r_{1}} + E_{2} p_{r_{2}} + E_{3} p_{r_{3}} \\ p_{r}^{'} & = & E_{1} p_{r_{1}}^{'} + E_{2} p_{r_{2}}^{'} + E_{3} p_{r_{3}}^{'} \end{matrix}

\begin{matrix} p_{r} + p_{r}^{'} & = & E_{1} p_{r_{1}} + E_{2} p_{r_{2}} + E_{3} p_{r_{3}} + E_{1} p_{r_{1}}^{'} + E_{2} p_{r_{2}}^{'} + E_{3} p_{r_{3}}^{'} \\ φ_{P} (p_{r} + p_{r}^{'}) & = & (p_{r_{1}} + p_{r_{1}}^{'}, p_{r_{2}} + p_{r_{2}}^{'}, p_{r_{3}} + p_{r_{3}}^{'}) A_{1} \\ = & (p_{r_{1}}, p_{r_{2}}, p_{r_{3}}) A_{1} + (p_{r_{1}}^{'}, p_{r_{2}}^{'}, p_{r_{3}}^{'}) A_{1} \\ = & φ_{P} (p_{r}) + φ_{P} (p_{r}^{'}) \end{matrix}

and we take

μ \in F_{q}

\begin{matrix} φ_{P} (μ p_{r}) & = & φ_{P} (E_{1} p_{r_{1}} + E_{2} p_{r_{2}} + E_{3} p_{r_{3}}) \\ = & (μ p_{r_{1}}, μ p_{r_{2}}, μ p_{r_{3}}) A_{1} \\ = & μ (p_{r_{1}}, p_{r_{2}}, p_{r_{3}}) A_{1} \\ = & μ φ_{P} (p_{r}) . \end{matrix}

So,

φ_{P}

is an

F_{q}

-linear map. Now, we will prove that

φ_{P}

is a bijection.

Then, we have

\begin{matrix} φ_{P} (p_{r}) & = & φ_{P} (p_{r}^{'}) \\ φ_{P} (\sum_{i = 1}^{3} E_{i} p_{r_{i}}) & = & φ_{P} (\sum_{i = 1}^{3} E_{i} p_{r_{i}}^{'}) \\ (p_{r_{1}}, p_{r_{2}}, p_{r_{3}}) A_{1} & = & (p_{r_{1}}^{'}, p_{r_{2}}^{'}, p_{r_{3}}^{'}) A_{1} \end{matrix}

where

p_{r_{i}}, p_{r_{i}}^{'} \in F_{q}^{β}

for

1 \leq i \leq 3

. This implies that

p_{r_{i}} = p_{r_{i}}^{'}, .

Then

p_{r} = p_{r}^{'}

. Henceforth,

φ_{P}

is one-one. Take any

(p_{r_{1}}, p_{r_{2}}, p_{r_{3}}) A_{1} \in F_{q}^{3 β}

; then there exists a corresponding element

p_{r} \in P

such that

φ_{P} (p_{r}) = (p_{r_{1}}, p_{r_{2}}, p_{r_{3}}) A_{1}

. Therefore,

φ_{P}

is an onto function. Hence,

φ_{P}

is a bijective map.

Moreover, we have

\begin{matrix} d_{L} (p_{r}, p_{r}^{'}) & = & w_{L} (p_{r} - p_{r}^{'}) \\ = & w_{H} (φ_{P} (p_{r} - p_{r}^{'})) \\ = & w_{H} (φ_{P} (p_{r}) - φ_{P} (p_{r}^{'})) \\ = & d_{H} (φ_{P} (p_{r}), φ_{P} (p_{r}^{'})) . \end{matrix}

Hence,

φ_{P}

is a distance preserving map. □

Proposition 4.

Let

B_{β} = \oplus_{i = 1}^{3} E_{i} B_{β_{i}}

be a linear code of length β over P. Then

(i).: $φ_{P} (B_{β}) = B_{β_{1}} \otimes B_{β_{2}} \otimes B_{β_{3}} .$
(ii).: $B_{β^{⊥}} = \oplus_{i = 1}^{3} E_{i} B_{β_{i}^{⊥}}$ , further, $B_{β}$ is a self-orthogonal code over P if and only if each $B_{β_{i}}$ is self-orthogonal code over $F_{q}$ and $B_{β}$ is a self-dual code over P if and only if each $B_{β_{i}}$ is self-dual code over $F_{q}$ , for $i = 1, 2, 3 .$

Proof

1. Let

s = (μ_{0}^{1}, μ_{1}^{1}, \dots, μ_{β - 1}^{1}, μ_{0}^{2}, μ_{1}^{2}, \dots, μ_{β - 1}^{2}, μ_{0}^{3}, μ_{1}^{3}, \dots, μ_{β - 1}^{3}) \in φ_{P} (B_{β})

and

t_{j} = \sum_{i = 1}^{3} μ_{j}^{i} E_{i}

, for

1 \leq j \leq β - 1 .

t = (t_{0}, t_{1}, \dots, t_{β - 1}) \in B_{β} .

Since

φ_{P}

is a bijective map,

(μ_{0}^{i}, μ_{1}^{i}, \dots, μ_{β - 1}^{i}) \in B_{β_{i}}

by definition of

B_{β_{i}}

for

i = 1, 2, 3,

and this implies that

s \in B_{β_{1}} \otimes B_{β_{2}} \otimes B_{β_{3}} .

Hence,

φ_{P} (B_{β}) \subseteq B_{β_{1}} \otimes B_{β_{2}} \otimes B_{β_{3}}

Conversely, let

s = (μ_{0}^{1}, μ_{1}^{1}, \dots, μ_{β - 1}^{1}, μ_{0}^{2}, μ_{1}^{2}, \dots, μ_{β - 1}^{2}, μ_{0}^{3}, μ_{1}^{3}, \dots, μ_{β - 1}^{3} \in B_{β_{1}} \otimes B_{β_{2}} \otimes B_{β_{3}}

then

(μ_{0}^{i}, μ_{1}^{i}, \dots, μ_{β - 1}^{i}) \in B_{β_{i}}

for

i = 1, 2, 3 .

We choose

t_{j} = \sum_{i = 1}^{3} μ_{j}^{i} E_{i}

for

1 \leq j \leq β - 1,

then

t = (t_{0}, t_{1}, \dots, t_{β - 1}) \in B_{β}

and

φ_{P} (t) = s .

Hence,

s \in φ (B_{β})

. Moreover, since

φ_{P}

is a bijective map,

∣ B_{β} ∣ = ∣ φ_{P} (B_{β}) ∣ .

, then

∣ B_{β} ∣ = ∣ B_{β_{1}} \otimes B_{β_{2}} \otimes B_{β_{3}} ∣ .

2. Let

D_{j} = {t_{j} \in F_{q}^{β} ∣ \sum_{i = 1}^{3} E_{i} t_{i} \in B_{β}^{⊥}

for some

t_{j} \in F_{q}^{n}

i \neq j

and

1 \leq i, j \leq 3}

. Then,

B_{β}^{⊥}

is uniquely represented as

B_{β}^{⊥} = E_{1} D_{1} \oplus E_{2} D_{2} \oplus E_{3} D_{3}

. Since

D_{1}

= {

t_{1} \in F_{q}^{β}

such that

\sum_{i = 1}^{3} E_{i} t_{i} \in B_{β}^{⊥}

, for some

t_{i} \in F_{q}^{β}

i \neq 1

and

1 \leq i \leq 3

}. Clearly,

B_{β_{1}} D_{1} = 0

, so

D_{1} \subseteq B_{β_{1}}^{⊥} .

Let

E_{1} \in B_{β_{1}}^{⊥}

; then

E_{1} x_{1} = 0

for any

c = \sum_{i = 1}^{4} E_{i} x_{i} \in B_{β} .

E_{1} E_{1} c = E_{1} E_{1} x_{1} = 0

and this implies that

E_{1} E_{1} \in B_{β_{1}}^{⊥}

. We have

E_{1} \in D_{1}

by the unique representation of

B_{β}^{⊥}

, so

B_{β_{1}}^{⊥} \subseteq D_{1} .

Similarly, we can show

B_{β_{j}}^{⊥} = D_{j}^{⊥}

for

i = 1, 2, 3 .

Thus

B_{β}^{⊥} = \oplus_{i = 1}^{3} E_{i} B_{β_{i}}^{⊥}

. Moreover,

B_{β}

is a self-orthogonal code over P if and only if

B_{β} \subseteq B_{β}^{⊥}

. This shows that

E_{1} B_{β_{1}} \oplus E_{2} B_{β_{2}} \oplus E_{3} B_{β_{3}} \subseteq E_{1} B_{β_{1}}^{⊥} \oplus B_{β_{2}} E_{2}^{⊥} \oplus E_{3} B_{β_{3}}^{⊥}

, for

i = 1, 2, 3 .

Hence,

B_{β}

is self-orthogonal code over P if and only if each

B_{β_{i}}

is self orthogonal code over

F_{q}

. Similarly,

B_{β}

is self-dual code over P if and only if each

B_{β_{i}}

is self-dual code over

F_{q}

, for

i = 1, 2, 3 .

□

Proposition 5.

Let

B_{β}

be a linear code of length β over P. If

B_{β}

is a self-orthogonal, then

φ_{P} (B_{β})

is self-orthogonal.

Proof.

Let

p_{r}, p_{r}^{'} \in B_{β}

. Now

p_{r} = (p_{r_{0}}, p_{r_{1}}, \dots, p_{r_{β - 1}}), p_{r}^{'} = (p_{r_{0}}^{'}, p_{r_{1}}^{'}, \dots, p_{r_{β - 1}}^{'})

, where

p_{j} = \sum_{i = 1}^{3} E_{i} p_{r_{i, j}}, p_{j}^{'} = \sum_{i = 1}^{3} E_{i} p_{r_{i, j}}^{'}, p_{r_{i, j}}, p_{r_{i, j}}^{'} \in F_{q}

, for

i = 1, 2, 3

and

j = 0, 1, 2, \dots, β - 1

. Next, let us consider that

p_{r} \cdot p_{r}^{'} = 0

. Then, we obtain

\begin{matrix} \sum_{i = 1}^{β - 1} p_{j} p_{j}^{'} & = & 0 \\ \Rightarrow \sum_{j = 0}^{β - 1} (\sum_{i = 1}^{3} E_{i} p_{r_{i, j}}) (\sum_{i = 1}^{3} E_{i} p_{r_{i, j}}^{'}) & = & 0 . \end{matrix}

Since

{(E_{i})}^{2} = E_{i}

, we have

\begin{matrix} \sum_{j = 0}^{β - 1} \sum_{i = 0}^{3} E_{i} p_{r_{i, j}} p_{r_{i, j}}^{'} & = & \sum_{i = 0}^{3} \sum_{j = 0}^{β - 1} E_{i} p_{r_{i, j}} p_{r_{i, j}}^{'} = 0 . \end{matrix}

Therefore,

\begin{matrix} \sum_{j = 0}^{n - 1} p_{r_{i, j}} p_{r_{i, j}}^{'} & = & 0, \end{matrix}

where

i = 1, 2, 3 .

Also,

\begin{matrix} φ_{P} (p_{r}) φ_{P} (p_{r}^{'}) & = & \sum_{j = 0}^{β - 1} \sum_{i = 0}^{3} p_{r_{i, j}} p_{r_{i, j}}^{'} \\ = & \sum_{i = 0}^{3} \sum_{j = 0}^{β - 1} p_{r_{i, j}} p_{r_{i, j}}^{'} \\ = & 0 . \end{matrix}

This implies that,

\begin{matrix} φ_{P} (C^{⊥}) & \subseteq & φ_{P} {(C)}^{⊥} . \end{matrix}

Since

φ_{P}

is a bijection, then

∣ φ_{P} (C^{⊥}) ∣ = ∣ φ_{P} {(C)}^{⊥} |

. Hence,

φ_{P} (C^{⊥}) = φ_{P} {(C)}^{⊥}

. Now, C is self-orthogonal if and only if

C \subseteq C^{⊥}

. Henceforth,

φ_{P} (C) \subseteq φ_{P} (C^{⊥}) = φ_{P} {(C)}^{⊥}

if and only if

φ_{P} (C)

is self-orthogonal. □

An element

q_{r}

of Q is of the form

q_{r} = b_{1} + u b_{2} + v b_{3} + u v b_{4} + v^{2} b_{5} + u v^{2} b_{6}

, for

b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6} \in F_{q}

. With the help of Chinese remainder theorem, it is clear to observe that

Q = ζ_{1} F_{q} \oplus ζ_{2} F_{q} \oplus ζ_{3} F_{q} \oplus ζ_{4} F_{q} \oplus ζ_{5} F_{q} \oplus ζ_{6} F_{q}

. Hence, Q is semi-local, commutative, and nonchain ring. Moreover, each

q_{r}

has a unique representation,

q_{r} = \sum_{j = 1}^{6} δ_{j} b_{j} = \sum_{j = 1}^{6} ζ_{j} q_{r_{j}}

, where

b_{j}

q_{r_{j}}

\in F_{q}

, for

1 \leq j \leq 6 .

