Quantum Codes as an Application of Constacyclic Codes

Mohd Arif Raza

^1,*

Mohammad Fareed Ahmad

Adel Alahmadi

³,

Widyan Basaffar

³,

Manish K. Gupta

⁴

Nadeem ur Rehman

Abdul Nadim Khan

¹,

Hatoon Shoaib

and

Patrick Sole

⁵

Research Group of Algebraic Structures and Applications, Department of Mathematics, Faculty of Science & Arts-Rabigh, King Abdulaziz University, Rabigh 21911, Saudi Arabia

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

Research Group of Algebraic Structures and Applications, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

⁴

Director Academics Office, Kaushalya: The Skill University, Ahmedabad 382424, India

⁵

I2M, (CNRS, Aix-Marseille University, Centrale Marseille), 163 Avenue de Luminy, 13009 Marseilles, France

Author to whom correspondence should be addressed.

Axioms2024,13(10), 697;https://doi.org/10.3390/axioms13100697

Submission received: 16 August 2024 /Revised: 14 September 2024 /Accepted: 20 September 2024 /Published: 8 October 2024

(This article belongs to the SectionAlgebra and Number Theory)

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Abstract

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring

R = F_{p} + w_{1} F_{p} + w_{2} F_{p} + w_{2}^{2} F_{p} + w_{1} w_{2} F_{p} + w_{1} w_{2}^{2} F_{p}

, where

w_{1}^{2} - α^{2} = 0

w_{1} w_{2} = w_{2} w_{1}

w_{2}^{3} - β^{2} w_{2} = 0

, and

α, β \in F_{p} ∖ {0}

, for a primep. We begin by introducing a Gray map defined over

R

, which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over

R

using Calderbank–Shore–Steane (CSS) construction.

Keywords:

linear codes;Gray map;CSS construction;quantum codes

MSC:

11T71; 94B05; 94B15

1. Introduction

In contrast to classical information theory, quantum information theory is a relatively emerging field [1,2,3]. The concept of quantum error-correcting codes (QECCs) was initially introduced by Shor [4] and Steane [5], with a construction method outlined by Calderbank et al. [6]. Subsequently, researchers have explored various approaches to utilize classical error-correcting codes to create new quantum codes (QECCs). The quantum code database remains quite limited when compared to classical block codes. The existing database [7] encompasses finite fields of order up to 9, but it focuses exclusively on QECCs for

p = 2

. Some static tables of quantum codes are available in [8,9], building upon the work in [10]. The online tables [9] might have been overlooked by many researchers.

The field of quantum error-correcting codes has seen remarkable growth since the initial realization that such codes could safeguard quantum information, which is analogous to how classical error-correcting codes protects classical information. Shor’s [4] pioneering work led to the discovery of the first quantum error-correcting code. In 1998, Calderbank et al. [6] provided a systematic method for constructing quantum codes from classical error-correcting codes. Many researchers have concentrated on using Calderbank–Shor–Steane (CSS) construction to produce quantum codes from linear codes that contain their duals (see [11,12,13,14]).

Qian et al. [15] initially presented the construction of quantum codes from cyclic codes of odd length over the chain ring

F_{2} + u F_{2}

, where

u^{2} = 0

. Subsequently, Kai and Zhu [16] introduced a technique for generating quantum codes from cyclic codes of odd length over the finite chain ring

F_{4} + u F_{4}

. Qian [17] proposed a novel approach for constructing quantum error-correcting codes from cyclic codes over the finite non-chain ring

F_{2} + v F_{2}

, where

v^{2} = v

of any length. Motivated by this study, Ashraf and Mohammad [18] obtained quantum codes from cyclic codes over the non-chain ring

F_{q} + u F_{q} + v F_{q} + u v F_{q}

, where

u^{2} = u

v^{2} = v

u v = v u

q = p^{n}

, andp is an odd prime.

Constacyclic codes, a robust extension of cyclic codes over finite non-chain rings, have proven to be a prolific source of new quantum codes. Recent research by coding theorists has explored constacyclic codes extensively. Distinguished investigations include Li et al. [19] over

F_{p} + u F_{p} + v F_{p} + u v F_{p}

, with

u^{2} - u = 0

v^{2} - v = 0

, and

u v - v u = 0

; Ma et al.’s [20] contributions over

F_{p} + v F_{p} + v^{2} F_{p}

, with

v^{3} = v

; and Gao and Wang’s [21] over

F_{p} + u F_{p}

, where

u^{2} = 1

. These studies have led to the construction of numerous significantly improved quantum codes, all originating from dual-containing constacyclic codes.

In light of these developments, it becomes evident that constacyclic codes over finite non-chain rings represent a valuable resource for generating new and better quantum codes. Therefore, this article delves into the exploration of constacyclic codes within the framework of the non-chain ring

R = F_{p} + w_{1} F_{p} + w_{2} F_{p} + w_{2}^{2} F_{p} + w_{1} w_{2} F_{p} + w_{1} w_{2}^{2} F_{p}

, where

w_{1}^{2} - α^{2} = 0

w_{1} w_{2} = w_{2} w_{1}

w_{2}^{3} - β^{2} w_{2} = 0

, and

α, β \in F_{p} ∖ {0}

, for a primep. The objective is to find new quantum codes over the finite field

F_{p}

. The article makes two significant contributions:

Comprehensive study of the structure of constacyclic codes with the lengthl over $R$ .
The construction of better quantum codes concerning their parameters, surpassing those previously documented in the literature.

A noteworthy aspect of this research involves the presentation of computational findings [22], highlighting the substantial impact of this work on the development of new quantum codes.

2. Preliminaries

Let

F_{p}

be a finite field of orderp (an odd prime). A subspace

C_{0}

F_{p}^{m}

is called a linear code of lengthm over

F_{p}

, and its members are called the codewords. Let

R = F_{p} + w_{1} F_{p} + w_{2} F_{p} + w_{2}^{2} F_{p} + w_{1} w_{2} F_{p} + w_{1} w_{2}^{2} F_{p}

, where

w_{1}^{2} - α^{2} = 0

w_{1} w_{2} = w_{2} w_{1}

w_{2}^{3} - β^{2} w_{2} = 0

, and

α, β \in F_{p} ∖ {0}

be a finite commutative ring. Remember that a linear codeC over the ring

R

of lengthn is essentially an

R

-submodule of the module

R^{n}

. One can also view an element

c = (c_{0}, c_{1}, \dots, c_{n - 1})

inC as a polynomial

c (z) = c_{0} + c_{1} z + \dots + c_{n - 1} z^{n - 1}

within the ring

\frac{R [z]}{〈 z^{n} - Λ 〉}

. A linear codeC is called a

Λ

-constacyclic code of lengthn over

R

if and only if it is an

R

-submodule in the module

\frac{R [z]}{〈 z^{n} - Λ 〉}

. Many researchers have extensively explored constacyclic codes over finite fields and finite commutative Frobenius rings [23,24,25,26,27]. Consider the elements of

R

as follows:

\begin{matrix} κ_{1} & = \frac{1}{2 α β^{2}} (α + w_{1}) (β^{2} - w_{2}^{2}), κ_{2} = \frac{1}{2 α β^{2}} (α - w_{1}) (β^{2} - w_{2}^{2}), \\ κ_{3} & = \frac{1}{4 α β^{2}} (α + w_{1}) (w_{2}^{2} - β w_{2}), κ_{4} = \frac{1}{4 α β^{2}} (α - w_{1}) (w_{2}^{2} - β w_{2}), \\ κ_{5} & = \frac{1}{4 α β^{2}} (α + w_{1}) (w_{2}^{2} + β w_{2}), κ_{6} = \frac{1}{4 α β^{2}} (α - w_{1}) (w_{2}^{2} + β w_{2}) . \end{matrix}

We can verify that

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

, and

κ_{i} κ_{j} = δ_{i j}

(Kronecker delta) for

i, j \in {1, 2 \dots, 6}

. Consequently, the set

{κ_{1}, κ_{2}, \dots, κ_{6}}

forms a set of non-zero pairwise orthogonal idempotent elements in

R

. This implies that

R

can be expressed as a sum of submodules as follows:

\begin{matrix} R & = κ_{1} R \oplus κ_{2} R \oplus κ_{3} R \oplus κ_{4} R \oplus κ_{5} R \oplus κ_{6} R \\ ≅ κ_{1} F_{p} \oplus κ_{2} F_{p} \oplus κ_{3} F_{p} \oplus κ_{4} F_{p} \oplus κ_{5} F_{p} \oplus κ_{6} F_{p} . \end{matrix}

Therefore, any element

r = a + w_{1} b + w_{2} c + w_{2}^{2} d + w_{1} w_{2} e + w_{1} w_{2}^{2} f \in R

can be uniquely written as

\begin{matrix} r & = a + w_{1} b + w_{2} c + w_{2}^{2} d + w_{1} w_{2} e + w_{1} w_{2}^{2} f \\ = κ_{1} \bar{a} + κ_{2} \bar{b} + κ_{3} \bar{c} + κ_{4} \bar{d} + κ_{5} \bar{e} + κ_{6} \bar{f}, \end{matrix}

(1)

where

\begin{matrix} \bar{a} & = a + α b, \\ \bar{b} & = a - α b, \\ \bar{c} & = a + α b - β c + β^{2} d - α β e + α β^{2} f, \\ \bar{d} & = a - α b - β c + β^{2} d + α β e - α β^{2} f, \\ \bar{e} & = a + α b + β c + β^{2} d + α β e + α β^{2} f, \\ \bar{f} & = a - α b + β c + β^{2} d - α β e - α β^{2} f, \end{matrix}

are the elements of

F_{p}

Suppose that

G L_{n} (F_{p})

is the group of invertible matrices of ordern over

F_{p}

and let

N \in G L_{6} (F_{p})

in such a way that

N N^{T} = k I_{6}

, where

N^{T}

is the transpose of the matrix

N

I_{6}

is the identity matrix of order 6, and

k \in F_{p} - {0}

. With the above notation, we define a Gray map associated with an invertible matrix

N

as follows:

\nabla : R ⟶ F_{p}^{6} such that \nabla (r) : = (\bar{a}, \bar{b}, \bar{c}, \bar{d}, \bar{e}, \bar{f}) N .

