Quantum Codes as an Application of Constacyclic Codes

Abstract

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring $\mathcal{R}={\mathbb{F}}_p+w_1{\mathbb{F}}_p+w_2{\mathbb{F}}_p+w_2^2{\mathbb{F}}_p+w_1{w}_2{\mathbb{F}}_p+w_1{w}_2^2{\mathbb{F}}_p$, where $w_1^2-{\alpha }^2=0$, $w_1{w}_2=w_2{w}_1$, $w_2^3-{\beta }^2{w}_2=0$, and $\alpha ,\beta \in {\mathbb{F}}_p\setminus \left\{0\right\}$, for a prime p. We begin by introducing a Gray map defined over $\mathcal{R}$, 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 $\mathcal{R}$ using Calderbank–Shore–Steane (CSS) construction.

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 ${\mathbb{F}}_2+u{\mathbb{F}}_2$, 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 ${\mathbb{F}}_4+u{\mathbb{F}}_4$. Qian [17] proposed a novel approach for constructing quantum error-correcting codes from cyclic codes over the finite non-chain ring ${\mathbb{F}}_2+v{\mathbb{F}}_2$, 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 ${\mathbb{F}}_q+u{\mathbb{F}}_q+v{\mathbb{F}}_q+uv{\mathbb{F}}_q$, where $u^2=u$, $v^2=v$, $uv=vu$, $q=p^n$, and p 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 ${\mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$, with $u^2-u=0$, $v^2-v=0$, and $uv-vu=0$; Ma et al.’s [20] contributions over ${\mathbb{F}}_p+v{\mathbb{F}}_p+v^2{\mathbb{F}}_p$, with $v^3=v$; and Gao and Wang’s [21] over ${\mathbb{F}}_p+u{\mathbb{F}}_p$, 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 $\mathcal{R}={\mathbb{F}}_p+w_1{\mathbb{F}}_p+w_2{\mathbb{F}}_p+w_2^2{\mathbb{F}}_p+w_1{w}_2{\mathbb{F}}_p+w_1{w}_2^2{\mathbb{F}}_p$, where $w_1^2-{\alpha }^2=0$, $w_1{w}_2=w_2{w}_1$, $w_2^3-{\beta }^2{w}_2=0$, and $\alpha ,\beta \in {\mathbb{F}}_p\setminus \left\{0\right\}$, for a prime p. The objective is to find new quantum codes over the finite field ${\mathbb{F}}_p$. The article makes two significant contributions:

• Comprehensive study of the structure of constacyclic codes with the length l over $\mathcal{R}$.
• 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 ${\mathbb{F}}_p$ be a finite field of order p (an odd prime). A subspace $C_0$ of ${\mathbb{F}}_p^m$ is called a linear code of length m over ${\mathbb{F}}_p$, and its members are called the codewords. Let $\mathcal{R}={\mathbb{F}}_p+w_1{\mathbb{F}}_p+w_2{\mathbb{F}}_p+w_2^2{\mathbb{F}}_p+w_1{w}_2{\mathbb{F}}_p+w_1{w}_2^2{\mathbb{F}}_p$, where $w_1^2-{\alpha }^2=0$, $w_1{w}_2=w_2{w}_1$, $w_2^3-{\beta }^2{w}_2=0$, and $\alpha ,\beta \in {\mathbb{F}}_p\setminus \left\{0\right\}$ be a finite commutative ring. Remember that a linear code C over the ring $\mathcal{R}$ of length n is essentially an $\mathcal{R}$-submodule of the module ${\mathcal{R}}^n$. One can also view an element $c=(c_0,c_1,\dots ,c_{n-1})$ in C as a polynomial $c\left(z\right)=c_0+c_{1}z+\cdots +c_{n-1}{z}^{n-1}$ within the ring ${\displaystyle \frac{\mathcal{R}\left[z\right]}{⟨z^n-\Lambda ⟩}}$. A linear code C is called a $\Lambda$-constacyclic code of length n over $\mathcal{R}$ if and only if it is an $\mathcal{R}$-submodule in the module ${\displaystyle \frac{\mathcal{R}\left[z\right]}{⟨z^n-\Lambda ⟩}}$. Many researchers have extensively explored constacyclic codes over finite fields and finite commutative Frobenius rings [23,24,25,26,27]. Consider the elements of $\mathcal{R}$ as follows:

$$\begin{array}{cc}\hfill {\kappa }_1& ={\displaystyle \frac{1}{2\alpha {\beta }^2}}(\alpha +w_1)({\beta }^2-w_2^2),{\kappa }_2={\displaystyle \frac{1}{2\alpha {\beta }^2}}(\alpha -w_1)({\beta }^2-w_2^2),\hfill \\ \hfill {\kappa }_3& ={\displaystyle \frac{1}{4\alpha {\beta }^2}}(\alpha +w_1)(w_2^2-\beta w_2),{\kappa }_4={\displaystyle \frac{1}{4\alpha {\beta }^2}}(\alpha -w_1)(w_2^2-\beta w_2),\hfill \\ \hfill {\kappa }_5& ={\displaystyle \frac{1}{4\alpha {\beta }^2}}(\alpha +w_1)(w_2^2+\beta w_2),{\kappa }_6={\displaystyle \frac{1}{4\alpha {\beta }^2}}(\alpha -w_1)(w_2^2+\beta w_2).\hfill \end{array}$$

We can verify that ${\kappa }_1+{\kappa }_2+{\kappa }_3+{\kappa }_4+{\kappa }_5+{\kappa }_6=1$, and ${\kappa }_i{\kappa }_j={\delta }_{ij}$ (Kronecker delta) for $i,j\in \{1,2\dots ,6\}$. Consequently, the set $\{{\kappa }_1,{\kappa }_2,\dots ,{\kappa }_6\}$ forms a set of non-zero pairwise orthogonal idempotent elements in $\mathcal{R}$. This implies that $\mathcal{R}$ can be expressed as a sum of submodules as follows:

$$\begin{array}{cc}\hfill \mathcal{R}& ={\kappa }_1\mathcal{R}\oplus {\kappa }_2\mathcal{R}\oplus {\kappa }_3\mathcal{R}\oplus {\kappa }_4\mathcal{R}\oplus {\kappa }_5\mathcal{R}\oplus {\kappa }_6\mathcal{R}\hfill \\ \hfill & \cong {\kappa }_1{\mathbb{F}}_p\oplus {\kappa }_2{\mathbb{F}}_p\oplus {\kappa }_3{\mathbb{F}}_p\oplus {\kappa }_4{\mathbb{F}}_p\oplus {\kappa }_5{\mathbb{F}}_p\oplus {\kappa }_6{\mathbb{F}}_p.\hfill \end{array}$$

Therefore, any element $r=a+w_{1}b+w_{2}c+w_2^{2}d+w_1{w}_{2}e+w_1{w}_2^{2}f\in \mathcal{R}$ can be uniquely written as

$$\begin{array}{cc}\hfill r& =a+w_{1}b+w_{2}c+w_2^{2}d+w_1{w}_{2}e+w_1{w}_2^{2}f\hfill \\ \hfill & ={\kappa }_1\overline{a}+{\kappa }_2\overline{b}+{\kappa }_3\overline{c}+{\kappa }_4\overline{d}+{\kappa }_5\overline{e}+{\kappa }_6\overline{f},\hfill \end{array}$$

(1)

where

$$\begin{array}{cc}\hfill \overline{a}& =a+\alpha b,\hfill \\ \hfill \overline{b}& =a-\alpha b,\hfill \\ \hfill \overline{c}& =a+\alpha b-\beta c+{\beta }^{2}d-\alpha \beta e+\alpha {\beta }^{2}f,\hfill \\ \hfill \overline{d}& =a-\alpha b-\beta c+{\beta }^{2}d+\alpha \beta e-\alpha {\beta }^{2}f,\hfill \\ \hfill \overline{e}& =a+\alpha b+\beta c+{\beta }^{2}d+\alpha \beta e+\alpha {\beta }^{2}f,\hfill \\ \hfill \overline{f}& =a-\alpha b+\beta c+{\beta }^{2}d-\alpha \beta e-\alpha {\beta }^{2}f,\hfill \end{array}$$

are the elements of ${\mathbb{F}}_p$.