We define the Gray map

φ_{Q} : Q ⟶ F_{q}^{6}

(2)

φ_{Q} (q_{r}) = φ_{Q} (ζ_{1} q_{r_{1}} + ζ_{2} q_{r_{1}} + ζ_{3} q_{r_{3}} + ζ_{4} q_{r_{4}} + ζ_{5} q_{r_{5}} + ζ_{6} q_{r_{6}}) = (q_{r_{1}}, q_{r_{2}}, q_{r_{3}}, q_{r_{4}}, q_{r_{5}}, q_{r_{6}}) A_{2} = s A_{2},

where

A_{2} \in G L_{6} (F_{q})

is a fixed matrix and

G L_{6} (F_{q})

is the linear group of all

6 \times 6

invertible matrices over the field

F_{q}

such that

A_{2} A_{2}^{T} = λ I_{6 \times 6}, A_{2}^{T}

is the transpose of

A_{2}

and

λ \in F_{q} ∖ {0}

and we use

s

for the vector

(q_{r_{1}}, q_{r_{2}}, q_{r_{3}}, q_{r_{4}}, q_{r_{5}}, q_{r_{6}})

The above-described map (2) can be extended component-wise as

φ_{Q} : Q^{β} ⟶ F_{q}^{6 γ}

(q_{0}, q_{1}, \dots, q_{γ - 1}) ⟶ ((q_{r_{0, 1}}, q_{r_{0, 2}}, q_{r_{0, 3}}, q_{r_{0, 4}}, q_{r_{0, 5}}, q_{r_{0, 6}}) A_{2}, (q_{r_{1, 1}}, q_{r_{1, 2}}, q_{r_{1, 3}}, q_{r_{1, 4}}, q_{r_{1, 5}}, q_{r_{1, 6}}) A_{2}, \dots, (q_{r_{γ - 1, 1}}, q_{r_{γ - 1, 2}}, q_{r_{γ - 1, 3}}, q_{r_{γ - 1, 4}}, q_{r_{γ - 1, 5}}, q_{r_{γ - 1, 6}})

A_{2}) = s_{0} A_{2}, s_{1} A_{2}, s_{2} A_{2}, \dots, s_{γ - 1} A_{2}

. We write

q_{r} = (q_{0}, q_{1}, \dots, q_{γ - 1})

and

q_{j} = ζ_{1} q_{r_{j, 1}} + ζ_{2} q_{r_{j, 2}} + ζ_{3} q_{r_{j, 3}} + ζ_{4} q_{r_{j 4}} + ζ_{5} q_{r_{j, 5}} + ζ_{6} q_{r_{j, 6}} \in Q

, where

j = 0, 1, 2, \dots, γ - 1

. We denote

w_{L} (q_{j}) = w_{H} (φ_{Q} (q_{j}))

to represent the Lee weight of

q_{j}

, where

w_{H}

stands for the Hamming weight over

F_{q}

. Let

E_{γ}

be a linear code of length

γ

over Q; we define

\begin{matrix} E_{γ_{1}} & = & {q_{r_{1}} \in F_{q}^{γ} | \sum_{j = 1}^{6} ζ_{j} q_{r_{j}} \in E_{γ}; q_{r_{2}}, q_{r_{3}}, q_{r_{4}}, q_{r_{5}}, q_{r_{6}}, \in F_{q}^{γ}, 1 \leq j \leq 6}, \\ E_{γ_{2}} & = & {q_{r_{2}} \in F_{q}^{γ} | \sum_{j = 1}^{6} ζ_{j} q_{r_{j}} \in E_{γ}; q_{r_{1}}, q_{r_{3}}, q_{r_{4}}, q_{r_{5}}, q_{r_{6}}, \in F_{q}^{γ}, 1 \leq j \leq 6}, \\ E_{γ_{3}} & = & {q_{r_{3}} \in F_{q}^{γ} | \sum_{j = 1}^{6} ζ_{j} q_{r_{j}} \in E_{γ}; q_{r_{1}}, q_{r_{2}}, q_{r_{4}}, q_{r_{5}}, q_{r_{6}}, \in F_{q}^{γ}, 1 \leq j \leq 6}, \\ E_{γ_{4}} & = & {q_{r_{4}} \in F_{q}^{γ} | \sum_{j = 1}^{6} ζ_{j} q_{r_{j}} \in E_{γ}; q_{r_{1}}, q_{r_{2}}, q_{r_{3}}, q_{r_{5}}, q_{r_{6}}, \in F_{q}^{γ}, 1 \leq j \leq 6}, \\ E_{γ_{5}} & = & {q_{r_{5}} \in F_{q}^{γ} | \sum_{j = 1}^{6} ζ_{j} q_{r_{j}} \in E_{γ}; q_{r_{1}}, q_{r_{2}}, q_{r_{3}}, q_{r_{4}}, q_{r_{6}}, \in F_{q}^{γ}, 1 \leq j \leq 6}, \\ E_{γ_{6}} & = & {q_{r_{6}} \in F_{q}^{γ} | \sum_{j = 1}^{6} ζ_{j} q_{r_{j}} \in E_{γ}; q_{r_{1}}, q_{r_{2}}, q_{r_{3}}, q_{r_{4}}, q_{r_{5}}, \in F_{q}^{γ}, 1 \leq j \leq 6} . \end{matrix}

Then,

E_{γ_{j}}

is a linear code of length

γ

over

F_{q}

, for

j = 1, 2, 3, 4, 5, 6 .

We come to the following conclusions for Q using similar justifications to those used in the case of P.

Proposition 6.

The Gray map

φ_{Q}

is a linear and distance preserving map from

(Q^{γ}, d_{L})

(F_{q}^{6 γ}, d_{H})

, where

d_{L} = d_{H}

Proposition 7.

Let

E_{γ} = \oplus_{j = 1}^{6} ζ_{j} E_{γ_{j}}

be a linear code of length γ over Q. Then,

(i).: $φ_{Q} (E_{γ}) = E_{γ_{1}} \otimes E_{γ_{2}} \otimes E_{γ_{3}} \otimes E_{γ_{4}} \otimes E_{γ_{5}} \otimes E_{γ_{6}} .$
(ii).: $E_{γ}^{⊥} = \oplus_{j = 1}^{6} ζ_{j} E_{γ_{j}}^{⊥}$ ; further, $E_{γ}$ is self-orthogonal if and only if $E_{γ_{j}}$ is self-orthogonal, and $E_{γ}$ is self-dual if and only if $E_{γ_{j}}$ is self-dual, for $j = 1, 2, 3, 4, 5, 6 .$

Proposition 8.

Let

E_{γ}

be a linear code of length γ over Q. If

E_{γ}

is a self-orthogonal, then

φ_{Q} (E_{γ})

is self-orthogonal.

4. Gray Image over $F_{q} PQ$

In the present section, we describe the Gray map over

F_{q} P Q

and its related results. In

F_{q} P Q

, every element can be written as

(a, p_{r}, q_{r}) = (a, E_{1} p_{r_{1}} + E_{2} p_{r_{2}} + E_{3} p_{r_{3}}, ζ_{1} q_{r_{1}} + ζ_{2} q_{r_{2}} + ζ_{3} q_{r_{3}} + ζ_{4} q_{r_{4}} + ζ_{5} q_{r_{5}} + ζ_{6} q_{r_{6}})

, where

a, p_{r}, q_{r}

are in

F_{q}, P, Q

, respectively. With the help of the above-described Gray maps (1) and (2), we define new Gary map over

F_{q} P Q

φ : F_{q} P Q ⟶ F_{q}^{10}

(3)

such that

\begin{matrix} φ (a, p_{r}, q_{r}) & = & (a, (p_{r_{1}}, p_{r_{2}}, p_{r_{3}}) A_{1}, (q_{r_{1}}, q_{r_{2}}, q_{r_{3}}, q_{r_{4}}, q_{r_{5}}, q_{r_{6}}) A_{2}) = (a, e A_{1}, s A_{2}) . \end{matrix}

Gray map

φ

is an

F_{q}

-linear and we can easily extend component-wise over

F_{q}^{α} P^{β} Q^{γ}

in the following manner:

φ : F_{q}^{α} \times P^{β} \times Q^{γ} ⟶ F_{q}^{α + 3 β + 6 γ}

is defined by

\begin{matrix} (a_{0}, a_{1}, \dots, a_{α - 1}, p_{0}, p_{1}, \dots, p_{β - 1}, q_{0}, q_{1}, \dots, q_{γ - 1}) & = & (a_{0}, a_{1}, \dots, a_{α - 1}, (p_{r_{0, 1}}, p_{r_{0, 2}}, p_{r_{0, 3}}) A_{1}, \\ (p_{r_{1, 1}}, p_{r_{1, 2}}, p_{r_{1, 3}}) A_{1}, \dots, (p_{r_{β - 1, 1}}, p_{r_{β - 1, 2}}, \\ p_{r_{β - 1, 3}}) A_{1}, (q_{r_{0, 1}}, q_{r_{0, 2}}, q_{r_{0, 3}}, q_{r_{0, 4}}, q_{r_{0, 5}}, \\ q_{r_{0, 6}}) A_{2}, (q_{r_{1, 1}}, q_{r_{1, 2}}, q_{r_{1, 3}}, q_{r_{1, 4}}, q_{r_{1, 5}}, q_{r_{1, 6}}) A_{2}, \\ \dots, (q_{r_{γ - 1, 1}}, q_{r_{γ - 1, 2}}, q_{r_{γ - 1, 3}}, q_{r_{γ - 1, 4}}, q_{r_{γ - 1, 5}}, \\ q_{r_{γ - 1, 6}}) A_{2}), \\ = & (a, e_{0} A_{1}, e_{1} A_{1}, \dots, e_{β - 1} A_{1}, s_{0} A_{2}, s_{1} A_{2}, \\ \dots, s_{γ - 1} A_{2}), \end{matrix}

where

a = (a_{0}, a_{1}, \dots, a_{α - 1}) \in F_{q}^{α}, p_{r} = (p_{0}, p_{1}, \dots, p_{β - 1}) \in P^{β}

and

q_{r} = (q_{0}, q_{1}, \dots, q_{γ - 1}) \in Q^{γ}

. Here, each

p_{i} = E_{1} p_{r_{i}, 1} + E_{2} p_{r_{i}, 2} + E_{3} p_{r_{i}, 3}

q_{j} = ζ_{1} q_{r_{j}, 1} + ζ_{2} q_{r_{j}, 2} + ζ_{3} q_{r_{j}, 3} + ζ_{4} q_{r_{j}, 4} + ζ_{5} q_{r_{j}, 5} + ζ_{6} q_{r_{j}, 6}

are in P and Q respectively, where

i = 0, 1, \dots, β - 1

and

j = 0, 1, 2, \dots, γ - 1

. In the same manner as in [20], we define the Lee weight for the element as

w_{L} (a^{'}, p_{r}^{'}, q_{r}^{'}) = w_{H} (a^{'}) + w_{L} (p_{r}^{'}) + w_{L} (q_{r}^{'}), \forall (a^{'}, p_{r}^{'}, q_{r}^{'}) \in F_{q}^{α} \times P^{β} \times Q^{γ},

where

w_{H}

represents the Hamming weight and

w_{L}

represents the Lee weight. Lee distance between the elements

x^{'}, y^{'} \in F_{q}^{α} \times P^{β} \times Q^{γ}

is defined as

d_{L} (x^{'}, y^{'}) = w_{L} (x^{'} - y^{'}) = w_{H} (φ (x^{'}, y^{'})) .

Next, we give the results on the Gray map over

F_{q} P Q

Proposition 9.