We can extend the Gray map ∇ for each component individually, as follows:

\nabla : R^{l} ⟶ F_{p}^{6 l} such that

\begin{matrix} \nabla (r_{0}, r_{1}, \dots, r_{l - 1}) = & (({\bar{a}}_{0}, {\bar{b}}_{0}, {\bar{c}}_{0}, {\bar{d}}_{0}, {\bar{e}}_{0}, {\bar{f}}_{0}) N, ({\bar{a}}_{1}, {\bar{b}}_{1}, {\bar{c}}_{1}, {\bar{d}}_{1}, {\bar{e}}_{1}, {\bar{f}}_{1}) N \\ , \dots, ({\bar{a}}_{l - 1}, {\bar{b}}_{l - 1}, {\bar{c}}_{l - 1}, {\bar{d}}_{l - 1}, {\bar{e}}_{l - 1}, {\bar{f}}_{l - 1}) N), \end{matrix}

(2)

where

r_{i} = κ_{1} \bar{a_{i}} + κ_{2} \bar{b_{i}} + κ_{3} \bar{c_{i}} + κ_{4} \bar{d_{i}} + κ_{5} \bar{e_{i}} + κ_{6} \bar{f_{i}} \in R, for i \in {0, 1, \dots, l - 1}

. Here, we introduce the Lee weight for the vector

r \in R

w_{L} (r) = w_{H} (\nabla (r))

, where

w_{L}

(resp.

w_{H}

) denotes the Lee weight (resp. the Hamming weight). The Lee weight of

w_{L} (r = (r_{0}, r_{1}, \dots, r_{l - 1})) = w_{L} (r_{0}) + w_{L} (r_{1}) + \dots + w_{L} (r_{l - 1})

and the Lee distance fromr to

r^{'} \in R^{l}

, is established as

d_{L} (r, r^{'}) = w_{L} (r - r^{'}) = w_{H} (\nabla (r - r^{'}))

. The Lee distance

d_{L} (C)

for the codeC is defined as follows:

d_{L} (C) = min {d_{L} (r, r^{'}) | r \neq r^{'}} .

It is notable that the Gray map ∇ is a linear map over

F_{p}

that preserves distances and mapping vectors from

R^{l}

F_{p}^{6 l}

. Since the Gray map ∇ is bijective, it follows that

\nabla (C)

forms a

[6 l, k, d_{H}]

linear code over

F_{p}

, where

d_{L}

is equal to

d_{H}

The Euclidean inner product of any two vectors,

r = (r_{0}, r_{1}, \dots, r_{l - 1})

and

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

R^{l}

is defined as

r \cdot r^{'} = r_{0} r_{0}^{'} + r_{1} r_{1}^{'} + \dots + r_{l - 1} r_{l - 1}^{'}

. The dual code ofC is formulated as

C^{⊥} = {r \in R^{l} | r \cdot r^{'} = 0 \forall r^{'} \in C}

. A codeC is called dual-containing if

C^{⊥} \subseteq C

, self-orthogonal if

C \subseteq C^{⊥}

, and self-dual if

C^{⊥} = C

Example 1.

Let

R_{3} = \frac{F_{3} [w_{1}, w_{2}]}{〈 w_{1}^{2} - 1, w_{2}^{3} - w_{2}, w_{1} w_{2} - w_{2} w_{1} 〉}

be a finite commutative non-chain ring. Then, we have

w_{1}^{2} - 1 = (w_{1} - 1) (w_{1} + 1)

and

w_{2}^{3} - w_{2} = w_{2} (w_{2} - 1) (w_{2} + 1)

. Thus, the orthogonal idempotent elements in

R_{3}

are

\begin{matrix} κ_{1} & = 2 (1 + w_{1}) (1 + 2 w_{2}^{2}), κ_{2} = \frac{1}{2} (1 - w_{1}) (1 - w_{2}^{2}) = 2 (1 + 2 w_{1}) (1 + 2 w_{2}^{2}), \\ κ_{3} & = (1 + w_{1}) (w_{2}^{2} + 2 w_{2}), κ_{4} = \frac{1}{4} (1 - w_{1}) (w_{2}^{2} - w_{2}) = (1 + 2 w_{1}) (w_{2}^{2} + 2 w_{2}), \\ κ_{5} & = (1 + w_{1}) (w_{2}^{2} + w_{2}), κ_{6} = \frac{1}{4} (1 - w_{1}) (w_{2}^{2} + w_{2}) = (1 + 2 w_{1}) (w_{2}^{2} + w_{2}), \end{matrix}

where

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

. By Chinese Remainder Theorem, we have

R_{3} = κ_{1} R_{3} \oplus κ_{2} R_{3} \oplus κ_{3} R_{3} \oplus κ_{4} R_{3} \oplus κ_{5} R_{3} \oplus κ_{6} R_{3} ≅ κ_{1} F_{3} \oplus κ_{2} F_{3} \oplus κ_{3} F_{3} \oplus κ_{4} F_{3} \oplus κ_{5} F_{3} \oplus κ_{6} F_{3} .

Therefore, any element

r = a + w_{1} b + w_{2} c + w_{2}^{2} d + w_{1} w_{2} e + w_{1} w_{2}^{2} f \in R

can be expressed as follows:

\begin{matrix} r = & a + w_{1} b + w_{2} c + w_{2}^{2} d + w_{1} w_{2} e + w_{1} w_{2}^{2} f \\ = & (a + b) κ_{1} + (a + 2 b) κ_{2} + (a + b + 2 c + d + 2 e + f) κ_{3} + \\ (a + 2 b + 2 c + d + e + 2 f) κ_{4} + (a + b + c + d + e + f) κ_{5} + \\ (a + 2 b + c + d + 2 e + 2 f) κ_{6} . \end{matrix}

Hence, the Gray map

\nabla : R_{3} ⟶ F_{3}^{6}

can be established as follows:

\begin{matrix} \nabla (r) : = & (a + b, a + 2 b, a + b + 2 c + d + 2 e + f, a + 2 b + 2 c + d + e + 2 f, \\ a + b + c + d + e + f, a + 2 b + c + d + 2 e + 2 f) N_{1}, \end{matrix}

where

a, b, c, d, e, f \in F_{3}

, and

N_{1} \in G L_{6} (F_{3})

, where

N_{1} = [\begin{matrix} 1 & 2 & 1 & 1 & 0 & 1 \\ 1 & 1 & 2 & 1 & 1 & 0 \\ 0 & 1 & 1 & 2 & 1 & 1 \\ 2 & 1 & 0 & 1 & 2 & 1 \\ 2 & 2 & 2 & 0 & 1 & 1 \\ 2 & 0 & 1 & 1 & 1 & 2 \end{matrix}],

which has the property that

N_{1} N_{1}^{T} = 2 I_{6} .

Example 2.

Let

R_{5} = \frac{F_{5} [w_{1}, w_{2}]}{〈 w_{1}^{2} - α^{2}, w_{2}^{3} - β^{2} w_{2}, w_{1} w_{2} - w_{2} w_{1} 〉}

be a finite commutative non-chain ring, where

α = 2

and

β = 3

are non-zero elements of

F_{5}

. Then, we have

w_{1}^{2} - α^{2} = (w_{1} - 2) (w_{1} + 2)

and

w_{2}^{3} - β^{2} w_{2} = w_{2} (w_{2} - 3) (w_{2} + 3)

. The orthogonal idempotent elements in

R_{5}

are as follows:

\begin{matrix} κ_{1} & = 4 (2 + w_{1}) (1 + w_{2}^{2}), κ_{2} = \frac{1}{2 α β^{2}} (α - w_{1}) (β^{2} - w_{2}^{2}) = 4 (2 - w_{1}) (1 + w_{2}^{2}), \\ κ_{3} & = 2 (2 + w_{1}) (w_{2}^{2} - 3 w_{2}), κ_{4} = \frac{1}{4 α β^{2}} (α - w_{1}) (w_{2}^{2} - β w_{2}) = 2 (2 - w_{1}) (w_{2}^{2} - 3 w_{2}), \\ κ_{5} & = 2 (2 + w_{1}) (w_{2}^{2} + 3 w_{2}), κ_{6} = \frac{1}{4 α β^{2}} (α - w_{1}) (w_{2}^{2} + β w_{2}) = 2 (2 - w_{1}) (w_{2}^{2} + 3 w_{2}), \end{matrix}

where

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

. By Chinese Remainder Theorem, we have

R_{5} = κ_{1} R_{5} \oplus κ_{2} R_{5} \oplus κ_{3} R_{5} \oplus κ_{4} R_{5} \oplus κ_{5} R_{5} \oplus κ_{6} R_{5} ≅ κ_{1} F_{5} \oplus κ_{2} F_{5} \oplus κ_{3} F_{5} \oplus κ_{4} F_{5} \oplus κ_{5} F_{5} \oplus κ_{6} F_{5} .