Suppose that $G{L}_n\left({\mathbb{F}}_p\right)$ is the group of invertible matrices of order n over ${\mathbb{F}}_p$ and let $\mathbf{N}\in G{L}_6\left({\mathbb{F}}_p\right)$ in such a way that $\mathbf{N}{\mathbf{N}}^T=k{I}_6$, where ${\mathbf{N}}^T$ is the transpose of the matrix $\mathbf{N}$, $I_6$ is the identity matrix of order 6, and $k\in {\mathbb{F}}_p-\left\{0\right\}$. With the above notation, we define a Gray map associated with an invertible matrix $\mathbf{N}$ as follows:

$$\nabla :\mathcal{R}⟶{\mathbb{F}}_p^6\mathrm{such}\mathrm{that}\nabla \left(r\right):=(\overline{a},\overline{b},\overline{c},\overline{d},\overline{e},\overline{f})\mathbf{N}.$$

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

$$\nabla :{\mathcal{R}}^l⟶{\mathbb{F}}_p^{6l}\mathrm{such}\mathrm{that}$$

$$\begin{array}{cc}\hfill \nabla (r_0,r_1,\dots ,r_{l-1})=& (({\overline{a}}_0,{\overline{b}}_0,{\overline{c}}_0,{\overline{d}}_0,{\overline{e}}_0,{\overline{f}}_0)\mathbf{N},({\overline{a}}_1,{\overline{b}}_1,{\overline{c}}_1,{\overline{d}}_1,{\overline{e}}_1,{\overline{f}}_1)\mathbf{N}\hfill \\ \hfill & ,\dots ,({\overline{a}}_{l-1},{\overline{b}}_{l-1},{\overline{c}}_{l-1},{\overline{d}}_{l-1},{\overline{e}}_{l-1},{\overline{f}}_{l-1})\mathbf{N}),\hfill \end{array}$$

(2)

where $r_i={\kappa }_1\overline{a_i}+{\kappa }_2\overline{b_i}+{\kappa }_3\overline{c_i}+{\kappa }_4\overline{d_i}+{\kappa }_5\overline{e_i}+{\kappa }_6\overline{f_i}\in \mathcal{R},\mathrm{for}i\in \{0,1,\dots ,l-1\}$. Here, we introduce the Lee weight for the vector $r\in \mathcal{R}$ as $w_L\left(r\right)=w_H(\nabla \left(r\right))$, where $w_L$ (resp. $w_H$) denotes the Lee weight (resp. the Hamming weight). The Lee weight of $w_L(r=(r_0,r_1,\dots ,r_{l-1}))=w_L\left(r_0\right)+w_L\left(r_1\right)+\cdots +w_L\left(r_{l-1}\right)$ and the Lee distance from r to $r^{\prime }\in {\mathcal{R}}^l$, is established as $d_L(r,r^{\prime })=w_L(r-r^{\prime })=w_H(\nabla (r-r^{\prime }))$. The Lee distance $d_L\left(C\right)$ for the code C is defined as follows:

$$d_L\left(C\right)=min\left\{d_L(r,r^{\prime })\right|r\ne r^{\prime }\}.$$

It is notable that the Gray map ∇ is a linear map over ${\mathbb{F}}_p$ that preserves distances and mapping vectors from ${\mathcal{R}}^l$ to ${\mathbb{F}}_p^{6l}$. Since the Gray map ∇ is bijective, it follows that $\nabla \left(C\right)$ forms a $[6l,k,d_H]$ linear code over ${\mathbb{F}}_p$, 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 $r^{\prime }=(r_0^{\prime },r_1^{\prime },\dots ,r_{l-1}^{\prime })$ in ${\mathcal{R}}^l$ is defined as $r·r^{\prime }=r_0{r}_0^{\prime }+r_1{r}_1^{\prime }+\cdots +r_{l-1}{r}_{l-1}^{\prime }$. The dual code of C is formulated as $C^{\perp }=\{r\in {\mathcal{R}}^l|r·r^{\prime }=0\forall r^{\prime }\in C\}$. A code C is called dual-containing if $C^{\perp }\subseteq C$, self-orthogonal if $C\subseteq C^{\perp }$, and self-dual if $C^{\perp }=C$.

Example 1.