Let φ be the above described Gray map. Then

(i).: φ is an $F_{q}$ -linear and distance preserving map from $F_{q}^{α} P^{β} Q^{γ}$ to $F_{q}^{α + 3 β + 6 γ}$ .
(ii).: If $E$ is a linear code of length $(α + β + γ)$ over $F_{q} P Q$ , then Gray image $φ (E)$ of $E$ is also a linear code with the parameters [ $α + 3 β + 6 γ, k, d_{H}$ ] over $F_{q}$ .

Proof.

(i). We take two arbitrary elements

x^{'}

and

y^{'}

F_{q}^{α} P^{β} Q^{γ}

such that

x^{'} = (a^{1}, p_{r}^{1}, q_{r}^{1})

and

y^{'} = (a^{2}, p_{r}^{2}, q_{r}^{2})

. Here,

\begin{matrix} a^{1} & = & a_{0}^{1}, a_{1}^{1}, a_{2}^{1}, \dots, a_{α - 1}^{1}, \\ a^{2} & = & a_{0}^{2}, a_{1}^{2}, a_{2}^{2}, \dots, a_{α - 1}^{2}, \\ p_{r}^{1} & = & E_{1} p_{r_{1}}^{1} + E_{2} p_{r_{2}}^{1} + E_{3} p_{r_{3}}^{1}, \\ p_{r}^{2} & = & E_{1} p_{r_{1}}^{2} + E_{2} p_{r_{2}}^{2} + E_{3} p_{r_{3}}^{2}, \\ q_{r}^{1} & = & ζ_{1} q_{r_{1}}^{1} + ζ_{2} q_{r_{2}}^{1} + ζ_{3} q_{r_{3}}^{1} + ζ_{4} q_{r_{4}}^{1} + ζ_{5} q_{r_{5}}^{1} + ζ_{6} q_{r_{6}}^{1}, \\ q_{r}^{2} & = & ζ_{1} q_{r_{1}}^{2} + ζ_{2} q_{r_{2}}^{2} + ζ_{3} q_{r_{3}}^{2} + ζ_{4} q_{r_{4}}^{2} + ζ_{5} q_{r_{5}}^{2} + ζ_{6} q_{r_{6}}^{2}, \end{matrix}

and

a^{1}, a^{2}, p_{r}^{1}, p_{r}^{2}

, and

q_{r}^{1}, q_{r}^{2}

are in

F_{q}^{α}, P^{β}

, and

Q^{γ}

, respectively. Also, we have

\begin{matrix} p_{r_{i}}^{1} & = & (p_{r_{i}, 0}^{1}, p_{r_{i}, 1}^{1}, p_{r_{i}, 2}^{1}, \dots, p_{r_{i}, β - 1}^{1}), \\ p_{r_{i}}^{2} & = & (p_{r_{i}, 0}^{2}, p_{r_{i}, 1}^{2}, p_{r_{i}, 2}^{2}, \dots, p_{r_{i}, β - 1}^{2}) \in F_{q}^{β}, \\ q_{r_{j}}^{1} & = & (q_{r_{j}, 0}^{1}, q_{r_{j}, 1}^{1}, q_{r_{j}, 2}^{1}, \dots, q_{r_{j}, γ - 1}^{1}), \\ q_{r_{j}}^{2} & = & (q_{r_{j}, 0}^{2}, q_{r_{j}, 1}^{2}, q_{r_{j}, 2}^{2}, \dots, q_{r_{j}, γ - 1}^{2}) \in F_{q}^{γ} . \end{matrix}

where

1 \leq i \leq 3

and

1 \leq j \leq 6 .

We take

\begin{matrix} φ (x^{'} + y^{'}) & = & (a^{1} + a^{2}, p_{r_{1}}^{1} + p_{r_{1}}^{2}, p_{r_{2}}^{1} + p_{r_{2}}^{2}, p_{r_{3}}^{1} + p_{r_{3}}^{2}, q_{r_{1}}^{1} + q_{r_{1}}^{2}, q_{r_{2}}^{1} + q_{r_{2}}^{2}, q_{r_{3}}^{1} + q_{r_{3}}^{2}, q_{r_{4}}^{1} + q_{r_{4}}^{2}, \\ q_{r_{5}}^{1} + q_{r_{5}}^{2}, q_{r_{6}}^{1} + q_{r_{6}}^{2}) \\ = & (a^{1}, p_{r_{1}}^{1}, p_{r_{2}}^{1}, p_{r_{3}}^{1}, q_{r_{1}}^{1}, q_{r_{2}}^{1}, q_{r_{3}}^{1}, q_{r_{4}}^{1}, q_{r_{5}}^{1}, q_{r_{6}}^{1}) + (a^{2}, p_{r_{1}}^{2}, p_{r_{2}}^{2}, p_{r_{3}}^{2}, q_{r_{1}}^{2}, q_{r_{2}}^{2}, q_{r_{3}}^{2}, q_{r_{4}}^{2}, q_{r_{5}}^{2}, q_{r_{6}}^{2}) \\ = & φ (x^{'}) + φ (y^{'}) \end{matrix}

and also

\begin{matrix} φ (ω x^{'}) & = & (ω a^{1}, ω p_{r_{1}}^{1}, ω p_{r_{2}}^{1}, ω p_{r_{3}}^{1}, ω q_{r_{1}}^{1}, ω q_{r_{2}}^{1}, ω q_{r_{3}}^{1}, ω q_{r_{4}}^{1}, ω q_{r_{5}}^{1}, ω q_{r_{6}}^{1}) \\ = & ω φ (x^{'}), \end{matrix}

where

ω \in F_{q}

. Hence,

φ

is an

F_{q}

-linear map.

For the remaining part, we will use the fact that

φ

is a linear map, so we have

d_{L} (x^{'}, y^{'}) = w_{L} (x^{'} - y^{'}) = w_{H} (x^{'} - y^{'}) = d_{H} (x^{'}, y^{'}) .

Therefore, the result is follows.

(ii). This is directly by the definition of Gray map

φ

. □

In the next step, we define the quasi cyclic code and generalized quasi cyclic code as follows:

Definition 8.

Suppose

ω \in F_{q}^{m n}

such that

ω = (ω_{1}, ω_{2}, \dots, ω_{n})

, where

ω_{i} \in F_{q}^{m}

for

i = 1, 2, \dots, n

. Let ξ be the cyclic shift from

F_{q}^{m}

F_{q}^{m}

and defined as

ξ (a_{0}, a_{1}, \dots, a_{m - 1}) = (a_{m - 1}, a_{0}, \dots, a_{m - 2}) .

We define another map from

F_{q}^{m n}

F_{q}^{m n}

such that

Ω (ω_{1}, ω_{2}, \dots, ω_{n}) = (ξ (ω_{1}), ξ (ω_{2}), \dots, ξ (ω_{n})),

where

ω \in F_{q}^{m n}

such that

ω = (ω_{1}, ω_{2}, \dots, ω_{n})

. From here, a code

E

is known as a quasi-cyclic code of index n if

Ω (E) = E

Definition 9.

Let

ω^{'} \in F_{q}^{m_{1}} \times F_{q}^{m_{2}} \times F_{q}^{m_{3}} \times \dots \times F_{q}^{m_{n}}

such that

ω^{'} = (ω_{1}^{'}, ω_{2}^{'}, \dots, ω_{n}^{'})

, where

ω_{i}^{'} \in F_{q}^{m_{i}}

such that

i = 1, 2, 3, \dots, n

. Now, again, let ξ be the cyclic shift from

F_{q}^{m_{i}}

F_{q}^{m_{i}}

ξ : F_{q}^{m_{i}} ⟶ F_{q}^{m_{i}}

and defined as

ξ (a_{0}, a_{1}, \dots, a_{m_{i} - 1}) = (a_{m_{i} - 1}, a_{0}, \dots, a_{m_{i} - 2}) .

Next, we define another map as

Ω_{g} : F_{q}^{m_{1}} \times F_{q}^{m_{2}} \times \dots \times F_{q}^{m_{n}} ⟶ F_{q}^{m_{1}} \times F_{q}^{m_{2}} \times \dots \times F_{q}^{m_{n}}

such that

Ω_{g} (ω_{1}^{'}, ω_{2}^{'}, \dots, ω_{n}^{'}) = (ξ (ω_{1}^{'}), ξ (ω_{2}^{'}), \dots, ξ (ω_{n}^{'})) .

A code

E

is called a generalized quasi-cyclic code if

Ω_{g} (E) = E

In view of the above definition, we prove the following result:

Theorem 1.

Let ξ be the cyclic shift over

F_{q} P Q

, and let φ and

Ω_{g}

be the mappings described above. Prove that

φ_{ξ} = Ω_{g} φ

Proof.

Let

c = (a_{0}, a_{1}, \dots, a_{α - 1}, p_{0}, p_{1}, \dots, p_{β - 1}, q_{0}, q_{1}, \dots, q_{γ - 1}) \in F_{q}^{α} P^{β} Q^{γ}

, where each

p_{i} = E_{1} p_{r_{i}, 1} + E_{2} p_{r_{i}, 2} + E_{3} p_{r_{i}, 3} \in P

and

q_{j} = ζ_{1} q_{r_{j}, 1} + ζ_{2} q_{r_{j}, 2} + ζ_{3} q_{r_{j}, 3} + ζ_{4} q_{r_{j}, 4} + ζ_{5} q_{r_{j}, 5} + ζ_{6} q_{r_{j}, 6} \in Q

for

i = 0, 1, 2, \dots, β - 1

and

j = 0, 1, 2, \dots, γ - 1

. Now, we take

\begin{matrix} φ_{ξ} (c) & = & φ_{ξ} (a_{0}, a_{1}, \dots, a_{α - 1}, p_{0}, p_{1}, \dots, p_{β - 1}, q_{0}, q_{1}, \dots, q_{γ - 1}) \\ = & φ (a_{α - 1}, a_{0}, a_{1}, \dots, a_{α - 2}, p_{β - 1}, p_{0}, p_{1}, \dots, p_{β - 2}, q_{γ - 1}, q_{0}, q_{1}, \dots, q_{γ - 2}) \\ = & (a_{α - 1}, a_{0}, a_{1}, \dots, a_{α - 2}, (p_{r_{β - 1, 1}}, p_{r_{β - 1, 2}}, p_{r_{β - 1, 3}}) A_{1}, (p_{r_{0}, 1}, p_{r_{0}, 2}, p_{r_{0}, 3}) A_{1}, (p_{r_{1}, 1}, p_{r_{1}, 2}, p_{r_{1}, 3}) A_{1}, \\ \dots, (p_{r_{β - 2}, 1}, p_{r_{β - 2, 2}}, p_{r_{β - 2, 3}}) A_{1}, (q_{r_{γ - 1, 1}}, q_{r_{γ - 1, 2}}, q_{r_{γ - 1, 3}}, q_{r_{γ - 1, 4}}, q_{r_{γ - 1, 5}}, q_{r_{γ - 1, 6}}) A_{2}, (q_{r_{0}, 1}, q_{r_{0}, 2}, \\ q_{r_{0}, 3}, q_{r_{0}, 4}, q_{r_{0}, 5}, q_{r_{0}, 6}) A_{2}, (q_{r_{1}, 1}, q_{r_{1}, 2}, q_{r_{1}, 3}, q_{r_{1}, 4}, q_{r_{1}, 5}, q_{r_{1}, 6}) A_{2}, \dots, (q_{r_{γ - 2, 1}}, q_{r_{γ - 2, 2}}, q_{r_{γ - 2, 3}}, \\ q_{r_{γ - 2}, 4}, q_{r_{γ - 2, 5}}, \dots, q_{r_{γ - 2}, 6}) A_{2}) . \end{matrix}