Therefore, any element

r = a + w_{1} b + w_{2} c + w_{2}^{2} d + w_{1} w_{2} e + w_{1} w_{2}^{2} f \in R

can be expressed as follows:

\begin{matrix} r = & a + w_{1} b + w_{2} c + w_{2}^{2} d + w_{1} w_{2} e + w_{1} w_{2}^{2} f \\ = & (a + 2 b) κ_{1} + (a + 3 b) κ_{2} + (a + 2 b + 2 c + 4 d + 4 e + 3 f) κ_{3} + \\ (a + 3 b + 2 c + 4 d + e + 2 f) κ_{4} + (a + 2 b + 3 c + 4 d + e + 3 f) κ_{5} + \\ (a + 3 b + 3 c + 4 d + 4 e + 2 f) κ_{6} . \end{matrix}

The Gray map

\nabla : R_{5} ⟶ F_{5}^{6}

can be established as follows:

\begin{matrix} \nabla (r) : = & (a + 2 b, a + 3 b, a + 2 b + 2 c + 4 d + 4 e + 3 f, a + 3 b + 2 c + 4 d + e + 2 f, \\ a + 2 b + 3 c + 4 d + e + 3 f, a + 3 b + 3 c + 4 d + 4 e + 2 f) N_{2}, \end{matrix}

where

a, b, c, d, e, f \in F_{5}

, and

N_{2} \in G L_{6} (F_{5})

, where

N_{2} = (\begin{matrix} 3 & 1 & 1 & 1 & 1 & 1 \\ 1 & 3 & 1 & 1 & 1 & 1 \\ 1 & 1 & 3 & 1 & 1 & 1 \\ 1 & 1 & 1 & 3 & 1 & 1 \\ 1 & 1 & 1 & 1 & 3 & 1 \\ 1 & 1 & 1 & 1 & 1 & 3 \end{matrix}),

which has the property that

N_{2} N_{2}^{T} = 4 I_{6} .

Theorem 1.

The Gray map

\nabla : R^{l} ⟶ F_{p}^{6 l}

defined in Equation (2) is linear and isometric.

Proof.

To prove that ∇ is a linear map, assume that

z = a_{1} κ_{1} + a_{2} κ_{2} + a_{3} κ_{3} + a_{4} κ_{4} + a_{5} κ_{5} + a_{6} κ_{6}

and

y = b_{1} κ_{1} + b_{2} κ_{2} + b_{3} κ_{3} + b_{4} κ_{4} + b_{5} κ_{5} + b_{6} κ_{6}

are any two elements of

R

and

λ

is a non-zero scalar in

F_{p}

. Then, we have

\begin{matrix} \nabla (z + y) & = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}, a_{4} + b_{4}, a_{5} + b_{5}, a_{6} + b_{6}) N \\ = [(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) + (b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6})] N \\ = (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) N + (b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}) N \\ = \nabla (z) + \nabla (y), \\ \nabla (λ \cdot z) & = (λ a_{1}, λ a_{2}, λ a_{3}, λ a_{4}, λ a_{5}, λ a_{6}) N \\ = λ (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) N \\ = λ \nabla (z) . \end{matrix}

This ensures that ∇ is a linear map. To prove that ∇ is an isometry, we shall show that the Lee distance and the Hamming distance of codeC are the same. As

z, y \in R^{l}

, then by definition of the Lee distance, we see that

\begin{matrix} d_{L} (z, y) = w t_{H} (\nabla (z - y)) = w t_{H} (\nabla (z) - \nabla (y)) = d_{H} (\nabla (z), \nabla (y)) . \end{matrix}

(3)

Therefore, the Gray map ∇ is an isometry. □

Theorem 2.

Let C be a linear code with parameters

[l, k, d_{L}]

over

R

(i): Then, $\nabla (C)$ is a linear code with parameters $[6 l, k, d_{H}]$ over $F_{p}$ , where $d_{L}$ and $d_{H}$ are the same.
(ii): The image $\nabla (C)$ is self-orthogonal over $F_{p}$ , provided C is self-orthogonal over $R$ .
(iii): The image $\nabla (C)$ is a dual-containing code over $F_{p}$ , provided C is a dual-containing code over $R$ .
(iv): C is a self-dual code over $R$ if and only if $\nabla (C)$ is a self-dual code over $F_{p}$ .

Proof.

(i): The proof follows by Theorem 1.
(ii): IfC is self-orthogonal over $R$ . Then, for any codewords $z = (z_{1}, z_{2}, \dots, z_{l})$ and $y = (y_{1}, y_{2}, \dots, y_{l})$ inC, where $z_{i} = a_{1}^{i} κ_{1} + a_{2}^{i} κ_{2} + a_{3}^{i} κ_{3} + a_{4}^{i} κ_{4} + a_{5}^{i} κ_{5} + a_{6}^{i} κ_{6}$ and $y_{i} = b_{1}^{i} κ_{1} + b_{2}^{i} κ_{2} + b_{3}^{i} κ_{3} + b_{4}^{i} κ_{4} + b_{5}^{i} κ_{5} + b_{6}^{i} κ_{6}$ are elements of $R$ for $1 \leq i \leq l$ , we have $z \cdot y = 0$ . This suggests that $a_{j}^{1} b_{j}^{1} + a_{j}^{2} b_{j}^{2} + \dots + a_{j}^{l} b_{j}^{l} = 0$ for $1 \leq j \leq 6$ . Let $z^{'}, y^{'} \in \nabla (C)$ be any two elements, then some $z, y \in C$ exists such that $z^{'} = \nabla (z)$ and $y^{'} = \nabla (y)$ , i.e.,
$\begin{matrix} z^{'} & = (\nabla (z_{1}), \nabla (z_{2}), \dots, \nabla (z_{l})) \\ = ((a_{1}^{1}, a_{2}^{1}, a_{3}^{1}, a_{4}^{1}, a_{5}^{1}, a_{6}^{1}) N, (a_{1}^{2}, a_{2}^{2}, a_{3}^{2}, a_{4}^{2}, a_{5}^{2}, a_{6}^{2}) N, \dots, (a_{1}^{l}, a_{2}^{l}, a_{3}^{l}, a_{4}^{l}, a_{5}^{l}, a_{6}^{l}) N), \\ y^{'} & = (\nabla (y_{1}), \nabla (y_{2}), \dots, \nabla (y_{l})) \\ = ((b_{1}^{1}, b_{2}^{1}, b_{3}^{1}, b_{4}^{1}, b_{5}^{1}, b_{6}^{1}) N, (b_{1}^{2}, b_{2}^{2}, b_{3}^{2}, b_{4}^{2}, b_{5}^{2}, b_{6}^{2}) N, \dots, (b_{1}^{l}, b_{2}^{l}, b_{3}^{l}, b_{4}^{l}, b_{5}^{l}, b_{6}^{l}) N), \end{matrix}$
where $N \in G L_{6} (F_{p})$ such that $N N^{T} = λ I_{6}$ , $λ \in F_{p} - {0}$ . Now, we have
$\begin{matrix} z^{'} \cdot y^{'} & = \nabla (z) \cdot \nabla (y) = \nabla (z) \cdot \nabla {(y)}^{T} \\ = \sum_{i = 1}^{l} (a_{1}^{i}, a_{2}^{i}, a_{3}^{i}, a_{4}^{i}, a_{5}^{i}, a_{6}^{i}) N N^{T} . (b_{1}^{i}, b_{2}^{i}, b_{3}^{i}, b_{4}^{i}, b_{5}^{i}, b_{6}^{i}) \\ = \sum_{i = 1}^{l} (a_{1}^{i}, a_{2}^{i}, a_{3}^{i}, a_{4}^{i}, a_{5}^{i}, a_{6}^{i}) λ I_{6} . (b_{1}^{i}, b_{2}^{i}, b_{3}^{i}, b_{4}^{i}, b_{5}^{i}, b_{6}^{i}) \\ = \sum_{i = 1}^{l} λ (a_{1}^{i} b_{1}^{i} + a_{2}^{i} b_{2}^{i} + a_{3}^{i} b_{3}^{i} + a_{4}^{i} b_{4}^{i} + a_{5}^{i} b_{5}^{i} + a_{6}^{i} b_{6}^{i}) \\ = \sum_{i = j}^{6} λ (a_{j}^{1} b_{j}^{1} + a_{j}^{2} b_{j}^{2} + \dots + a_{j}^{l} b_{j}^{l}) = 0 . \end{matrix}$
Thus, we have $z^{'} \cdot y^{'} = \nabla (z) \cdot \nabla (y) = 0$ for all $z^{'}, y^{'} \in \nabla (C)$ ifC is self-orthogonal over $R$ . Hence, $\nabla (C)$ is a self-orthogonal code of length $6 l$ over $F_{p}$ , providedC is a self-orthogonal code over $R$ .
(iii): Suppose that $C^{⊥} \subseteq C$ , then by the linearity of ∇, we have $\nabla (C^{⊥}) \subseteq \nabla (C)$ . To prove that $\nabla (C)$ is dual-containing, it remains to show that $\nabla (C^{⊥}) = \nabla {(C)}^{⊥}$ . For this, let $z = (z_{1}, z_{2}, \dots, z_{l}) \in C$ and $y = (y_{1}, y_{2}, \dots, y_{l}) \in C^{⊥}$ , where $z_{i} = a_{1}^{i} κ_{1} + a_{2}^{i} κ_{2} + a_{3}^{i} κ_{3} + a_{4}^{i} κ_{4} + a_{5}^{i} κ_{5} + a_{6}^{i} κ_{6}$ and $y_{i} = b_{1}^{i} κ_{1} + b_{2}^{i} κ_{2} + b_{3}^{i} κ_{3} + b_{4}^{i} κ_{4} + b_{5}^{i} κ_{5} + b_{6}^{i} κ_{6}$ are elements of $R$ for $1 \leq i \leq l$ . Now, $x \cdot y = 0$ gives that $a_{j}^{1} b_{j}^{1} + a_{j}^{2} b_{j}^{2} + \dots + a_{j}^{l} b_{j}^{l} = 0$ for $1 \leq j \leq 6$ . Consider
$\begin{matrix} \nabla (z) & = ((a_{1}^{1}, a_{2}^{1}, a_{3}^{1}, a_{4}^{1}, a_{5}^{1}, a_{6}^{1}) N, (a_{1}^{2}, a_{2}^{2}, a_{3}^{2}, a_{4}^{2}, a_{5}^{2}, a_{6}^{2}) N, \dots, (a_{1}^{l}, a_{2}^{l}, a_{3}^{l}, a_{4}^{l}, a_{5}^{l}, a_{6}^{l}) N), \\ \nabla (y) & = ((b_{1}^{1}, b_{2}^{1}, b_{3}^{1}, b_{4}^{1}, b_{5}^{1}, b_{6}^{1}) N, (b_{1}^{2}, b_{2}^{2}, b_{3}^{2}, b_{4}^{2}, b_{5}^{2}, b_{6}^{2}) N, \dots, (b_{1}^{l}, b_{2}^{l}, b_{3}^{l}, b_{4}^{l}, b_{5}^{l}, b_{6}^{l}) N), \end{matrix}$
Now, $\nabla (z) \cdot \nabla (y) = 0$ suggests that $\nabla (y) \in \nabla {(C)}^{⊥}$ . Thus, we have $\nabla (C^{⊥}) \subseteq \nabla {(C)}^{⊥}$ . Contrarily, ∇ is a bijective linear map, so the sizes of $\nabla (C^{⊥})$ and $\nabla {(C)}^{⊥}$ are the same. Thus, $\nabla (C^{⊥}) = \nabla {(C)}^{⊥}$ . Hence, $\nabla (C)$ is a dual-containing code over $F_{p}$ providedC is a dual-containing code over $R$ .
(iv): It follows from part (iii).