Let ${\mathcal{R}}_3={\displaystyle \frac{{\mathbb{F}}_3[w_1,w_2]}{⟨w_1^2-1,w_2^3-w_2,w_1{w}_2-w_2{w}_1⟩}}$ 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 ${\mathcal{R}}_3$ are

$$\begin{array}{cc}\hfill {\kappa }_1& =2(1+w_1)(1+2w_2^2),{\kappa }_2={\displaystyle \frac{1}{2}}(1-w_1)(1-w_2^2)=2(1+2w_1)(1+2w_2^2),\hfill \\ \hfill {\kappa }_3& =(1+w_1)(w_2^2+2w_2),{\kappa }_4={\displaystyle \frac{1}{4}}(1-w_1)(w_2^2-w_2)=(1+2w_1)(w_2^2+2w_2),\hfill \\ \hfill {\kappa }_5& =(1+w_1)(w_2^2+w_2),{\kappa }_6={\displaystyle \frac{1}{4}}(1-w_1)(w_2^2+w_2)=(1+2w_1)(w_2^2+w_2),\hfill \end{array}$$

where ${\kappa }_1+{\kappa }_2+{\kappa }_3+{\kappa }_4+{\kappa }_5+{\kappa }_6=1$. By Chinese Remainder Theorem, we have ${\mathcal{R}}_3={\kappa }_1{\mathcal{R}}_3\oplus {\kappa }_2{\mathcal{R}}_3\oplus {\kappa }_3{\mathcal{R}}_3\oplus {\kappa }_4{\mathcal{R}}_3\oplus {\kappa }_5{\mathcal{R}}_3\oplus {\kappa }_6{\mathcal{R}}_3\cong {\kappa }_1{\mathbb{F}}_3\oplus {\kappa }_2{\mathbb{F}}_3\oplus {\kappa }_3{\mathbb{F}}_3\oplus {\kappa }_4{\mathbb{F}}_3\oplus {\kappa }_5{\mathbb{F}}_3\oplus {\kappa }_6{\mathbb{F}}_3.$ Therefore, any element $r=a+w_{1}b+w_{2}c+w_2^{2}d+w_1{w}_{2}e+w_1{w}_2^{2}f\in \mathcal{R}$ can be expressed as follows:

$$\begin{array}{cc}\hfill r=& a+w_{1}b+w_{2}c+w_2^{2}d+w_1{w}_{2}e+w_1{w}_2^{2}f\hfill \\ \hfill =& (a+b){\kappa }_1+(a+2b){\kappa }_2+(a+b+2c+d+2e+f){\kappa }_3+\hfill \\ \hfill & (a+2b+2c+d+e+2f){\kappa }_4+(a+b+c+d+e+f){\kappa }_5+\hfill \\ \hfill & (a+2b+c+d+2e+2f){\kappa }_6.\hfill \end{array}$$

Hence, the Gray map $\nabla :{\mathcal{R}}_3⟶{\mathbb{F}}_3^6$ can be established as follows:

$$\begin{array}{cc}\hfill \nabla \left(r\right):=& (a+b,a+2b,a+b+2c+d+2e+f,a+2b+2c+d+e+2f,\hfill \\ \hfill & a+b+c+d+e+f,a+2b+c+d+2e+2f){\mathbf{N}}_1,\hfill \end{array}$$

where $a,b,c,d,e,f\in {\mathbb{F}}_3$, and ${\mathbf{N}}_1\in G{L}_6\left({\mathbb{F}}_3\right)$, where

$${\mathbf{N}}_1=\left[\begin{array}{cccccc}1& 2& 1& 1& 0& 1\\ 1& 1& 2& 1& 1& 0\\ 0& 1& 1& 2& 1& 1\\ 2& 1& 0& 1& 2& 1\\ 2& 2& 2& 0& 1& 1\\ 2& 0& 1& 1& 1& 2\end{array}\right],$$

which has the property that ${\mathbf{N}}_1{\mathbf{N}}_1^T=2I_6.$

Example 2.