After that, we will obtain

\begin{matrix} Ω_{g} φ (E) & = & Ω_{g} φ (a_{0}, a_{1}, \dots, a_{α - 1}, p_{0}, p_{1}, \dots, p_{β - 1}, q_{0}, q_{1}, \dots, q_{γ - 1}) \\ = & Ω (a_{0}, a_{1}, \dots, a_{α - 1}, (p_{r_{0}, 1}, p_{r_{0}, 2}, p_{r_{0}, 3}) A_{1}, (p_{r_{1}, 1}, p_{r_{1}, 2}, p_{r_{1}, 3}) A_{1}, \dots, (p_{r_{β - 1}, 1}, p_{r_{β - 1}, 2}, p_{r_{β - 1}, 3}) A_{1}, \\ (q_{r_{0}, 1}, q_{r_{0}, 2}, q_{r_{0}, 3}, q_{r_{0}, 4}, q_{r_{0}, 5}, q_{r_{0}, 6}) A_{2}, (q_{r_{1}, 1}, q_{r_{1}, 2}, q_{r_{1}, 3}, q_{r_{1}, 4}, q_{r_{1}, 5}, q_{r_{1}, 6}) A_{2}, \dots, (q_{r_{γ - 1}, 1}, q_{r_{γ - 1}, 2}, \\ q_{r_{γ - 1}, 3}, q_{r_{γ - 1}, 4}, q_{r_{γ - 1}, 5}, q_{r_{γ - 1}, 6}) A_{2}) \\ = & (a_{α - 1}, a_{0}, a_{1}, \dots, a_{α - 2}, (p_{r_{β - 1}, 1}, p_{r_{β - 1}, 2}, p_{r_{β - 1}, 3}) A_{1}, (p_{r_{0}, 1}, p_{r_{0}, 2}, p_{r_{0}, 3}) A_{1}, (p_{r_{1}, 1}, p_{r_{1}, 2}, p_{r_{1}, 3}) A_{1}, \\ \dots, (p_{r_{β - 2}, 1}, p_{r_{β - 2}, 2}, p_{r_{β - 2}, 3}) A_{1}, (q_{r_{γ - 1}, 1}, q_{r_{γ - 1}, 2}, q_{r_{γ - 1}, 3}, q_{r_{γ - 1}, 4}, q_{r_{γ - 1}, 5}, q_{r_{γ - 1}, 6}) A_{2}, (q_{r_{0}, 1}, q_{r_{0}, 2}, \\ q_{r_{0}, 3}, q_{r_{0}, 4}, q_{r_{0}, 5}, q_{r_{0}, 6}) A_{2}, (q_{r_{1}, 1}, q_{r_{1}, 2}, q_{r_{1}, 3}, q_{r_{1}, 4}, q_{r_{1}, 5}, q_{r_{1}, 6}) A_{2}, \dots, (q_{r_{γ - 2}, 1}, q_{r_{γ - 2}, 2}, q_{r_{γ - 2}, 3}, \\ q_{r_{γ - 2}, 4}, q_{r_{γ - 2}, 5}, \dots, q_{r_{γ - 2}, 6}) A_{2}) . \end{matrix}

Hence, we conclude that,

φ_{ξ} = Ω_{g} φ

. □

In view of Theorem 1, we obtain following result.

Theorem 2.

Let

E

be a linear code of length

(α + β + γ)

over

F_{q} P Q

. Then, the Gray image of a

F_{q} P Q

-cyclic code of length

(α + β + γ)

is a generalized quasi-cyclic code with an index of 10 over

F_{q}

5. Main Results

In this section, we describe the structural properties of cyclic codes over

F_{q}, P, Q

, and

F_{q} P Q

as well as obtain quantum error-correcting codes over

F_{q} P Q

5.1. Cyclic Codes over $F_{q}$

Theorem 3

([28], Theorem 12.9). Let A be a cyclic code of length α of over

F_{q}

. Then there exists a unique polynomial

a (κ) \in \frac{F_{q} [κ]}{〈 κ^{α} - 1 〉}

such that

A_{α} = 〈 a (κ) 〉

and

a (κ) | (κ^{α} - 1)

. Moreover, the dimension of

A_{α}

r = α - d e g (a)

with

{a (κ), κ a (κ), \dots, κ^{r - 1} a (κ)}

as a basis.

5.2. Cyclic Codes over P

Theorem 4.

Let

B_{β} = \oplus_{i = 1}^{3} E_{i} B_{β_{i}}

be a linear code of length β over P. Then

B_{β}

is a cyclic code of length β over P if and only if each

B_{β_{i}}

is a cyclic code over

F_{q}

, where

i = 1, 2, 3

Proof.

For any

p_{r} = (p_{0}, p_{1}, p_{2}, \dots, p_{β - 1}) \in B_{β}

. We can also have,

p_{i} = E_{1} p_{r_{i, 1}} + E_{2} p_{r_{i, 2}} + E_{3} p_{r_{i, 3}},

where

p_{r_{i, 1}}, p_{r_{i, 2}}, p_{r_{i, 3}} \in F_{q}

and also

p_{r_{1}} = (p_{r_{1, 0}}, p_{r_{1, 1}}, p_{r_{1, 2}}, \dots, p_{r_{1, β - 1}}) \in B_{β_{1}},

p_{r_{2}} = (p_{r_{2, 0}}, p_{r_{2, 1}}, p_{r_{2, 2}}, \dots, p_{r_{2, β - 1}}) \in B_{β_{2}},

p_{r_{3}} = (p_{r_{3, 0}}, p_{r_{3, 1}}, p_{r_{3, 2}}, \dots, p_{r_{3, β - 1}}) \in B_{β_{2}} .

Here,

p_{r_{1}}, p_{r_{2}}, p_{r_{3}}

are in

B_{β_{1}}, B_{β_{2}}, B_{β_{3}}

respectively. Next, let us consider that

B_{β_{1}}, B_{β_{2}}, B_{β_{3}}

are cyclic code over

F_{q} .

It means that

ξ (p_{r_{1}}) = (p_{r_{1, β - 1}}, p_{r_{1, 0}}, p_{r_{1, 1}}, p_{r_{1, 2}}, \dots, p_{r_{1, β - 2}}) \in B_{β_{1}},

ξ (p_{r_{2}}) = (p_{r_{2, β - 1}}, p_{r_{2, 0}}, p_{r_{2, 1}}, p_{r_{2, 2}}, \dots, p_{r_{2, β - 2}}) \in B_{β_{2}},

ξ (p_{r_{3}}) = (p_{r_{3, β - 1}}, p_{r_{3, 0}}, p_{r_{3, 1}}, p_{r_{3, 2}}, \dots, p_{r_{3, β - 2}}) \in B_{β_{3}} .

Hence, we have

ξ (p_{i}) = E_{1} ξ (p_{r_{i, 1}}) + E_{2} ξ (p_{r_{i, 2}}) + E_{3} ξ (p_{r_{i, 3}}) \in B_{β} .

This gives

E_{1} ξ (p_{r_{1}}) + E_{2} ξ (p_{r_{2}}) + E_{3} ξ (p_{r_{3}}) = ξ (p_{r}) .

Thus, we obtain,

ξ (p_{r}) \in B_{β}

. This implies that

B_{β}

is a cyclic code over P.

Conversely, we consider that

B_{β}

is a cyclic code over P. Suppose

p_{r_{i}} = E_{1} p_{r_{i, 1}} + E_{2} p_{r_{i, 2}} + E_{3} p_{r_{i, 3}},

where

p_{r_{i, 1}}, p_{r_{i, 2}}, p_{r_{i, 3}} \in F_{q}

. Then, for any

p_{r_{1}} = (p_{r_{1, 0}}, p_{r_{1, 1}}, p_{r_{1, 2}}, \dots, p_{r_{1, β - 1}}) \in B_{β_{1}},

p_{r_{2}} = (p_{r_{2, 0}}, p_{r_{2, 1}}, p_{r_{2, 2}}, \dots, p_{r_{2, β - 1}}) \in B_{β_{2}},

p_{r_{3}} = (p_{r_{3, 0}}, p_{r_{3, 1}}, p_{r_{3, 2}}, \dots, p_{r_{3, β - 1}}) \in B_{β_{3}} .

Here,

p_{r_{1}}, p_{r_{2}}, p_{r_{3}}

are in

B_{β_{1}}, B_{β_{2}}, B_{β_{3}}

, respectively. Thus,

p_{r} = (p_{0}, p_{1}, \dots, p_{β - 1}) \in B_{β}

. By the hypothesis,

ξ (p_{r}) \in B_{β}

because

E_{1} ξ (p_{r_{1}}) + E_{2} ξ (p_{r_{2}}) + E_{3} ξ (p_{r_{3}}) = ξ (p_{r})

Then, we have

E_{1} ξ (p_{r_{1}}) + E_{2} ξ (p_{r_{2}}) + E_{3} ξ (p_{r_{3}}) \in B_{β} .

Therefore,

ξ (p_{r_{1}}) \in B_{β, 1}, ξ (p_{r_{2}}) \in B_{β, 2}, ξ (p_{r_{3}}) \in B_{β, 3} .

This shows that

B_{β_{1}}, B_{β_{2}}

and

B_{β_{3}}

are cyclic codes over

F_{q}

. □

Corollary 1.

Let

B_{β} = \oplus_{i = 1}^{3} E_{i} B_{β_{i}}

be a cyclic code of length β over P. Then,

B_{β}^{⊥} = \oplus_{i = 1}^{3} E_{i} B_{β_{i}}^{⊥}

is also a cyclic code of length β over P if and only if

B_{β_{i}}^{⊥}

are cyclic codes of length β over

F_{q}

, for

i = 1, 2, 3

Theorem 5.

Let

B_{β} = \oplus_{i = 1}^{3} E_{i} B_{β_{i}}

be a cyclic code of length β over P and

h_{i} (κ)

be the generator polynomial of the cyclic code

B_{β_{i}}

, where

i = 1, 2, 3 .

Then,

B_{β} = 〈 E_{1} h_{1} (κ), E_{2} h_{2} (κ), E_{3} h_{3} (κ) 〉

and

∣ B_{β} ∣ = q^{3 β - \sum_{i = 1}^{3} d e g (h_{i})}

Proof.

Let

B_{β}

be a cyclic code of length

β

over P. Then, by Theorem 4,

B_{β_{1}} = 〈 h_{1} (κ) 〉 \subseteq \frac{F_{q} [κ]}{〈 κ^{β} - 1 〉},

B_{β_{2}} = 〈 h_{2} (κ) 〉 \subseteq \frac{F_{q} [κ]}{〈 κ^{β} - 1 〉},

B_{β_{3}} = 〈 h_{3} (κ) 〉 \subseteq \frac{F_{q} [κ]}{〈 κ^{β} - 1 〉},

and also

B_{β} = E_{1} B_{β_{1}} \oplus E_{2} B_{β_{2}} \oplus E_{3} B_{β_{3}},

where

h_{1} (κ) \in B_{β_{1}}, h_{2} (κ) \in B_{β_{2}}, h_{3} (κ) \in B_{β_{3}} .

Therefore,

\begin{matrix} B_{β} & \subseteq & 〈 E_{1} h_{1} (κ), E_{2} h_{2} (κ), E_{3} h_{3} (κ) 〉 \\ \subseteq & \frac{P [κ]}{〈 κ^{β} - 1 〉} . \end{matrix}

We take any

\begin{matrix} E_{1} k_{r_{1}} (κ) h_{1} (κ) + E_{2} k_{r_{2}} (κ) h_{2} (κ) + E_{3} k_{r_{3}} (κ) h_{3} (κ) & \in & 〈 E_{1} h_{1} (κ), E_{2} h_{2} (κ), E_{3} h_{3} (κ) 〉 \\ \subseteq & \frac{P [κ]}{〈 κ^{β} - 1 〉} . \end{matrix}

Here,

k_{r_{1}} (κ), k_{r_{2}} (κ), k_{r_{3}} (κ) \in \frac{P [κ]}{〈 κ^{β} - 1 〉}

and also

h_{1} (κ), h_{2} (κ), h_{3} (κ) \in F_{q} [κ]

such that

\begin{matrix} E_{1} k_{r_{1}} (κ) & = & E_{1} h_{1} (κ), \\ E_{2} k_{r_{2}} (κ) & = & E_{2} h_{2} (κ), \\ E_{3} k_{r_{3}} (κ) & = & E_{3} h_{3} (κ) . \end{matrix}

This means that

〈 E_{1} h_{1} (κ), E_{2} h_{2} (κ), E_{3} h_{3} (κ) 〉 \subseteq B_{β}

. From the above discussion, we conclude that

〈 E_{1} h_{1} (κ), E_{2} h_{2} (κ), E_{3} h_{3} (κ) 〉 = B_{β}

. But,

B_{β} = | B_{β_{1}} | | B_{β_{2}} | | B_{β_{3}} | .