□

Theorem 3

([11]).Let

C = κ_{1} \bar{C_{1}} \oplus κ_{2} \bar{C_{2}} \oplus κ_{3} \bar{C_{3}} \oplus κ_{4} \bar{C_{4}} \oplus κ_{5} \bar{C_{5}} \oplus κ_{6} \bar{C_{6}}

be a linear code over

R

. Then:

(i): $C^{⊥} = κ_{1} {\bar{C_{1}}}^{⊥} \oplus κ_{2} {\bar{C_{2}}}^{⊥} \oplus κ_{3} {\bar{C_{3}}}^{⊥} \oplus κ_{4} {\bar{C_{4}}}^{⊥} \oplus κ_{5} {\bar{C_{5}}}^{⊥} \oplus κ_{6} {\bar{C_{6}}}^{⊥}$ ;
(ii): C is self-dual over $R$ if $\bar{C_{i}}$ are self-dual codes over $F_{p}$ for $1 \leq i \leq 6$ .

Here, we define the direct sum and the direct product as defined by Dinh et al. [24] in the following ways:

\begin{matrix} D_{1} \oplus D_{2} = {d_{1} + d_{2} | d_{j} \in D_{j}; j = 1, 2}, \end{matrix}

(4)

\begin{matrix} D_{1} \otimes D_{2} = {(d_{1}, d_{2}) | d_{j} \in D_{j}; j = 1, 2} . \end{matrix}

(5)

Suppose thatC is a linear code with lengthl over

R

. Consider the following sets:

\begin{matrix} \bar{C_{1}} & = {\bar{a} \in F_{p}^{l} | κ_{1} \bar{a} + κ_{2} \bar{b} + κ_{3} \bar{c} + κ_{4} \bar{d} + κ_{5} \bar{e} + κ_{6} \bar{f} \in C; for some \bar{b}, \bar{c}, \bar{d}, \bar{e}, \bar{f} \in F_{p}^{l}}; \\ \bar{C_{2}} & = {\bar{b} \in F_{p}^{l} | κ_{1} \bar{a} + κ_{2} \bar{b} + κ_{3} \bar{c} + κ_{4} \bar{d} + κ_{5} \bar{e} + κ_{6} \bar{f} \in C; for some \bar{a}, \bar{c}, \bar{d}, \bar{e}, \bar{f} \in F_{p}^{l}}; \\ \bar{C_{3}} & = {\bar{c} \in F_{p}^{l} | κ_{1} \bar{a} + κ_{2} \bar{b} + κ_{3} \bar{c} + κ_{4} \bar{d} + κ_{5} \bar{e} + κ_{6} \bar{f} \in C; for some \bar{a}, \bar{b}, \bar{d}, \bar{e}, \bar{f} \in F_{p}^{l}}; \\ \bar{C_{4}} & = {\bar{d} \in F_{p}^{l} | κ_{1} \bar{a} + κ_{2} \bar{b} + κ_{3} \bar{c} + κ_{4} \bar{d} + κ_{5} \bar{e} + κ_{6} \bar{f} \in C; for some \bar{a}, \bar{b}, \bar{c}, \bar{e}, \bar{f} \in F_{p}^{l}}; \\ \bar{C_{5}} & = {\bar{e} \in F_{p}^{l} | κ_{1} \bar{a} + κ_{2} \bar{b} + κ_{3} \bar{c} + κ_{4} \bar{d} + κ_{5} \bar{e} + κ_{6} \bar{f} \in C; for some \bar{a}, \bar{b}, \bar{c}, \bar{d}, \bar{f} \in F_{p}^{l}}; \\ \bar{C_{6}} & = {\bar{f} \in F_{p}^{l} | κ_{1} \bar{a} + κ_{2} \bar{b} + κ_{3} \bar{c} + κ_{4} \bar{d} + κ_{5} \bar{e} + κ_{6} \bar{f} \in C; for some \bar{a}, \bar{b}, \bar{c}, \bar{d}, \bar{e} \in F_{p}^{l}} . \end{matrix}

It can be seen that

\bar{C_{i}}

for

1 \leq i \leq 6

is a linear code with lengthl over

F_{p}

. Therefore, we can express a linear codeC with lengthl over

R

C = κ_{1} \bar{C_{1}} \oplus κ_{2} \bar{C_{2}} \oplus κ_{3} \bar{C_{3}} \oplus κ_{4} \bar{C_{4}} \oplus κ_{5} \bar{C_{5}} \oplus κ_{6} \bar{C_{6}}

. If

G_{i}

is the generator matrix of

\bar{C_{i}}

for

1 \leq i \leq 6

, then the generator matrix

\nabla (G)

of the Gray image

\nabla (C)

is given as follows:

\nabla (G) = [\begin{matrix} \nabla (κ_{1} G_{1}) \\ \nabla (κ_{2} G_{2}) \\ \nabla (κ_{3} G_{3}) \\ \nabla (κ_{4} G_{4}) \\ \nabla (κ_{5} G_{5}) \\ \nabla (κ_{6} G_{6}) \end{matrix}] .

3. $Λ$ -Constacyclic Codes over $R$

A constacyclic code is an important class of linear error-correcting codes. It is a generalization of cyclic codes, which are themselves a subset of linear codes. Suppose that

Λ = κ_{1} a + κ_{2} b + κ_{3} c + κ_{4} d + κ_{5} e + κ_{6} f

is a unit element in

R

. Then, a linear code C with lengthl over

R

is called a

Λ

-constacyclic code if, for any codeword

c = (c_{0}, c_{1}, \dots, c_{l - 1})

inC, it satisfies the property that

ω_{Λ} (c) = (Λ c_{l - 1}, c_{0}, \dots, c_{l - 2})

is again a member ofC. In particular, if

Λ = 1

, then

Λ

-constacyclic codeC becomes a cyclic code, and if

Λ = - 1

, thenC becomes a negacyclic code.

Lemma 1.

Let

Λ = κ_{1} a + κ_{2} b + κ_{3} c + κ_{4} d + κ_{5} e + κ_{6} f \in R

be a non-zero element. Then, the element

Λ \in R

is a unit element in

R

a, b, c, d, e, f

are unit elements in

F_{p}

. Moreover, when

Λ \in R

is a unit element, then its inverse is given by

Λ^{- 1} = κ_{1} a^{- 1} + κ_{2} b^{- 1} + κ_{3} c^{- 1} + κ_{4} d^{- 1} + κ_{5} e^{- 1} + κ_{6} f^{- 1}

Proof.

Suppose that

Λ = κ_{1} a + κ_{2} b + κ_{3} c + κ_{4} d + κ_{5} e + κ_{6} f \in R

is a unit element. Then, an element

Λ_{1} = κ_{1} a_{1} + κ_{2} b_{1} + κ_{3} c_{1} + κ_{4} d_{1} + κ_{5} e_{1} + κ_{6} f_{1} \in R

exists such that

Λ Λ_{1} = 1

. Using the idempotent orthogonality of

κ_{i}

for

1 \leq i \leq 6

, we have

κ_{1} a a_{1} + κ_{2} b b_{1} + κ_{3} c c_{1} + κ_{4} d d_{1} + κ_{5} e e_{1} + κ_{6} f f_{1} = 1

. Putting the values of

κ_{i}

for

1 \leq i \leq 6

and comparing the constant term and coefficients of

w_{1}, w_{2}, w_{2}^{2}, w_{1} w_{2}, w_{1} w_{2}^{2}

, we obtain

\begin{matrix} a a_{1} + b b_{1} & = 2, \\ a a_{1} - b b_{1} & = 0, \\ - c c_{1} - d d_{1} + e e_{1} + f f_{1} & = 0, \\ c c_{1} + d d_{1} + e e_{1} + f f_{1} & = 4, \\ - c c_{1} + d d_{1} + e e_{1} - f f_{1} & = 0, \\ c c_{1} - d d_{1} + e e_{1} - f f_{1} & = 0 . \end{matrix}

Solving these equations, we obtain

a a_{1} = 1

b b_{1} = 1

c c_{1} = 1

d d_{1} = 1

e e_{1} = 1

, and

f f_{1} = 1

. Therefore, we have

Λ^{- 1} = κ_{1} a^{- 1} + κ_{2} b^{- 1} + κ_{3} c^{- 1} + κ_{4} d^{- 1} + κ_{5} e^{- 1} + κ_{6} f^{- 1}

The converse part can be performed in a similar way. □

Theorem 4.

Let

C = κ_{1} \bar{C_{1}} \oplus κ_{2} \bar{C_{2}} \oplus κ_{3} \bar{C_{3}} \oplus κ_{4} \bar{C_{4}} \oplus κ_{5} \bar{C_{5}} \oplus κ_{6} \bar{C_{6}}

be a linear code over

R

and

Λ = κ_{1} Λ_{1} + κ_{2} Λ_{2} + κ_{3} Λ_{3} + κ_{4} Λ_{4} + κ_{5} Λ_{5} + κ_{6} Λ_{6} \in R

be a unit element. Then, C is aΛ-constacyclic code over

R

\bar{C_{i}}

is a

Λ_{i}

-constacyclic code for

1 \leq i \leq 6

over

F_{p}

Proof.

Suppose thatC is a

Λ

-constacyclic code with lengthl over

R

. If

c = (r_{0}, r_{1}, \dots, r_{l - 1}) \in C

, where

r_{j} = κ_{1} a_{1, j} + κ_{2} a_{2, j} + κ_{3} a_{3, j} + κ_{4} a_{4, j} + κ_{5} a_{5, j} + κ_{6} a_{6, j}

such that

a_{i, j} \in F_{p}

for

1 \leq i \leq 6

and

0 \leq j \leq l - 1

, then we have

(a_{i, 0}, a_{i, 1}, \dots, a_{i, l - 1}) \in \bar{C_{i}}

. Thus, the

Λ

-constacyclic shift ofc is

ω_{Λ} (c) = (Λ r_{l - 1}, r_{0}, \dots, r_{l - 2}) \in C

, where

Λ r_{l - 1} = κ_{1} Λ_{1} a_{1, l - 1} + κ_{2} Λ_{2} a_{2, l - 1} + κ_{3} Λ_{3} a_{3, l - 1} + κ_{4} Λ_{4} a_{4, l - 1} + κ_{5} Λ_{5} a_{5, l - 1} + κ_{6} Λ_{6} a_{6, l - 1} .

Therefore, we obtain

ω_{Λ} (c) = \sum_{i = 1}^{6} κ_{i} (Λ_{i} a_{i, l - 1}, a_{i, 0}, a_{i, 1}, \dots, a_{i, l - 2}) \in C

, which leads to

(Λ_{i} a_{i, l - 1}, a_{i, 0}, a_{i, 1}, \dots, a_{i, l - 2}) \in \bar{C_{i}} .

Therefore,

\bar{C_{i}}

is a

Λ_{i}

-constacyclic code for

1 \leq i \leq 6

of lengthl over

F_{p}

Conversely, assume that

\bar{C_{i}}

is a

Λ_{i}

-constacyclic code of lengthl over

F_{p}

for

1 \leq i \leq 6

. Then, for a vector

\bar{a_{i}} = (a_{i, 0}, a_{i, 1}, \dots, a_{i, l - 1}) \in \bar{C_{i}}

, we have

ω_{Λ_{i}} \bar{(a_{i})} = (Λ_{i} a_{i, l - 1}, a_{i, 0}, a_{i, 1}, \dots, a_{i, l - 2}) \in \bar{C_{i}}

. Thus, we have

\sum_{i = 1}^{6} κ_{i} ω_{Λ_{i}} \bar{(a_{i})} = \sum_{i = 1}^{6} κ_{i} (Λ_{i} a_{i, l - 1}, a_{i, 0}, a_{i, 1}, \dots, a_{i, l - 2}) = (Λ r_{l - 1}, r_{0}, \dots, r_{l - 2}) = ω_{Λ} (c) .

Therefore, if

\bar{C_{i}}

is a

Λ_{i}

-constacyclic code for

1 \leq i \leq 6

of lengthl over

F_{p}

, thenC is a

Λ

-constacyclic code over

R

. □

Theorem 5.

Let

C = κ_{1} \bar{C_{1}} \oplus κ_{2} \bar{C_{2}} \oplus κ_{3} \bar{C_{3}} \oplus κ_{4} \bar{C_{4}} \oplus κ_{5} \bar{C_{5}} \oplus κ_{6} \bar{C_{6}}

be aΛ-constacyclic code over

R

and

p_{i} (z) \in \frac{F_{p} [z]}{〈 z^{l} - Λ_{i} 〉}

a unique monic polynomial of the lowest degree such that

\bar{C_{i}} = 〈 p_{i} (z) 〉

and

p_{i} (z) | (z^{l} - Λ_{i})

for

1 \leq i \leq 6

. Then,

C = 〈 p (z) 〉

, where

p (z) = κ_{1} p_{1} (z) + κ_{2} p_{2} (z) + κ_{3} p_{3} (z) + κ_{4} p_{4} (z) + κ_{5} p_{5} (z) + κ_{6} p_{6} (z)

and

p (z) | (z^{l} - Λ)

Proof.

Suppose that

C = κ_{1} \bar{C_{1}} \oplus κ_{2} \bar{C_{2}} \oplus κ_{3} \bar{C_{3}} \oplus κ_{4} \bar{C_{4}} \oplus κ_{5} \bar{C_{5}} \oplus κ_{6} \bar{C_{6}}

is a

Λ

-constacyclic code with lengthl over

R

, then each

\bar{C_{i}}

is a

Λ_{i}

-constacyclic code over

F_{p}

for

1 \leq i \leq 6

. Therefore,

\bar{C_{i}} \subseteq \frac{F_{p} [z]}{〈 z^{l} - Λ_{i} 〉}

is a principal ideal generated by a monic polynomial

p_{i} (z) \in \frac{F_{p} [z]}{〈 z^{l} - Λ_{i} 〉}

of lowest degree such that

p_{i} (z) | (z^{l} - Λ_{i})

for

1 \leq i \leq 6

. Thus,

κ_{i} p_{i} (z)

are the generator polynomials ofC.

If we take

p (z) = κ_{1} p_{1} (z) + κ_{2} p_{2} (z) + κ_{3} p_{3} (z) + κ_{4} p_{4} (z) + κ_{5} p_{5} (z) + κ_{6} p_{6} (z)

, then

〈 p (z) 〉 \subseteq C

. Furthermore, we see that

κ_{i} p (z) = κ_{i} p_{i} (z) \in 〈 p (z) 〉

implies that

C \subseteq 〈 p (z) 〉

. Thus, we conclude that

C = 〈 p (z) 〉

Moreover, we have

p_{i} (z) \in \frac{F_{p} [z]}{〈 z^{l} - Λ_{i} 〉}

such that

p_{i} (z) | (z^{l} - Λ_{i})

. Thus, polynomials

q_{i} (z) \in F_{p} [z]

exist such that

(z^{l} - Λ_{i}) = p_{i} (z) q_{i} (z)

for

1 \leq i \leq 6

. Thus, we have

p (z) (\sum_{i = 1}^{6} κ_{i} q_{i} (z)) = \sum_{i = 1}^{6} κ_{i} p_{i} (z) q_{i} (z) = κ_{i} (z^{l} - Λ_{i}) = z^{l} - Λ .

Thus, we conclude that

p (z) | (z^{l} - Λ)

. □

Corollary 1.

Let

C = \oplus_{i = 1}^{6} κ_{i} \bar{C_{i}}

be aΛ-constacyclic code over

R

, and

\bar{C_{i}} = 〈 p_{i} (z) 〉

such that

z^{l} - Λ_{i} = p_{i} (z) q_{i} (z)

for

1 \leq i \leq 6

. Then:

(i): $C^{⊥} = \oplus_{i = 1}^{6} κ_{i} {\bar{C_{i}}}^{⊥}$ is a $Λ^{- 1}$ -constacyclic code over $R$ ;
(ii): $C^{⊥} = 〈 \sum_{i = 1}^{6} κ_{i} q_{i}^{*} (z) 〉$ , where $q_{i}^{*} (z)$ is the reciprocal polynomial of $q_{i} (z)$ , which is defined as $q_{i}^{*} (z) = z^{deg (q_{i} (z))} q_{i} (z^{- 1})$ for $1 \leq i \leq 6$ ;
(iii): $| C^{⊥} | = p^{\sum_{i = 1}^{6} deg (p_{i} (z))}$ .

4. Dual-Containing $Λ$ -Constacyclic Codes

The dual-containing code is a very important class of code for the construction of quantum error-correcting codes.

Definition 1.

Suppose that C is aΛ-constacyclic code of length l over

R

, whereΛ is a unit element of

R

. Then, C is said to be dual-containing if

C^{⊥} \subseteq C

Proposition 1.

Let C be aΛ-constacyclic code over

R

, where

Λ = κ_{1} Λ_{1} + κ_{2} Λ_{2} + κ_{3} Λ_{3} + κ_{4} Λ_{4} + κ_{5} Λ_{5} + κ_{6} Λ_{6} \in R

. If C is a non-trivial dual-containing code, then

Λ_{i} = \pm 1

for

1 \leq i \leq 6

, i.e.,

Λ \in {\pm κ_{1} \pm κ_{2} \pm κ_{3} \pm κ_{4} \pm κ_{5} \pm κ_{6}} \in R

Remark 1.

Suppose that C is aΛ-constacyclic code over

R

, then from Proposition 1 we conclude that:

(i): If $Λ = 1$ , then $Λ_{i} = 1$ and $C_{i}$ is a cyclic code over $F_{p}$ for $1 \leq i \leq 6$ .
(ii): If $Λ = - 1$ , then $Λ_{i} = - 1$ and $C_{i}$ is a negacyclic code over $F_{p}$ for $1 \leq i \leq 6$ .
(iii): If $Λ_{i} = 1$ and $Λ_{j} = - 1$ , then $C_{i}$ is a cyclic code, and $C_{j}$ is a negacyclic code over $F_{p}$ for $1 \leq i \neq j \leq 6$ .

Example 3.

Let C be a

(w_{1} - w_{2}^{2} - w_{1} w_{2}^{2})

-constacyclic code over

R

, then

Λ_{1} = 1

implies that

C_{1}

is a cyclic code, and

Λ_{j} = - 1

further implies that

C_{j}

is a negacyclic code for

2 \leq j \leq 6

over

F_{p}

Example 4.

Let C be a

(1 - 2 w_{2}^{2})

-constacyclic code over

R

, then

Λ_{i} = 1

implies that

C_{i}

is a cyclic code for

i = 1, 2

, and

Λ_{j} = - 1

further implies that

C_{j}

is a negacyclic code for

3 \leq j \leq 6

over

F_{p}

Example 5.