This yields that

\begin{matrix} | B_{β} | & = & q^{β - d e g (h_{1} (κ))} q^{β - d e g (h_{2} (κ))} q^{β - d e g (h_{3} (κ))} \\ = & q^{3 β - (d e g (h_{1} (κ)) + d e g (h_{2} (κ)) + d e g (h_{3} (κ)))} \\ | B_{β} | & = & q^{3 β - \sum_{i = 1}^{3} d e g (h_{i})} . \end{matrix}

□

Theorem 6.

Let

B_{β} = \oplus_{i = 1}^{3} E_{i} B_{β_{i}}

be a cyclic code of length β over P and

h_{i} (κ)

be the generator polynomial of the cyclic code

B_{β_{i}}

, where

i = 1, 2, 3 .

Suppose there exists a unique polynomial

h (κ) \in P (κ)

such that

B_{β} = 〈 h (κ) 〉

and

h (κ)

divides

κ^{β} - 1

and also

h (κ) = E_{1} h_{1} (κ) + E_{2} h_{2} (κ) + E_{3} h_{3} (κ)

Proof.

In view of Theorem 5, let

C = 〈 E_{1} h_{1} (κ), E_{2} h_{2} (κ), E_{3} h_{3} (κ) 〉

and

h_{1} (κ), h_{2} (κ), h_{3} (κ)

be the monic generator polynomials of

B_{β_{1}}, B_{β_{2}}, B_{β_{3}}

, respectively. Next, let us consider that

h (κ) = E_{1} h_{1} (κ) + E_{2} h_{2} (κ) + E_{3} h_{3} (κ)

. Obviously,

〈 h (κ) 〉 \subseteq B_{β}

. Now,

\begin{matrix} E_{1} h_{1} (κ) & = & E_{1} h (κ), \\ E_{2} h_{2} (κ) & = & E_{2} h (κ), \\ E_{3} h_{3} (κ) & = & E_{3} h (κ) . \end{matrix}

Also, it means that

B_{β} \subseteq 〈 h (κ) 〉

. This is clear from above discussion that

B_{β} = 〈 h (κ) 〉

. But

h_{1} (κ), h_{2} (κ), h_{3} (κ)

are the monic divisor of

(κ^{β} - 1)

. There are

t_{r_{1}} (κ), t_{r_{2}} (κ), t_{r_{3}} (κ) \in \frac{F_{q} [κ]}{〈 κ^{β} - 1 〉}

. This implies that

[E_{1} t_{r_{1}} (κ) + E_{2} t_{r_{2}} (κ) + E_{3} t_{r_{3}} (κ)] h (κ) = κ^{β} - 1 .

Therefore,

h (κ) / (κ^{β} - 1)

. Hence,

h (κ)

is unique by the uniqueness of

h_{1} (κ), h_{2} (κ), h_{3} (κ)

. □

Corollary 2.

Let

B_{β} = \oplus_{i = 1}^{3} E_{i} B_{β_{i}}

be a cyclic code of length β over P,

h_{i} (κ)

be the generator polynomial of the cyclic code

B_{β_{i}}

and

e_{i}^{*} (κ)

is the reciprocal polynomials of

e_{i} (κ)

such that

κ^{β} - 1 = e_{i} (κ) h_{i} (κ)

for

i = 1, 2, 3 .

Then,

B_{β}^{⊥} = 〈 E_{1} e_{1}^{*} (κ), E_{2} e_{2}^{*} (κ), E_{3} e_{3}^{*} (κ) 〉

and

∣ B_{β}^{⊥} ∣ = q^{\sum_{i = 1}^{3} d e g (h_{i})}

5.3. Cyclic Codes over Q

We arrive at the following conclusions for cyclic codes over Q using similar justifications to those used in the case of cyclic codes over P.

Theorem 7.

Let

E_{γ} = \oplus_{j = 1}^{6} ζ_{j} E_{γ_{j}}

be a linear code of length γ over Q. Then

E_{γ}

is a cyclic code of length γ over Q if and only if each

E_{γ_{j}}

is a cyclic code over

F_{q}

, where

j = 1, 2, 3, 4, 5, 6

Corollary 3.

Let

E_{γ} = \oplus_{j = 1}^{6} ζ_{j} E_{γ_{j}}

be a cyclic code of length γ over Q. Then,

E_{γ}^{⊥} = \oplus_{j = 1}^{6} ζ_{j} (E_{γ_{j}}^{⊥}

is also a cyclic code of length γ over Q if and only if each

E_{γ_{j}}^{⊥}

is a cyclic code of length γ over

F_{q}

, for

i = 1, 2, 3, 4, 5, 6

Theorem 8.

Let

E_{γ} = \oplus_{j = 1}^{6} ζ_{j} E_{γ_{j}}

be a cyclic code of length γ over Q and

ℓ_{j} (κ)

be the generator polynomial of the cyclic code

E_{γ_{j}}

, where

j = 1, 2, 3, 4, 5, 6 .

Then,

E_{γ} = 〈 ℓ (κ) 〉

and

∣ E_{γ} ∣ = q^{3 γ - \sum_{j = 1}^{6} d e g (ℓ_{j})}

, where

ℓ (κ) = ζ_{1} ℓ_{1} (κ) + ζ_{2} ℓ_{2} (κ) + ζ_{3} ℓ_{3} (κ) + ζ_{4} ℓ_{4} (κ) + ζ_{5} ℓ_{5} (κ) + ζ_{6} ℓ_{6} (κ)

Theorem 9.

Let

E_{γ} = \oplus_{j = 1}^{6} ζ_{j} E_{γ_{j}}

be a cyclic code of length γ over Q and

ℓ_{j} (κ)

be the generator polynomial of the cyclic code

E_{γ_{j}}

, where

j = 1, 2, 3, 4, 5, 6 .

Suppose there exists a unique polynomial

ℓ (κ) \in Q (κ)

such that

E_{γ} = 〈 ℓ (κ) 〉

and

ℓ (κ)

divides

κ^{γ} - 1

and also

ℓ (κ) = ζ_{1} ℓ_{1} (κ) + ζ_{2} ℓ_{2} (κ) + ζ_{3} ℓ_{3} (κ) + ζ_{4} ℓ_{4} (κ) + ζ_{5} ℓ_{5} (κ) + ζ_{6} ℓ_{6} (κ)

Corollary 4.

Let

E_{γ} = \oplus_{j = 1}^{6} ζ_{j} E_{γ_{j}}

be a cyclic code of length γ over Q. Suppose

ℓ_{j} (κ)

is the generator polynomial of the cyclic code

E_{γ_{j}}

and

k_{j}^{*} (κ)

is the reciprocal polynomials of

k_{j} (κ)

such that

κ^{γ} - 1 = k_{j} (κ) ℓ_{j} (κ)

for

j = 1, 2, 3, 4, 5, 6 .

Then,

E_{γ}^{⊥} = 〈 ζ_{1} k_{1}^{*} (κ), ζ_{2} k_{2}^{*} (κ), ζ_{3} k_{3}^{*} (κ), ζ_{4} k_{4}^{*} (κ), ζ_{5} k_{5}^{*} (κ), ζ_{6} k_{6}^{*} (κ), 〉

and

| (E_{γ}^{⊥} | = q^{\sum_{j = 1}^{6} d e g (ℓ_{j}^{*})}

5.4. Cyclic Codes over $F_{q} P Q$

In the present section, we discuss the generator polynomial of

E

over

F_{q} P Q

. We begin with the following result:

Theorem 10.

Let

E

be a cyclic code over

F_{q} P Q

. Then,

E = 〈 (a (κ) | 0 | 0), (0 | h (κ) | 0), (r_{1} (κ) | r_{2} (κ) | ℓ (κ)) 〉,

where

a (κ) | (κ^{α} - 1), h (κ) | (κ^{β} - 1), ℓ (κ) | (κ^{γ} - 1)

and also

r_{1} (κ) \in \frac{F_{q} [κ]}{〈 κ^{α} - 1 〉}

r_{2} (κ) \in \frac{F_{q} [κ]}{〈 κ^{β} - 1 〉}

Proof.

In view of Theorems 3, 5 and 8, we define

\begin{matrix} A_{α} & = & 〈 a (κ) 〉, \\ B_{β} & = & 〈 h (κ) 〉, \\ E_{γ} & = & 〈 ℓ (κ) 〉, \end{matrix}

where

a (κ) | (κ^{α} - 1), h (κ) | (κ^{β} - 1), ℓ (κ) | (κ^{γ} - 1)

. Then, the proof is similar as in [20]. □

Definition 10

([25]). A

F_{q} P Q

-linear code

E

of length

(α + β + γ)

is called a separable code if

E = A_{α}^{'} \otimes B_{β}^{'} \otimes E_{γ}^{'}

while considering

A_{α}^{'}

B_{β}^{'}

and

E_{γ}^{'}

as punctured code of

E

by deleting the coordinate outside the

α, β

and γ components, respectively.

Lemma 1.

Let

E = 〈 (a (κ) | 0 | 0), (0 | h (κ) | 0), (r_{1} (κ) | r_{2} (κ) | ℓ (κ)) 〉

be a

F_{q} P Q

-cyclic code. Then,

(i): $d e g (r_{1} (κ)) \leq d e g (a (κ))$ , $d e g (r_{2} (κ)) \leq d e g (h (κ))$ and $a (κ) | h (κ) r_{1} (κ)$ , $h (κ) | ℓ (κ) r_{2} (κ)$ .
(ii): $A_{α}^{'} = 〈 g c d (a (κ), r_{1} (κ)) 〉$ , $B_{β}^{'} = 〈 g c d (h (κ), r_{2} (κ)) 〉$ and $E_{γ}^{'} = 〈 ℓ (κ) 〉$ .

Proof.

The proof is parallel to that of Lemmas 3.2, 3.3 and 3.4 of [20]. □

Lemma 2.

Let

E = 〈 (a (κ) | 0 | 0), (0 | h (κ) | 0), (r_{1} (κ) | r_{2} (κ) | ℓ (κ)) 〉

be a

F_{q} P Q

-cyclic code. Then,

(i): $a (κ) | r_{1} (κ)$ if and only if $r_{1} (κ) = 0$ .
(ii): $h (κ) | r_{2} (κ)$ if and only if $r_{2} (κ) = 0$ .

Proof.

Proof is parallel to that of Lemmas

5.8

and

5.9

of [20]. □

The following Lemma is a direct consequence of Lemma 2.

Lemma 3.

Let

E = 〈 (a (κ) | 0 | 0), (0 | h (κ) | 0), (r_{1} (κ) | r_{2} (κ) | ℓ (κ)) 〉

be a

F_{q} P Q

-cyclic code. Then, the following are equivalent:

(i): $E$ is separable.
(ii): $a (κ) | r_{1} (κ)$ , $h (κ) ∣ r_{2} (κ)$ .
(iii): $E = 〈 (a (κ) ∣ 0 ∣ 0), (0 ∣ h (κ) ∣ 0), (0 ∣ 0 ∣ ℓ (κ)) 〉$ .

Consequently, for a separable code, we have

\begin{matrix} A_{α}^{'} & = & 〈 g c d (a (κ), r_{1} (κ)) 〉 = 〈 a (κ) 〉 = A_{α}, \\ B_{β}^{'} & = & 〈 g c d (h (κ), r_{2} (κ)) 〉 = 〈 h (κ) 〉 = B_{β}, \\ E_{γ}^{'} & = & 〈 ℓ (κ) 〉 = E_{γ} . \end{matrix}

Theorem 11.

Let

E = A_{α} \otimes B_{β} \otimes E_{γ}

be a

F_{q} P Q

-linear code of length

(α + β + γ)

, where

A_{α}, B_{β}

and

E_{γ}

are linear code of

α, β

and γ, respectively. Then,

E

is a

F_{q} P Q

cyclic code of length

(α + β + γ)

if and only if

A_{α}, B_{β}

and

E_{γ}

are cyclic codes of length

α, β

and γ over

F_{q}, P

and Q, respectively.

Proof.