Let C be a

(1 - \frac{1}{2} w_{2} - \frac{3}{2} w_{2}^{2} - w_{1} w_{2} + w_{1} w_{2}^{2})

-constacyclic code over

R

, then

Λ_{i} = 1

implies that

C_{i}

is a cyclic code for

i = 1, 2, 3

, and

Λ_{j} = - 1

further implies that

C_{j}

is a negacyclic code for

j = 4, 5, 6

over

F_{p}

Lemma 2

([6]).Let

C_{j}

be a

Λ_{j}

-constacyclic code with generator polynomial

p_{j} (z)

over

F_{p}

. Then,

C_{j}

is a dual-containing code if

z^{n} - Λ_{j} \equiv 0 mod (p_{j} (z) p_{j}^{*} (z))

, where

Λ_{j} = \pm 1

and

p_{j}^{*} (z)

is the reciprocal polynomial of

p_{j} (z)

, for

j = 1, 2, \dots, 6

Lemma 3.

Let C be a linear code over

R

and

C^{⊥}

be the dual of C. If∇ is a Gray map as defined in Equation (2), then

\nabla (C^{⊥}) = \nabla {(C)}^{⊥}

. Moreover, if C is a self-orthogonal (self-dual) code over

R

, then

\nabla (C)

is a self-orthogonal (resp. self-dual) code over

F_{p}

Proof.

The set

K = {κ_{1}, κ_{2}, \dots, κ_{6}}

forms a basis for 6-dimensional vector space

R

over

F_{p}

. An element

r \in R

can be uniquely expressed as

r = κ_{1} a_{1} + κ_{2} a_{2} + κ_{3} a_{3} + κ_{4} a_{4} + κ_{5} a_{5} + κ_{6} a_{6}

, where

a_{i} \in F_{p}

for

1 \leq i \leq 6

. Then, we have

\nabla (r) = (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) M,

where

M \in G L_{6} (F_{p})

such that

M M^{T} = α I_{6}

and

α \in F_{p} - {0}

. Let

z = (z_{0}, z_{1}, \dots, z_{l - 1}) \in C \subseteq R^{l}

, where

z_{j} = κ_{1} a_{1}^{j} + κ_{2} a_{2}^{j} + κ_{3} a_{3}^{j} + κ_{4} a_{4}^{j} + κ_{5} a_{5}^{j} + κ_{6} a_{6}^{j} \in R

for

0 \leq j \leq l - 1

. Then, we have

z = κ_{1} a_{1} + κ_{2} a_{2} + κ_{3} a_{3} + κ_{4} a_{4} + κ_{5} a_{5} + κ_{6} a_{6}

, where

a_{i} = (a_{i}^{0}, a_{i}^{1}, \dots, a_{i}^{l - 1}) \in F_{p}^{l}

. Suppose that

y = κ_{1} b_{1} + κ_{2} b_{2} + κ_{3} b_{3} + κ_{4} b_{4} + κ_{5} b_{5} + κ_{6} b_{6} \in C^{⊥}

. Then, we obtain that

z \cdot y = 0

implies that

κ_{1} a_{1} b_{1} + κ_{2} a_{2} b_{2} + κ_{3} a_{3} b_{3} + κ_{4} a_{4} b_{4} + κ_{5} a_{5} b_{5} + κ_{6} a_{6} b_{6} = 0 .

SinceK is linearly independent, we obtain

a_{i} b_{i} = 0

for

1 \leq i \leq 6

. Also, we have

\nabla (z) = (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) M \in \nabla (C)

and

\nabla (y) = (b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}) M \in \nabla (C^{⊥})

. Consider

\begin{matrix} \nabla (z) \cdot \nabla (y) = \nabla (z) \cdot \nabla {(y)}^{T} = & (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) M M^{T} (b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}) \\ = & α (a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} + a_{4} b_{4} + a_{5} b_{5} + a_{6} b_{6}) \\ = & 0 . \end{matrix}

Therefore,

\nabla (y) \in \nabla {(C)}^{⊥}

, i.e.,

\nabla (C^{⊥}) \subseteq \nabla {(C)}^{⊥}

. Since the Gray map ∇ is bijective,

| \nabla (C^{⊥}) | = | \nabla {(C)}^{⊥} |

suggests that

\nabla (C^{⊥}) = \nabla {(C)}^{⊥}

. IfC is a self-orthogonal code, then

C \subseteq C^{⊥}

, and hence,

\nabla (C) \subseteq \nabla (C^{⊥}) = \nabla {(C)}^{⊥}

. Therefore,

\nabla (C)

is a self-orthogonal code. □

Theorem 6

([25]).Let

C_{1} = {[n, k_{1}, d_{1}]}_{q}

and

C_{2} = {[n, k_{2}, d_{2}]}_{q}

be two linear codes over

G F (q)

with

C_{2}^{⊥} \subseteq C_{1}

. Then, a QECC exists with parameters

{[[n, k_{1} + k_{2} - n, d]]}_{q}

, where

d = min {w t (v) : v \in (C_{1} ∖ C_{2}^{⊥}) \cup (C_{2} ∖ C_{1}^{⊥})} \geq min {d_{1}, d_{2}}

. Moreover, if

C_{1}

is a dual-containing code, then a QECC with parameters

{[[n, 2 k_{1} - n, d_{1}]]}_{q}

exists, where

d_{1} = min {w t (v) : v \in C_{1} ∖ C_{1}^{⊥}}

Theorem 7

([6]).Let C be aΛ-constacyclic code over

F_{p}

having a generator polynomial

p (z)

. Then, C is dual-containing if

(z^{l} - Λ) \equiv 0 mod (p (z) p^{*} (z))

, where

Λ = \pm 1

The dual-containing cyclic and negacyclic codes over

F_{p}

are provided by Theorem 7. Using this outcome, we can now ascertain the prerequisites and requirements for

Λ

-constacyclic codes over

R

to have their duals, as demonstrated in the following theorem.

Theorem 8.

Let

C = \oplus_{i = 1}^{6} κ_{i} \bar{C_{i}}

be aΛ-constacyclic code of length l over

R

, where

Λ = κ_{1} Λ_{1} + κ_{2} Λ_{2} + \dots + κ_{6} Λ_{6} \in R

and

C = 〈 p (z) 〉 = 〈 κ_{1} p_{1} (z) + κ_{2} p_{2} (z) + \dots + κ_{6} p_{6} (z) 〉

, where

p_{i} (z)

is the generating polynomial of code

\bar{C_{i}}

over

F_{p}

for

1 \leq i \leq 6

. Then, C is a dual-containing code if and only if

(z^{l} - Λ_{i}) \equiv 0 mod (p_{i} (z) p_{i}^{*} (z))

, where

Λ_{i} = \pm 1

for

1 \leq i \leq 6

Proof.

Suppose that

C = \oplus_{i = 1}^{6} κ_{i} \bar{C_{i}}

is a

Λ

-constacyclic code over

R

, where

Λ = κ_{1} Λ_{1} + κ_{2} Λ_{2} + \dots + κ_{6} Λ_{6} \in R

. Then, by Theorem 4, the code

\bar{C_{i}}

is a

Λ_{i}

-constacyclic code with generating polynomial

p_{i} (z)

over

F_{p}

. IfC is a dual-containing code, then we have

\oplus_{i = 1}^{6} κ_{i} {\bar{C_{i}}}^{⊥} \subseteq \oplus_{i = 1}^{6} κ_{i} \bar{C_{i}}

. Since this expression is unique, we have

{\bar{C_{i}}}^{⊥} \subseteq \bar{C_{i}}

. Therefore, by Lemma 2 we have

(z^{l} - Λ_{i}) \equiv 0 mod (p_{i} (z) \cdot p_{i}^{*} (z))

. □

Corollary 2.

Let

C = \oplus_{i = 1}^{6} κ_{i} \bar{C_{i}}

be aΛ-constacyclic code over

R

. Then, C is a dual-containing code over

R

if and only if

C_{i}

is a dual-containing code over

F_{p}

for

1 \leq i \leq 6

Theorem 9.

Let

C = \oplus_{i = 1}^{6} κ_{i} \bar{C_{i}}

be aΛ-constacyclic code of length l over

R

, and∇ be the Gray map. If

\nabla (C)

has parameters

[6 l, k, d_{H}]

, where

k = k_{1} + k_{2} + \dots + k_{6}

is the dimension of

\nabla (C)

and

d_{L}

is the Lee distance of C, if C is a dual-containing code, then a QECC exists with parameters

[6 l, 2 k - 6 l, d_{H}]

over

F_{p}

Proof.

Suppose thatC is a dual-containing code over

R

and ∇ is a Gray map. Then,

\nabla (C)

is also a dual-containing code with parameters

[6 l, k, d_{H}]

over

F_{p}

. Therefore, by Theorem 6, a QECC with parameters

{[[6 l, 2 k - 6 l, d_{H}]]}_{p}

exists over

F_{p}

. □

Example 6.

Let

R_{3} = \frac{F_{3} [w_{1}, w_{2}]}{〈 w_{1}^{2} - 1, w_{2}^{3} - w_{2}, w_{1} w_{2} - w_{2} w_{1} 〉}

be a finite non-chain ring. Suppose that

Λ = 2 w_{2}^{2} - 1

is a unit element in

R_{3}

. Then,

Λ_{1} = Λ_{2} = - 1

and

Λ_{3} = Λ_{4} = Λ_{5} = Λ_{6} = 1

. Thus, in

F_{3} [z]

, we have

\begin{matrix} z^{9} - 1 & = {(z + 2)}^{9} \\ z^{9} + 1 & = {(z + 1)}^{9} . \end{matrix}

Let

p_{1} (z) = p_{2} (z) = {(z + 1)}^{4}

p_{3} (z) = p_{4} (z) = (z + 2)

, and

p_{5} (z) = p_{6} (z) = 1

be the generator polynomials of

\bar{C_{i}}

for

1 \leq i \leq 6

, respectively. Then,

C = 〈 κ_{1} p_{1} (z) + κ_{2} p_{2} (z) + \dots + κ_{6} p_{6} (z) 〉

is a

(2 w_{2}^{2} - 1)

-constacyclic code of length9 over

R_{3}

. Let

N_{1} \in G L_{6} (F_{3})

, as given in Example 1, then

N_{1} N_{1}^{T} = 2 I_{6}

and the Gray image

\nabla (C)

has the parameters

[54, 44, 4]

. Moreover,

(x^{9} - Λ_{i}) \equiv 0 mod (p_{i} (z) p_{i}^{*} (z))

for

1 \leq i \leq 6

; thus, by Theorem 8, we find that C is a dual-containing code; so, by Theorem 9, we have a QECC

{[[54, 34, 4]]}_{3}

, which is a new QECC with this parameter.

Remark 2.

In the previous example, we have seen that the Gray image

\nabla (C)

is a linear code with parameters

[54, 44, 4]

over the field

F_{3}

. Specifically, for a code with a length9, the Gray image’s length is54, and its dimension is equal to the rational sum of the dimensions of the individual codes, yielding44 as a result. Let

G_{i}

denote the generator matrix of

\bar{C_{i}} = 〈 p_{i} (z) 〉

for

i \in {1, 2, \dots, 6}

. Then, the generator matrix for

\nabla (C)

is given inSection 2.

After providing the generator matrix

\nabla (G)

as input to the Magma Computation System [22], it was determined that the minimum distance of

\nabla (C)

is4. Based on this computation, it is crucial to note that the minimum distance of the Gray image is greater than the distance of each

C_{i}

. As in Example 6,

d_{H} (C_{1}) = d_{H} (C_{2}) = 3

d_{H} (C_{3}) = d_{H} (C_{4}) = 2

, and

d_{H} (C_{5}) = d_{H} (C_{6}) = 1

, while the Lee distance is4. Notably, employing the canonical Gray map rather than the Gray map∇ would result in a Lee distance of1 instead of4. Which underlines one of the primary advantages of using the Gray map∇.

Example 7.

Let

R_{5} = \frac{F_{5} [w_{1}, w_{2}]}{〈 w_{1}^{2} - 4, w_{2}^{3} - 4 w_{2}, w_{1} w_{2} - w_{2} w_{1} 〉}

be a non-chain ring. Suppose that

Λ = 1 + w_{2} (1 + 3 w_{1}) (1 + 3 w_{2})

is a unit element in

R_{5}

. Then,

Λ_{1} = Λ_{2} = Λ_{3} = 1

and

Λ_{4} = Λ_{5} = Λ_{6} = - 1

. Thus, in

F_{5} [z]

, we have

\begin{matrix} z^{15} - 1 & = {(z + 4)}^{5} {(z^{2} + z + 1)}^{5} \\ z^{15} + 1 & = {(z + 1)}^{5} {(z^{2} + 4 z + 1)}^{5} . \end{matrix}

Let

p_{1} (z) = (z^{2} + z + 1)

p_{2} (z) = {(z + 4)}^{2}

p_{3} (z) = 1

, and

p_{4} (z) = p_{5} (z) = p_{6} (z) = (z + 1)

be the generator polynomials of

\bar{C_{i}}

for

1 \leq i \leq 6

, respectively. Then,

C = 〈 κ_{1} p_{1} (z) + κ_{2} p_{2} (z) + \dots + κ_{6} p_{6} (z) 〉

is a

(1 + \frac{1}{2} (1 - w_{1}) (w_{2} - w_{2}^{2}))

-constacyclic code with length15 over

R_{5}

. Let

N_{2} \in G L_{6} (F_{5})

, as given in Example 2, then

N_{2} N_{2}^{T} = 4 I_{6}

and the Gray image

\nabla (C)

has the parameters

[90, 83, 3]

. Moreover,

(z^{15} - Λ_{i}) \equiv 0 mod (p_{i} (z) p_{i}^{*} (z))

for

1 \leq i \leq 6

; thus, by Theorem 8, we find that C is a dual-containing code; hence, by Theorem 9 we have a new QECC

{[[90, 76, 3]]}_{5}

, with this parameter. Again, here we can see that the distance of

\nabla (C) \geq d_{H} (\bar{C_{i}})

for

1 \leq i \leq 6

Example 8.

Let

R_{5} = \frac{F_{5} [w_{1}, w_{2}]}{〈 w_{1}^{2} + 1, w_{2}^{3} - 4 w_{2}, w_{1} w_{2} - w_{2} w_{1} 〉}

be a non-chain ring. Suppose that

Λ = 4

is a unit element in

R_{5}

. Then,

Λ_{1} = - 1 = Λ_{2}

and

Λ_{3} = 1 = Λ_{4} = Λ_{5} = Λ_{6}

. Thus, in

F_{5} [z]

, we have

\begin{matrix} z^{35} - 1 & = {(z + 4)}^{5} {(z^{6} + z^{5} + z^{4} + z^{3} + z^{2} + z + 1)}^{5} \\ z^{35} + 1 & = {(z + 1)}^{5} {(z^{6} + 4 z^{5} + z^{4} + 4 z^{3} + z^{2} + 4 z + 1)}^{5} . \end{matrix}

Let

p_{1} (z) = (z^{6} + 4 z^{5} + z^{4} + 4 z^{3} + z^{2} + 4 z + 1)

p_{2} (z) = {(z + 1)}^{2}

p_{3} (z) = 1

, and

p_{4} (z) = p_{5} (z) = p_{6} (z) = (z + 4)

be the generator polynomials of

\bar{C_{i}}

for

1 \leq i \leq 6

, respectively. Then,

C = 〈 κ_{1} p_{1} (z) + κ_{2} p_{2} (z) + \dots + κ_{6} p_{6} (z) 〉

is a Λ-constacyclic code with length35 over

R_{5}

. Let

N_{2} \in G L_{6} (F_{5})

, as given in Example 2, then

N_{2} N_{2}^{T} = 4 I_{6}

and the Gray image

\nabla (C)

has the parameters

[210, 199, 3]

. Moreover,

(z^{35} - Λ_{i}) \equiv 0 mod (p_{i} (z) p_{i}^{*} (z))

for

1 \leq i \leq 6

; thus, by Theorem 8, we find that C is a dual-containing code; so, by Theorem 9 we have an improved QECC

{[[210, 188, 3]]}_{5}

against the existing code

{[[210, 186, 3]]}_{5}

[14]. Here, we can see that the distance of

\nabla (C) \geq d_{H} (\bar{C_{i}})

for

1 \leq i \leq 6

Example 9.

Let

R_{5}

be a non-chain ring, as in Example 7. Suppose that

Λ = 4 + w_{2} (2 + w_{1}) (2 + w_{2})

is a unit element in

R_{5}

. Then,

Λ_{1} = - 1 = Λ_{2} = Λ_{3}

and

Λ_{4} = 1 = Λ_{5} = Λ_{6}

. Thus, in

F_{5} [z]

, we have

\begin{matrix} z^{45} - 1 & = {(z + 4)}^{5} {(z^{2} + z + 1)}^{5} {(z^{6} + z^{3} + 1)}^{5} \\ z^{45} + 1 & = {(z + 1)}^{5} {(z^{2} + 4 z + 1)}^{5} {(z^{6} + 4 z^{3} + 1)}^{5} . \end{matrix}

Let

p_{1} (z) = (z^{6} + 4 z^{3} + 1)

p_{2} (z) = (z + 1)

p_{3} (z) = {(z + 1)}^{2}

p_{4} (z) = 1

, and

p_{5} (z) = p_{6} (z) = (z + 4)

be the generator polynomials of

\bar{C_{i}}

for

1 \leq i \leq 6

, respectively. Then,

C = 〈 κ_{1} p_{1} (z) + κ_{2} p_{2} (z) + \dots + κ_{6} p_{6} (z) 〉

is a

(4 + w_{2} (2 + w_{1}) (2 + w_{2}))

-constacyclic code with length45 over

R_{5}

. Let

N_{2} \in G L_{6} (F_{5})

, as given in Example 2, then

N_{2} N_{2}^{T} = 4 I_{6}

and the Gray image

\nabla (C)

has the parameters

[270, 259, 3]

. Moreover,

(z^{45} - Λ_{i}) \equiv 0 mod (p_{i} (z) p_{i}^{*} (z))

for

1 \leq i \leq 6

; thus, by Theorem 8, we find that C is a dual-containing code; thus, by Theorem 9 we have an improved QECC with parameters

{[[270, 248, 3]]}_{5}

against the existing code

{[[270, 246, 3]]}_{5}

[28]. Here, we can see that the distance of

\nabla (C) \geq d_{H} (\bar{C_{i}})

for

1 \leq i \leq 6

Example 10.