First, we suppose that

E

is a

F_{q} P Q

-cyclic code of length

(α + β + γ)

and

c \in E

, where

c = (a_{0}, a_{1}, \dots, a_{α - 1}, p_{0}, p_{1}, \dots, p_{β - 1}, q_{0}, q_{1}, \dots, q_{γ - 1})

and also

\begin{matrix} (a_{0}, a_{1}, \dots, a_{α - 1}) & \in A_{α}, \\ (p_{0}, p_{1}, \dots, p_{β - 1}) & \in B_{β}, \\ (q_{0}, q_{1}, \dots, q_{γ - 1}) & \in E_{γ} . \end{matrix}

By the definition of cyclic code, we have

(a_{α - 1}, a_{0}, a_{1}, \dots, a_{α - 2}, p_{β - 1} p_{0}, p_{1}, \dots, p_{β - 2}, q_{γ - 1}, q_{0}, q_{1}, \dots, q_{γ - 2}) \in E

Now,

\begin{matrix} (a_{α - 1}, a_{0}, a_{1}, \dots, a_{α - 2}) & \in A_{α}, \\ (p_{β - 1}, p_{0}, p_{1}, \dots, p_{β - 2}) & \in B_{β}, \\ (q_{γ - 1}, q_{0}, q_{1}, \dots, q_{γ - 2}) & \in E_{γ} . \end{matrix}

Hence,

A_{α}, B_{β}

and

E_{γ}

are cyclic codes of length

α, β

and

γ

over

F_{q}, P

and Q, respectively.

For the converse part, we consider that

A_{α}, B_{β}

and

E_{γ}

are cyclic codes of length

α, β

and

γ

over

F_{q}, P

and Q, respectively, and next, we will prove that

E = A_{α} \otimes B_{β} \otimes E_{γ}

is a cyclic code over

F_{q} P Q

. Hence,

\begin{matrix} (a_{0}, a_{1}, \dots, a_{α - 1}) & \in A_{α}, \\ (p_{0}, p_{1}, \dots, p_{β - 1}) & \in B_{β}, \\ (q_{0}, q_{1}, \dots, q_{γ - 1}) & \in E_{γ} . \end{matrix}

But, all are cyclic, and we have

\begin{matrix} (a_{α - 1}, a_{0}, a_{1}, \dots, a_{α - 2}) & \in A_{α}, \\ (p_{β - 1}, p_{0}, p_{1}, \dots, p_{β - 2}) & \in B_{β}, \\ (q_{γ - 1}, q_{0}, q_{1}, \dots, q_{γ - 2}) & \in E_{γ} . \end{matrix}

Thus,

(a_{α - 1}, a_{0}, a_{1}, \dots, a_{α - 2}, p_{β - 1}, p_{0}, p_{1}, \dots, p_{β - 2}, q_{γ - 1}, q_{0}, q_{1}, \dots, q_{γ - 2}) \in A_{α} \otimes B_{β} \otimes E_{γ} = E .

Consequently,

E

is a

F_{q} P Q

-cyclic code of length

(α + β + γ) .

□

In view of Theorems 4, 7 and 11, we have the following result:

Corollary 5.

Let

E = A_{α} \otimes B_{β} \otimes E_{γ}

be a

F_{q} P Q

-linear code of length

(α + β + γ)

such that

A_{α}, B_{β}

and

E_{γ}

are the linear codes of length

α, β

and γ over

F_{q}

, P and Q, respectively. Then,

E

is a

F_{q} P Q

-cyclic code of length

(α + β + γ)

if and only if

A_{α}, B_{β_{i}}

and

E_{γ_{j}}

are the cyclic codes of length

α, β

and γ over

F_{q}

, P and Q for

i = 1, 2, 3

and

j = 1, 2, 3, 4, 5, 6 .

In Theorem 10, we studied the generator polynomial for a

F_{q} P Q

-cyclic code of length

(α + β + γ)

. Here, we examine the generator polynomial for a separable

F_{q} P Q

-cyclic code of length

(α + β + γ)

in the manner described below.

Theorem 12.

Let

E = A_{α} \otimes B_{β} \otimes E_{γ}

be a

F_{q} P Q

-cyclic code of length

(α + β + γ)

, where

A_{α} = 〈 a (κ) 〉

B_{β} = 〈 h (κ) 〉

and

E_{γ} = 〈 ℓ (κ) 〉

. Then,

C = 〈 a (κ) 〉 \otimes 〈 h (κ) 〉 \otimes 〈 ℓ (κ) 〉

Proof.

We have

\begin{matrix} A_{α} & = & 〈 a (κ) 〉, \\ B_{β} & = & 〈 h (κ) 〉, \\ E_{γ} & = & 〈 ℓ (κ) 〉 . \end{matrix}

Then, the proof directly follows. □

Example 1.

Let

α = 11, β = 7, γ = 8

and

Q_{11 + 7 + 8} = \frac{F_{27} [κ]}{〈 κ^{11} - 1 〉} \times \frac{P [κ]}{〈 κ^{7} - 1 〉} \times \frac{Q [κ]}{〈 κ^{8} - 1 〉}

, where

P = F_{27} [v] / 〈 v^{3} - α_{2}^{2} v 〉

and

Q = F_{27} [u, v] / 〈 u^{2} - α_{1}^{2}, v^{3} - α_{2}^{2} v 〉

. We take

F_{27} = \frac{F_{3}}{〈 κ^{3} + 2 κ + 1 〉}

. It can be easily seen that

κ^{3} + 2 κ + 1

is a irreducible in

F_{3}

and ω be a zero of polynomial in

F_{27}

, then,

κ^{11} - 1 = (κ + 2) (κ^{5} + 2 κ^{3} + κ^{2} + 2 κ + 2) (κ^{5} + κ^{4} + 2 κ^{3} + κ^{2} + 2) \in F_{27} [κ] .

Let

a (κ) = (κ + 2) (κ^{5} + 2 κ^{3} + κ^{2} + 2 κ + 2)

. Then,

A_{α} = 〈 a (κ) 〉

is a cyclic code of length 11 over

F_{27}

. Also, we have,

κ^{7} - 1 = (κ + 2) (κ^{2} + ω^{5} κ + 1) (κ^{2} + ω^{15} κ + 1) (κ^{2} + ω^{19} κ + 1) \in F_{27} [κ] .

Let

h_{1} (κ) = (κ + 2) (κ^{2} + ω^{5} κ + 1), h_{2} (κ) = (κ + 2) (κ^{2} + ω^{15} κ + 1)

and

h_{3} (κ) = (κ + 2) (κ^{2} + ω^{19} κ + 1)

. Thus

B_{β, i} = 〈 h_{i} (κ) 〉

are cyclic codes of length 7 over

F_{27}

, for

i = 1, 2, 3 .

Therefore,

B_{β} = 〈 h (κ) 〉

is a cyclic code of length 7 over P.

Now, we have

κ^{8} - 1 = (κ + 1) (κ + 2) (κ^{2} + κ + 2) (κ^{2} + 2 κ + 2) (κ^{2} + 1) \in F_{27} [κ] .

Let

ℓ_{1} (κ) = ℓ_{2} (κ) = (κ + 1) (κ + 2) (κ^{2} + κ + 2)

ℓ_{3} (κ) = ℓ_{4} (κ) = (κ + 1) (κ + 2) (κ^{2} + 2 κ + 2)

and

ℓ_{5} (κ) = ℓ_{6} (κ) = (κ + 1) (κ + 2) (κ^{2} + 1)

E_{γ, j} = 〈 ℓ_{j} 〉

are cyclic codes of length 8 over

F_{27}

, where

j = 1, 2, 3, 4, 5, 6 .

Thus,

E_{γ} = 〈 ℓ (κ) 〉

is a cyclic code of length 8 over Q, where

ℓ (κ) = ζ_{1} ℓ_{1} (κ) + ζ_{2} ℓ_{2} (κ) + ζ_{3} ℓ_{3} (κ) + ζ_{4} ℓ_{4} (κ) + ζ_{5} ℓ_{5} (κ) + ζ_{6} ℓ_{6} (κ)

. Hence,

E = 〈 (a (κ) ∣ 0 ∣ 0), (0 ∣ h (κ) ∣ 0), (r_{1} (κ) ∣ r_{2} (κ) ∣ ℓ (κ)) 〉

〈 a (κ) 〉 \otimes 〈 h (κ) 〉 \otimes 〈 ℓ (κ) 〉

is a separable

F_{q} P Q

-cyclic code of length

(11 + 7 + 8)

5.5. Quantum Error-Correcting Codes

In the present section, we will explore how to obtain quantum codes using the Calderbank–Shor–Steane (CSS) construction from [29], which utilizes dual-containing cyclic codes. The CSS construction is a powerful method for constructing quantum codes with desirable properties. By employing this construction, we can create quantum codes that outperform existing codes in terms of their parameters, such as dimension and minimum distance. We use a necessary and sufficient condition over the finite fields to obtain the condition for cyclic codes to contain their duals. It must be stated that the set of n-fold tensor product

{(C^{q})}^{\otimes n} = C^{q} \otimes C^{q} \otimes \dots \otimes C^{q}

(n-times) is a Hilbert space of dimension

q^{n}

, and also

C^{q}

is the Hilbert space of dimension q, where

C

is the complex field. A quantum code is the subspace of Hilbert space

{(C^{q})}^{\otimes n}

. A quantum code of length n over the field

F_{q}

(q is a power of a prime.) is denoted by

{[[n, k, d]]}_{q}

, where k is the dimension, and d is the minimum distance. We know that each quantum code satisfies the singleton bound, i.e.,

n - k + 2 \geq 2 d

. A quantum code is said to be MDS (maximum distance separable) if

n - k + 2 = 2 d

To construct better quantum codes compared to existing ones, we focus on two main conditions:

Higher Dimension (k): One way to improve a quantum code is by increasing its dimension, denoted as k. The dimension represents the number of encoded qubits or logical operators that can be stored in the code. By constructing a CSS code with a higher dimension compared to existing codes, we can encode more information in the same number of physical qubits, leading to increased storage capacity and computational capabilities.

Larger Minimum Distance (d): The minimum distance, denoted as d, of a quantum code determines its error-correcting capability. A larger minimum distance implies better error detection and correction properties. By constructing a CSS code with a larger minimum distance compared to existing codes, we enhance its ability to protect against errors and improve the overall reliability of the encoded information.

A quantum code

{[[n, k, d]]}_{q}

is better than the other quantum code

{[[n^{'}, k^{'}, d^{'}]]}_{q}

if one or both the following conditions hold:

1.: $\frac{k}{n} > \frac{k^{'}}{n^{'}}$ , where $d = d^{'}$ (larger code rate with same distance);
2.: $d > d^{'}$ where $\frac{k}{n} = \frac{k^{'}}{n^{'}}$ (larger distance with the same code rate).

In summary, the CSS construction, utilizing dual-containing cyclic codes, allows us to construct quantum codes. By carefully selecting the parameters of the codes involved, we can create better quantum codes compared to existing ones, with improved dimensions and minimum distances.

Lemma 4

([29]). [CSS Construction] If

E

is an [n, k, d] linear code with

(E^{⊥} \subseteq C

over

F_{q}

, then there exists a QEC code with parameters

{[[n, 2 k - n, d]]}_{q}

over

F_{q}

Lemma 5

([30]). A cyclic code

E

of length n over

F_{q}

with generator polynomial

f (κ)

that contains its dual if and only if

κ^{n} - 1 \equiv 0 m o d (f (κ) f^{*} (κ)),

where

f^{*} (κ)

is the reciprocal polynomial of

f (κ)

Proposition 10.

Let

E

be a

F_{q} P Q

-linear code of length (

α + β + γ

). Then, the Gray image of

E

, i.e.,

φ (E) = A_{α} \otimes B_{β_{1}} \otimes B_{β_{2}} \otimes B_{β_{3}} \otimes E_{γ_{1}} \otimes E_{γ_{2}} \otimes E_{γ_{3}} \otimes E_{γ_{4}} \otimes E_{γ_{5}} \otimes E_{γ_{6}}

is a linear code of length

(α + 3 β + 6 γ)

over

F_{q}

, where

A_{α}, B_{β_{i}}

and

E_{γ_{j}}

are linear codes of length

α, β

and γ over

F_{q}, P

and Q, respectively, for

i = 1, 2, 3

and

j = 1, 2, 3, 4, 5, 6

Proof.

The proof directly follows from the definition of a Gray map φ. □

Theorem 13.

Let

E

be a

F_{q} P Q

-linear code of length (

α + β + γ

). If

A_{α}^{⊥} \subseteq A_{α}, B_{β_{i}}^{⊥} \subseteq B_{β_{i}}

and

(E_{γ_{j}}^{⊥} \subseteq E_{γ_{j}}

for

i = 1, 2, 3

j = 1, 2, 3, 4, 5, 6 .

Then, there exists a quantum error-correcting code having parameters

[[α + 3 β + 6 γ, 2 k - (α + 3 β + 6 γ), d_{H}]]

, where

d_{H}

is the Hamming distance.

Proof.

The proof is similar to that in [25]. □

6. Applications

In this section, we mainly focus on the applications of separable

F_{q} P Q

-cyclic codes. Using the Gray images of cyclic codes over P, we obtain a number of optimal linear codes in Table 1. Additionally, we describe several quantum codes over

F_{q}, P, Q

F_{q} P

and

F_{q} P Q

. In Table 2, Table 3, Table 4, Table 5 and Table 6, we obtain MDS quantum codes, better quantum codes than the existing codes that appeared in some reference (see [25,31,32,33,34,35] for details) and new quantum codes, respectively. The Magma computation system [36] is used to complete all of the computations in these examples and tables, and we take

α_{1} = α_{2} = 1

in the rings P and Q.

The invertible matrices A used to construct the quantum codes are as follows:

$A_{11} = [\begin{matrix} 2 & 3 & 4 \\ 2 & 1 & 2 \\ 1 & 2 & 3 \end{matrix}] \in G L_{3} (F_{5})$	$A_{12} = [\begin{matrix} 2 & 1 & 2 \\ 5 & 2 & 1 \\ 1 & 2 & 5 \end{matrix}] \in G L_{3} (F_{7})$
$A_{13} = [\begin{matrix} 9 & 2 & 1 \\ 10 & 9 & 2 \\ 2 & 1 & 2 \end{matrix}] \in G L_{3} (F_{11})$	$A_{14} = [\begin{matrix} 2 & 1 & 2 \\ 15 & 2 & 1 \\ 1 & 2 & 15 \end{matrix}] \in G L_{3} (F_{17})$
$A_{21} = [\begin{matrix} 2 & 3 & 4 & 0 & 0 & 0 \\ 2 & 1 & 2 & 0 & 0 & 0 \\ 1 & 2 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 3 & 4 \\ 0 & 0 & 0 & 2 & 1 & 2 \\ 0 & 0 & 0 & 1 & 2 & 3 \end{matrix}] \in G L_{6} (F_{5})$	$A_{22} = [\begin{matrix} 2 & 1 & 2 & 0 & 0 & 0 \\ 5 & 2 & 1 & 0 & 0 & 0 \\ 1 & 2 & 5 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 & 2 \\ 0 & 0 & 0 & 5 & 2 & 1 \\ 0 & 0 & 0 & 1 & 2 & 5 \end{matrix}] \in G L_{6} (F_{7})$

Example 2.

β = 4

α_{2} = 1

q = 5

, and

P = F_{5} [v] / 〈 v^{3} - v 〉

Then,

κ^{4} - 1 = (κ + 1) (κ + 2) (κ + 3) (κ + 4) \in F_{5} [κ] .

take

h_{1} (κ) = κ^{2} + 1

h_{2} (κ) = h_{3} (κ) = (κ + 1)

. Also, we let

A_{11} = [\begin{matrix} 2 & 3 & 4 \\ 2 & 1 & 2 \\ 1 & 2 & 3 \end{matrix}]

such that

A_{1} A_{1}^{T} = 4 I_{3 \times 3}

, where

I_{3 \times 3}

is an identity matrix of order 3. Then,

B_{β}

be a cyclic code of length 4 over P and Gray image

φ_{P} (B_{β})

having the parameters

{[12, 8, 4]}_{5}

which is an optimal code as per the database [37].

Example 3.

Let

q = 5, α_{1} = α_{2} = 1, α = 10, β = 10, γ = 10

and

Q_{α + β + γ} = \frac{F_{5} [κ]}{(κ^{10} - 1)} \times \frac{P [κ]}{(κ^{10} - 1)} \times \frac{Q [κ]}{(κ^{10} - 1)}

, where

P = F_{5} [v] / 〈 v^{3} - v 〉

and

Q = F_{5} [u, v] / 〈 u^{2} - 1, v^{3} - v 〉

. Now,

κ^{10} - 1 = {(κ + 1)}^{5} {(κ + 4)}^{5} \in F_{10} [κ]

. We take

a (κ) = {(κ + 1)}^{2} (κ + 4) \in F_{5} [κ]

and

A_{α} = 〈 a (κ) 〉

be a cyclic code of length 10 over

F_{5} [κ]

with the parameters

{[10, 7, 3]}_{5} .

Again,

κ^{10} - 1 = {(κ + 1)}^{5} {(κ + 4)}^{5} \in F_{5} [κ] .

We take

h_{1} (κ) = {(κ + 1)}^{2} (κ + 4)

h_{2} (κ) = (κ + 1), h_{3} (κ) = 1

. Let

A_{11} = [\begin{matrix} 2 & 3 & 4 \\ 2 & 1 & 2 \\ 1 & 2 & 3 \end{matrix}]

such that

A_{1} A_{1}^{T} = 4 I_{3 \times 3}

, where

I_{3 \times 3}

is an identity matrix of order 3. Then,

B_{β}

is a cyclic code of length 10 on P and its Gray image have the parameters

{[30, 26, 3]}_{5}

over

F_{5}

. Next, let us consider that

κ^{10} - 1 = {(κ + 1)}^{5} {(κ + 4)}^{5} \in F_{5} [κ]

. We take

ℓ_{1} (κ) = ℓ_{4} (κ) = {(κ + 1)}^{2} (κ + 4) \in F_{5} [κ]

ℓ_{3} (κ) = ℓ_{5} (κ) = (κ + 1)

and

= ℓ_{2} (κ) = ℓ_{6} (κ) = 1

. Take

A_{21} = [\begin{matrix} 2 & 3 & 4 & 0 & 0 & 0 \\ 2 & 1 & 2 & 0 & 0 & 0 \\ 1 & 2 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 3 & 4 \\ 0 & 0 & 0 & 2 & 1 & 2 \\ 0 & 0 & 0 & 1 & 2 & 3 \end{matrix}]

such that

A_{1} A_{1}^{T} = 4 I_{6 \times 6}

, where

I_{6 \times 6}

is an identity matrix of order 6. Then,

E_{γ_{j}}

be the cyclic code of length 10 over Q and its Gray image has the parameters

{[60, 52, 3]}_{5}

over

F_{5}

. Then,

φ (E)

is a linear code having the parameters

{[100, 85, 3]}_{5}

over

F_{5}

. It is clear that,

κ^{10} - 1 \equiv 0 m o d (h_{i} (κ) h_{i}^{*} (κ)),

where

i = 1, 2, 3 .

Also,

κ^{10} - 1 \equiv 0 m o d (ℓ_{j} (κ) ℓ_{j}^{*} (κ)),

where

j = 1, 2, 3, 4, 5, 6 .

With the help of Lemma 5, we have

\begin{matrix} A_{α}^{⊥} & \subseteq & A_{α}; \\ B_{β_{i}}^{⊥} & \subseteq & B_{β_{i}}; \\ E_{γ_{j}}^{⊥} & \subseteq & E_{γ_{j}} . \end{matrix}

By using the Theorem 13, there exists a quantum error-correcting code with the parameters

{[[100, 70, 3]]}_{5}

. This is a new quantum code according to the database [38].

In Table 1, we obtain the optimal linear codes. In Table 2, we obtain MDS quantum error-correcting codes over

F_{q}

, and in Table 3 and Table 4, we obtain better quantum error-correcting codes than previously known quantum error-correcting codes. In Table 5 and Table 6, we obtain new quantum error-correcting codes over

F_{q} P

and

F_{q} P Q

7. Conclusions

In this work, cyclic codes over

F_{q} P Q

are introduced, where

P = \frac{F_{q} [v]}{〈 v^{3} - α_{2}^{2} v 〉}

and

Q = \frac{F_{q} [u, v]}{〈 u^{2} - α_{1}^{2}, v^{3} - α_{2}^{2} v 〉}

are nonchain finite rings and

α_{i}

are in

F_{q} / {0}

for

i \in {1, 2}

q = p^{m}

with

m \geq 1

a positive integer and p is an odd prime. We reviewed some characteristics of

F_{q} P Q

-cyclic codes and defined a Gray map over

F_{q} P Q

. As an application, we constructed quantum error-correcting (QEC) codes using

F_{q} P Q

-cyclic codes. This analysis can be applied to the product of finite rings in general.

Author Contributions

Conceptualization and supervision of the study, S.A.; investigation and preparation of the original draft of the manuscript, A.S.A., P.S., K.B.W., E.S.Ö. and M.J. All authors have read and agreed to the published version of the manuscript.

Funding

This study was carried out with financial support from Princess Nourah bint Abdulrahman University (PNU), Riyadh, Saudi Arabia.

Institutional Review Board Statement

Not applicable.

Data Availability Statement

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Acknowledgments

The authors are very thankful to the anonymous referees for their valuable comments and suggestions, which have improved the manuscript immensely. Moreover, the authors extend their appreciation to Princess Nourah bint Abdulrahman University (PNU), Riyadh, Saudi Arabia, for funding this research under Researchers Supporting Project Number (PNURSP2023R231).

Conflicts of Interest

The authors declare that they have no conflict of interest.

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Table 1. Gray images of cyclic codes over P.

n	$h_{1} (κ)$	$h_{2} (κ)$	$h_{3} (κ)$	$φ_{P} (C)$	Remarks
4	$(κ^{2} + 1)$	$κ + 1$	$κ + 1$	${[12, 8, 4]}_{5}$	optimal
5	${(κ + 4)}^{3}$	$κ + 4$	$κ + 4$	${[15, 10, 4]}_{5}$	optimal
6	$(κ^{2} + κ + 1)$	$κ + 1$	$κ + 1$	${[18, 14, 4]}_{5}$	optimal
7	$κ^{6} + κ^{5} + κ^{4} + κ^{3} + κ^{2} + κ + 1$	$κ + 4$	$κ + 4$	${[21, 13, 4]}_{5}$	…
8	$(κ + 2) (κ^{2} + 2)$	$κ + 2$	$κ + 1$	${[24, 19, 4]}_{5}$	optimal
10	$(κ + 1) {(κ + 4)}^{2}$	$κ + 1$	1	${[30, 26, 3]}_{5}$	optimal
12	$(κ + 4) (κ^{2} + 2 κ + 2)$	$κ + 1$	$κ + 1$	${[36, 31, 4]}_{5}$	optimal

Table 2. MDS Quantum codes over

F_{q}

Table 2. MDS Quantum codes over

F_{q}

$α$	$α (κ)$	Parameters of $A_{α}$	Quantum Codes	Remarks
3	$(κ + 2)$	${[3, 2, 2]}_{3}$	${[[3, 1, 3]]}_{3}$	$M D S C o d e$
4	$(κ + 1) (κ + 2)$	${[4, 2, 3]}_{5}$	${[[4, 0, 3]]}_{5}$	$M D S C o d e$
5	${(κ + 4)}^{2}$	${[5, 3, 3]}_{5}$	${[[5, 1, 3]]}_{5}$	$M D S C o d e$
6	$(κ + 1) (κ^{2} + κ + 1)$	${[6, 3, 4]}_{5}$	${[[6, 0, 4]]}_{5}$	$M D S C o d e$
7	${(κ + 6)}^{2}$	${[7, 5, 3]}_{7}$	${[[7, 3, 3]]}_{7}$	$M D S C o d e$
7	${(κ + 6)}^{3}$	${[7, 4, 3]}_{7}$	${[[7, 1, 4]]}_{7}$	$M D S C o d e$
8	$(κ + 1) (κ^{2} + 3 κ + 1)$	${[8, 5, 4]}_{7}$	${[[8, 2, 4]]}_{7}$	$M D S C o d e$

Table 3. Quantum codes over P.