Let

R_{7} = \frac{F_{7} [w_{1}, w_{2}]}{〈 w_{1}^{2} - 1, w_{2}^{3} - w_{2}, w_{1} w_{2} - w_{2} w_{1} 〉}

be a non-chain ring. Suppose that

Λ = 2 w_{2}^{2} - 1

is a unit element in

R_{7}

. Then,

Λ_{1} = - 1 = Λ_{2}

and

Λ_{3} = Λ_{4} = 1 = Λ_{5} = Λ_{6}

. Thus, in

F_{7} [z]

, we have

\begin{matrix} z^{9} - 1 & = (z + 3) (z + 5) (z + 6) (z^{3} + 3) (z^{3} + 5) \\ z^{9} + 1 & = (z + 1) (z + 2) (z + 4) (z^{3} + 2) (z^{3} + 4) . \end{matrix}

Let

p_{1} (z) = (z^{3} + 2)

p_{2} (z) = 1

, and

p_{3} (z) = p_{4} (z) = (z + 3) = p_{5} (z) = p_{6} (z)

be the generator polynomials of

\bar{C_{i}}

for

1 \leq i \leq 6

, respectively. Then,

C = 〈 κ_{1} p_{1} (z) + κ_{2} p_{2} (z) + \dots + κ_{6} p_{6} (z) 〉

is a

(2 w_{2}^{2} - 1)

-constacyclic code of length9 over

R_{7}

. Let

N_{3} \in G L_{6} (F_{7})

such that

N_{3} = [\begin{matrix} 3 & 2 & 2 & 2 & 2 & 2 \\ 2 & 3 & 2 & 2 & 2 & 2 \\ 2 & 2 & 3 & 2 & 2 & 2 \\ 2 & 2 & 2 & 3 & 2 & 2 \\ 2 & 2 & 2 & 2 & 3 & 2 \\ 2 & 2 & 2 & 2 & 2 & 3 \end{matrix}],

then

N_{3} N_{3}^{T} = I_{6}

and the Gray image

\nabla (C)

has the parameters

[54, 47, 3]

. Moreover,

(z^{9} - Λ_{i}) \equiv 0 mod (p_{i} (z) p_{i}^{*} (z))

for

1 \leq i \leq 6

; thus, by Theorem 8, we find that C is a dual-containing code; so, by Theorem 9, we have a new QECC with parameters

{[[54, 40, 3]]}_{7}

. Here, one can see that the distance of

\nabla (C) \geq d_{H} (\bar{C_{i}})

for

1 \leq i \leq 6

Note: InTable 1,

q, n

, and

Λ

represent the order of the field, the length of the code defined over

R

, and the unit element in

R

, respectively.

p_{i} (z)

is a generator polynomial of

C_{i}

for

i \in {1, 2, \dots, 6}

N_{1}, N_{2}, N_{3}

are the invertible matrices over

F_{3}

F_{5}

F_{7}

, respectively, used to define the Gray map ∇. The parameters of the corresponding Gray image (dual-containing code) are denoted by

\nabla (C)

[[n, k, d]]

and

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

represent the parameters of the new QECC and existing QECC, respectively.

5. Conclusions

This article focuses on the exploration of constacyclic codes in the context of non-chain rings

R = \frac{F_{p} [u, v]}{〈 w_{1}^{2} - α^{2}, w_{2}^{3} - β^{2} w_{2}, w_{1} w_{2} - w_{2} w_{1} 〉}

, where

α, β \in F_{p} - {0}

for a primep. From this investigation, numerous new and improved quantum codes have been derived. Substantial potential exists for discovering additional quantum codes within the finite field

F_{p}

by considering prime powers instead of primes. Applying the Gray map ∇ harnesses this potential. In a more general context, substituting the ring

R

with alternative commutative finite rings offers the prospect of developing many fresh quantum code constructions.

Author Contributions

Conceptualization, M.A.R. and P.S.; methodology, N.u.R., A.A. and M.F.A.; investigation, M.A.R., M.F.A., A.A. and M.K.G.; writing—original draft preparation, M.A.R., M.F.A., N.u.R., A.A. and A.N.K.; writing—review and editing, M.A.R., M.F.A., H.S., W.B. and N.u.R.; supervision, M.A.R. and A.A.; project administration, M.A.R., W.B., M.K.G., A.N.K., H.S. and P.S.; funding acquisition, M.A.R. and A.N.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research work was funded by the Institutional Fund Projects of Saudi Arabia under grant number IFPRC-167-130-2020. Therefore, the authors gratefully acknowledge technical and financial support from the Ministry of Education and King Abdulaziz University, Jeddah, Saudi Arabia.

Data Availability Statement

This article required no data set.

Acknowledgments

We appreciate the constructive feedback provided by the reviewers, which has greatly improved the quality of our work.

Conflicts of Interest

The authors declare no conflicts of interest.

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Table 1. Some new and improved QECCs over

F_{p}

from constacyclic codes over

R_{p}

for

(p = 3, 5, 7)

Table 1. Some new and improved QECCs over

F_{p}

from constacyclic codes over

R_{p}

for

(p = 3, 5, 7)

q	n	$Λ$	$p_{1} (z)$	$p_{2} (z)$	$p_{3} (z) = p_{4} (z)$	$p_{5} (x) = p_{6} (x)$	N	$\nabla (C)$	$[[n, k, d]]$	$[[n, k^{'}, d^{'}]]$
3	9	$- w_{1} + w_{2}^{2} + w_{1} w_{2}^{2}$	11	12	1	1	$N_{1}$	$[54, 52, 2]$	${[[54, 50, 2]]}_{3}$	${[[54, 46, 2]]}_{3}$ [12]
3	9	$2 w_{2}^{2} - 1$	11011	11011	12	1	$N_{1}$	$[54, 44, 4]$	${[[54, 34, 4]]}_{3}$	New QECC
3	11	$- w_{1} + w_{2}^{2} + w_{1} w_{2}^{2}$	102221	1	102122	1	$N_{1}$	$[66, 51, 3]$	${[[66, 36, 3]]}_{3}$	New QECC
3	11	$- w_{1} + w_{2}^{2} + w_{1} w_{2}^{2}$	102221	102122	102122	1	$N_{1}$	$[66, 46, 4]$	${[[66, 26, 4]]}_{3}$	New QECC
3	11	$- w_{1} + w_{2}^{2} + w_{1} w_{2}^{2}$	102221	1	102122	1022122	$N_{1}$	$[66, 41, 5]$	${[[66, 16, 5]]}_{3}$	New QECC
3	13	$2 w_{2}^{2} - 1$	1021	1021	1102	1	$N_{1}$	$[78, 66, 4]$	${[[78, 54, 4]]}_{3}$	New QECC
3	18	1	11011	11011	12	1	$N_{1}$	$[108, 98, 3]$	${[[108, 88, 3]]}_{3}$	New QECC
5	6	$- 1$	134	13	12	12	$N_{2}$	$[36, 29, 4]$	${[[36, 22, 4]]}_{5}$	New QECC
5	18	$- 1$	1003004	13	12	12	$N_{2}$	$[108, 97, 4]$	${[[108, 86, 4]]}_{5}$	New QECC
5	19	$- 1$	1	1033234341	1033234341	1033234341	$N_{2}$	$[114, 69, 6]$	${[[114, 24, 6]]}_{5}$	New QECC
5	19	$- 1$	1033234341	1033234341	1033234341	1033234341	$N_{2}$	$[114, 60, 7]$	${[[114, 6, 7]]}_{5}$	New QECC
5	20	$- w_{1} + w_{2}^{2} + w_{1} w_{2}^{2}$	10404	11	11	11	$N_{2}$	$[120, 111, 3]$	${[[120, 102, 3]]}_{5}$	${[[120, 96, 3]]}_{5}$ [14]
5	22	$- w_{1} - w_{2}^{2} + w_{1} w_{2}^{2}$	111212	1	13	12	$N_{2}$	$[132, 123, 4]$	${[[132, 114, 4]]}_{5}$	${[[132, 110, 4]]}_{5}$ [21]
5	25	$1 - 2 w_{2}^{2}$	1400041	1	11	11	$N_{2}$	$[150, 140, 3]$	${[[150, 130, 3]]}_{5}$	New QECC
5	25	$1 - 2 w_{2}^{2}$	140000000014	1	11	11	$N_{2}$	$[150, 135, 4]$	${[[150, 120, 4]]}_{5}$	New QECC
5	40	1	12342	1	12	13	$N_{2}$	$[240, 232, 3]$	${[[240, 224, 3]]}_{5}$	New QECC
7	7	$1 - 2 w_{2}^{2}$	1436	1	11	11	$N_{3}$	$[42, 35, 4]$	${[[42, 28, 4]]}_{7}$	New QECC
7	9	$2 w_{2}^{2} - 1$	1002	12	13	13	$N_{3}$	$[54, 46, 4]$	${[[54, 38, 4]]}_{7}$	New QECC
7	14	$- w_{1} + w_{2}^{2} + w_{1} w_{2}^{2}$	10201	1331	11	11	$N_{3}$	$[84, 73, 4]$	${[[84, 62, 4]]}_{7}$	New QECC
7	15	$1 - 2 w_{2}^{2}$	153356	1	12	14	$N_{3}$	$[90, 81, 4]$	${[[90, 72, 4]]}_{7}$	New QECC
7	18	$- w_{1} + w_{2}^{2} + w_{1} w_{2}^{2}$	102	13026	12	12	$N_{3}$	$[108, 98, 4]$	${[[108, 88, 4]]}_{7}$	New QECC

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Raza, M.A.; Ahmad, M.F.; Alahmadi, A.; Basaffar, W.; Gupta, M.K.; Rehman, N.u.; Khan, A.N.; Shoaib, H.; Sole, P. Quantum Codes as an Application of Constacyclic Codes.Axioms2024,13, 697. https://doi.org/10.3390/axioms13100697

AMA Style

Raza MA, Ahmad MF, Alahmadi A, Basaffar W, Gupta MK, Rehman Nu, Khan AN, Shoaib H, Sole P. Quantum Codes as an Application of Constacyclic Codes.Axioms. 2024; 13(10):697. https://doi.org/10.3390/axioms13100697

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Raza, Mohd Arif, Mohammad Fareed Ahmad, Adel Alahmadi, Widyan Basaffar, Manish K. Gupta, Nadeem ur Rehman, Abdul Nadim Khan, Hatoon Shoaib, and Patrick Sole. 2024. "Quantum Codes as an Application of Constacyclic Codes"Axioms 13, no. 10: 697. https://doi.org/10.3390/axioms13100697

APA Style

Raza, M. A., Ahmad, M. F., Alahmadi, A., Basaffar, W., Gupta, M. K., Rehman, N. u., Khan, A. N., Shoaib, H., & Sole, P. (2024). Quantum Codes as an Application of Constacyclic Codes.Axioms,13(10), 697. https://doi.org/10.3390/axioms13100697

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Abstract

1. Introduction

2. Preliminaries

3. $Λ$ -Constacyclic Codes over $R$

4. Dual-Containing $Λ$ -Constacyclic Codes

5. Conclusions

Author Contributions

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Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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Quantum Codes as an Application of Constacyclic Codes

Abstract

1. Introduction

2. Preliminaries

3.Λ-Constacyclic Codes overR

4. Dual-ContainingΛ-Constacyclic Codes

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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3. $Λ$ -Constacyclic Codes over $R$

4. Dual-Containing $Λ$ -Constacyclic Codes