$β$	$h_{1} (κ)$	$h_{2} (κ)$	$h_{3} (κ)$	Gray Image over P	Quantum Codes	Known Quantum Codes
25	${(κ + 4)}^{6}$	$(κ + 4)$	1	$[75, 68, 3]$	${[[75, 61, 3]]}_{5}$	$N e w Q u a n t u m c o d e$
30	${(κ + 4)}^{2} (κ^{2} + 4 κ + 1)$	$κ + 1$	1	$[90, 85, 3]$	${[[90, 80, 3]]}_{5}$	${[[90, 72, 2]]}_{5}$ [32]
40	${(κ + 1)}^{2} (κ^{2} + 3)$	$κ + 1$	1	$[120, 115, 3]$	${[[120, 110, 3]]}_{5}$	${[[120, 104, 2]]}_{5}$ [31]
60	${(κ + 1)}^{2} (κ^{2} + 3 κ + 4)$	$κ + 1$	1	$[180, 175, 3]$	${[[180, 170, 3]]}_{5}$	${[[168, 126, 3]]}_{5}$ [25]
7	${(κ + 6)}^{3}$	$(κ + 6)$	$κ + 6$	$[[21, 16, 4]]$	${[21, 11, 4]}_{7}$	…
21	${(κ + 3)}^{3} (κ + 5)$	$κ + 6$	1	$[63, 58, 3]$	${[[63, 53, 3]]}_{7}$	$N e w Q u a n t u m c o d e$
28	${(κ + 1)}^{2} (κ^{2} + 1)$	$κ + 1$	1	$[84, 79, 3]$	${[[84, 74, 3]]}_{7}$	${[[84, 72, 3]]}_{7}$ [33]
35	${(κ + 6)}^{3} (κ^{4} + κ^{3} + κ^{2} + κ + 1)$	$κ + 6$	$κ + 1$	$[105, 96, 4]$	${[[105, 87, 4]]}_{7}$	${[[108, 28, 3]]}_{7}$ [34]
77	${(κ + 6)}^{3} (κ^{10} + κ^{9} + κ^{8} +$	$κ + 1$	1	$[231, 216, 4]$	${[[231, 201, 4]]}_{7}$	${[[228, 196, 3]]}_{7}$ [35]
	$κ^{7} + κ^{6} + κ^{5} + κ^{4}$
	$+ κ^{3} + κ^{2} + κ + 1)$
11	${(κ + 10)}^{3}$	$(κ + 10)$	$(κ + 10)$	$[33, 28, 4]$	${[[33, 23, 4]]}_{11}$	$N e w Q u a n t u m c o d e$
11	${(κ + 10)}^{5}$	${(κ + 10)}^{3}$	$(κ + 10)$	$[[33, 24, 6]]$	${[[33, 15, 6]]}_{11}$	$N e w Q u a n t u m c o d e$
17	${(κ + 16)}^{3}$	$(κ + 16)$	$(κ + 16)$	$[51, 46, 4]$	${[[51, 41, 4]]}_{17}$	$N e w Q u a n t u m c o d e$

Table 4. Quantum codes over Q.

$γ$	$ℓ_{1} (κ) = ℓ_{4} (κ)$	$ℓ_{3} (κ) = ℓ_{5} (κ)$	$ℓ_{2} (κ) = ℓ_{6} (κ)$	Gray Image over P	Quantum Codes	Known Quantum Codes
5	${(κ + 4)}^{2}$	$κ + 4$	1	$[30, 24, 3]$	${[[30, 18, 3]]}_{5}$	$N e w Q u a n t u m c o d e$
8	$(κ + 2) (κ^{2} + 2)$	$κ + 2$	1	$[48, 40, 3]$	${[[48, 32, 3]]}_{5}$	…
10	${(κ + 1)}^{2} (κ + 4)$	$κ + 1$	1	$[60, 52, 3]$	${[[60, 44, 3]]}_{5}$	$N e w Q u a n t u m c o d e$
14	${(κ + 1)}^{3} (κ + 6)$	$κ + 1$	$κ + 1$	$[84, 72, 4]$	${[[84, 60, 4]]}_{7}$	${[[84, 60, 4]]}_{7}$ [34]
18	$(κ + 3) (κ^{3} + 2) (κ^{3} + 3)$	$κ + 3$	$κ + 3$	$[108, 90, 4]$	${[[108, 72, 4]]}_{7}$	${[[108, 28, 3]]}_{7}$ [34]
30	${(κ + 4)}^{2} (κ^{2} + 4 κ + 1)$	$κ + 4$	1	$[180, 170, 3]$	${[[180, 160, 3]]}_{5}$	${[[180, 156, 2]]}_{5}$ [32]
35	${(κ + 4)}^{2} (κ^{6} + κ^{5} + κ^{4} + κ^{3} + κ^{2} + κ + 1)$	$κ + 4$	1	$[210, 192, 3]$	${[[210, 174, 3]]}_{5}$	${[[210, 150, 2]]}_{5}$ [32]
38	$(κ^{3} + 3 κ^{2} + 4 κ + 1)$	$κ^{3} + 2 κ + 6$	1	$[228, 216, 3]$	${[[228, 204, 3]]}_{7}$	${[[228, 196, 3]]}_{7}$ [34]
42	$(κ + 1) {(κ + 3)}^{2}$	$κ + 1$	1	$[252, 244, 3]$	${[[252, 236, 3]]}_{7}$	$[{[252, 212, 3]}_{7}]$ [34]

Table 5. Quantum codes over

F_{q} P

Table 5. Quantum codes over

F_{q} P

$α$	$β$	$α (κ)$	$h_{1} (κ)$	$h_{2} (κ)$	$h_{3} (κ)$	Gray Image over $F_{q} P$	New Quantum Codes
10	5	${(κ + 1)}^{2} (κ + 4)$	${(κ + 4)}^{2}$	$κ + 4$	1	$[25, 19, 3]$	${[[25, 13, 3]]}_{5}$
15	10	$(κ^{2} + κ + 1) {(κ + 4)}^{2}$	${(κ + 1)}^{2} (κ + 4)$	$κ + 1$	1	$[45, 37, 3]$	${[[45, 29, 3]]}_{5}$
20	20	${(κ + 1)}^{2} (κ + 2)$	${(κ + 1)}^{2} (κ + 2)$	$κ + 1$	1	$[80, 73, 3]$	${[[80, 66, 3]]}_{5}$
30	20	${(κ + 1)}^{2} (κ^{2} + κ + 1)$	${(κ + 1)}^{2} (κ + 2)$	$κ + 1$	1	$[90, 82, 3]$	${[[90, 74, 3]]}_{5}$
7	7	${(κ + 6)}^{3}$	${(κ + 6)}^{3}$	$κ + 6$	$κ + 6$	$[28, 20, 4]$	${[[28, 12, 4]]}_{7}$
14	7	${(κ + 6)}^{3} (κ + 1)$	${(κ + 6)}^{3}$	$κ + 6$	$κ + 6$	$[35, 26, 4]$	${[[35, 17, 4]]}_{7}$
28	28	${(κ + 1)}^{3} (κ^{2} + 1)$	${(κ + 1)}^{3} (κ^{2} + 1)$	$κ + 1$	$κ + 1$	$[112, 100, 4]$	${[[112, 88, 4]]}_{7}$
42	56	${(κ + 1)}^{3} (κ + 2) (κ + 3)$	${(κ + 1)}^{3} (κ + 2) (κ + 3)$	$κ + 1$	$κ + 1$	$[210, 198, 4]$	${[[210, 186, 4]]}_{7}$

Table 6. Quantum codes over

F_{q} P Q

Table 6. Quantum codes over

F_{q} P Q

$α$	$β$	$γ$	$α (κ)$	$h_{1} (κ)$	$h_{2} (κ)$	$h_{3} (κ)$	$ℓ_{1} (κ) = ℓ_{4} (κ)$	$ℓ_{3} (κ) = ℓ_{5} (κ)$	$ℓ_{2} (κ) = ℓ_{6} (κ)$	Gray Image over $F_{q} PQ$	New Quantum Codes
5	5	5	${(κ + 4)}^{2}$	${(κ + 4)}^{2}$	$κ + 4$	1	${(κ + 4)}^{2}$	$κ + 4$	1	$[50, 39, 3]$	${[[50, 28, 3]]}_{5}$
10	10	10	${(κ + 1)}^{2} (κ + 4)$	${(κ + 1)}^{2} (κ + 4)$	$κ + 1$	1	${(κ + 1)}^{2} (κ + 4)$	$κ + 1$	1	$[100, 85, 3]$	${[[100, 70, 3]]}_{5}$
10	20	30	${(κ + 1)}^{2} (κ + 4)$	${(κ + 1)}^{2} (κ + 2)$	$κ + 1$	1	${(κ + 1)}^{2} (κ^{2} + κ + 1)$	$κ + 1$	1	$[250, 233, 3]$	${[[250, 216, 3]]}_{5}$
30	20	40	${(κ + 1)}^{2} (κ^{2} + κ + 1)$	${(κ + 1)}^{2} (κ + 2)$	$κ + 1$	1	${(κ + 1)}^{2} (κ^{2} + 2)$	$κ + 1$	1	$[330, 312, 3]$	${[[330, 294, 3]]}_{5}$
7	21	14	${(κ + 6)}^{3}$	${(κ + 3)}^{3} (κ + 5) (κ + 6)$	$κ + 3$	1	${(κ + 1)}^{3} (κ + 6)$	$κ + 1$	$κ + 1$	$[154, 132, 4]$	${[[154, 110, 4]]}_{7}$
14	28	35	${(κ + 1)}^{3} (κ + 6)$	${(κ + 1)}^{3} (κ^{2} + 1)$	$κ + 1$	$κ + 1$	${(κ + 6)}^{3} (κ^{4} + κ^{3} + κ^{2} + κ + 6)$	$κ + 6$	$κ + 6$	$[308, 279, 4]$	${[[308, 250, 4]]}_{7}$
48	84	70	$(κ + 3) (κ^{3} + κ + 3)$	${(κ + 1)}^{2} (κ^{2} + 2)$	$κ + 1$	1	${(κ + 1)}^{2} (κ^{4} + κ^{3} + κ^{2} + κ + 1)$	$κ + 1$	1	$[720, 697, 3]$	${[[720, 674, 3]]}_{7}$
63	56	98	${(κ + 3)}^{2} (κ^{3} + 3)$	${(κ + 1)}^{2} (κ^{2} + 3 κ + 1)$	$κ + 1$	1	${(κ + 1)}^{8} (κ + 6)$	$κ + 1$	1	$[819, 789, 3]$	${[[819, 759, 3]]}_{7}$

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Ali, S.; Alali, A.S.; Sharma, P.; Wong, K.B.; Öztas, E.S.; Jeelani, M. On Optimal and Quantum Code Construction from Cyclic Codes over $F$ _qPQ with Applications. Entropy 2023, 25, 1161. https://doi.org/10.3390/e25081161

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Ali S, Alali AS, Sharma P, Wong KB, Öztas ES, Jeelani M. On Optimal and Quantum Code Construction from Cyclic Codes over $F$ _qPQ with Applications. Entropy. 2023; 25(8):1161. https://doi.org/10.3390/e25081161

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Ali, Shakir, Amal S. Alali, Pushpendra Sharma, Kok Bin Wong, Elif Segah Öztas, and Mohammad Jeelani. 2023. "On Optimal and Quantum Code Construction from Cyclic Codes over $F$ _qPQ with Applications" Entropy 25, no. 8: 1161. https://doi.org/10.3390/e25081161

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Ali, S., Alali, A. S., Sharma, P., Wong, K. B., Öztas, E. S., & Jeelani, M. (2023). On Optimal and Quantum Code Construction from Cyclic Codes over $F$ _qPQ with Applications. Entropy, 25(8), 1161. https://doi.org/10.3390/e25081161

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On Optimal and Quantum Code Construction from Cyclic Codes over $F$ _qPQ with Applications

Abstract

1. Introduction

2. Preliminaries

3. Linear Codes and Gray Images over P and Q

4. Gray Image over $F_{q} PQ$

5. Main Results

5.1. Cyclic Codes over $F_{q}$

5.2. Cyclic Codes over P

5.3. Cyclic Codes over Q

5.4. Cyclic Codes over $F_{q} P Q$

5.5. Quantum Error-Correcting Codes

6. Applications

7. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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On Optimal and Quantum Code Construction from Cyclic Codes over FqPQ with Applications

Abstract

1. Introduction

2. Preliminaries

3. Linear Codes and Gray Images over P and Q

4. Gray Image over F q PQ

5. Main Results

5.1. Cyclic Codes over F q

5.2. Cyclic Codes over P

5.3. Cyclic Codes over Q

5.4. Cyclic Codes over F q P Q

5.5. Quantum Error-Correcting Codes

6. Applications

7. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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On Optimal and Quantum Code Construction from Cyclic Codes over $F$ _qPQ with Applications

4. Gray Image over $F_{q} PQ$

5.1. Cyclic Codes over $F_{q}$

5.4. Cyclic Codes over $F_{q} P Q$