Let ${\mathcal{R}}_5={\displaystyle \frac{{\mathbb{F}}_5[w_1,w_2]}{⟨w_1^2-{\alpha }^2,w_2^3-{\beta }^2{w}_2,w_1{w}_2-w_2{w}_1⟩}}$ be a finite commutative non-chain ring, where $\alpha =2$ and $\beta =3$ are non-zero elements of ${\mathbb{F}}_5$. Then, we have $w_1^2-{\alpha }^2=(w_1-2)(w_1+2)$ and $w_2^3-{\beta }^2{w}_2=w_2(w_2-3)(w_2+3)$. The orthogonal idempotent elements in ${\mathcal{R}}_5$ are as follows:

$$\begin{array}{cc}\hfill {\kappa }_1& =4(2+w_1)(1+w_2^2),{\kappa }_2={\displaystyle \frac{1}{2\alpha {\beta }^2}}(\alpha -w_1)({\beta }^2-w_2^2)=4(2-w_1)(1+w_2^2),\hfill \\ \hfill {\kappa }_3& =2(2+w_1)(w_2^2-3w_2),{\kappa }_4={\displaystyle \frac{1}{4\alpha {\beta }^2}}(\alpha -w_1)(w_2^2-\beta w_2)=2(2-w_1)(w_2^2-3w_2),\hfill \\ \hfill {\kappa }_5& =2(2+w_1)(w_2^2+3w_2),{\kappa }_6={\displaystyle \frac{1}{4\alpha {\beta }^2}}(\alpha -w_1)(w_2^2+\beta w_2)=2(2-w_1)(w_2^2+3w_2),\hfill \end{array}$$

where ${\kappa }_1+{\kappa }_2+{\kappa }_3+{\kappa }_4+{\kappa }_5+{\kappa }_6=1$. By Chinese Remainder Theorem, we have ${\mathcal{R}}_5={\kappa }_1{\mathcal{R}}_5\oplus {\kappa }_2{\mathcal{R}}_5\oplus {\kappa }_3{\mathcal{R}}_5\oplus {\kappa }_4{\mathcal{R}}_5\oplus {\kappa }_5{\mathcal{R}}_5\oplus {\kappa }_6{\mathcal{R}}_5\cong {\kappa }_1{\mathbb{F}}_5\oplus {\kappa }_2{\mathbb{F}}_5\oplus {\kappa }_3{\mathbb{F}}_5\oplus {\kappa }_4{\mathbb{F}}_5\oplus {\kappa }_5{\mathbb{F}}_5\oplus {\kappa }_6{\mathbb{F}}_5.$ Therefore, any element $r=a+w_{1}b+w_{2}c+w_2^{2}d+w_1{w}_{2}e+w_1{w}_2^{2}f\in \mathcal{R}$ can be expressed as follows:

$$\begin{array}{cc}\hfill r=& a+w_{1}b+w_{2}c+w_2^{2}d+w_1{w}_{2}e+w_1{w}_2^{2}f\hfill \\ \hfill =& (a+2b){\kappa }_1+(a+3b){\kappa }_2+(a+2b+2c+4d+4e+3f){\kappa }_3+\hfill \\ \hfill & (a+3b+2c+4d+e+2f){\kappa }_4+(a+2b+3c+4d+e+3f){\kappa }_5+\hfill \\ \hfill & (a+3b+3c+4d+4e+2f){\kappa }_6.\hfill \end{array}$$

The Gray map $\nabla :{\mathcal{R}}_5⟶{\mathbb{F}}_5^6$ can be established as follows:

$$\begin{array}{cc}\hfill \nabla \left(r\right):=& (a+2b,a+3b,a+2b+2c+4d+4e+3f,a+3b+2c+4d+e+2f,\hfill \\ \hfill & a+2b+3c+4d+e+3f,a+3b+3c+4d+4e+2f){\mathbf{N}}_2,\hfill \end{array}$$

where $a,b,c,d,e,f\in {\mathbb{F}}_5$, and ${\mathbf{N}}_2\in G{L}_6\left({\mathbb{F}}_5\right)$, where

$${\mathbf{N}}_2=\left(\begin{array}{cccccc}3& 1& 1& 1& 1& 1\\ 1& 3& 1& 1& 1& 1\\ 1& 1& 3& 1& 1& 1\\ 1& 1& 1& 3& 1& 1\\ 1& 1& 1& 1& 3& 1\\ 1& 1& 1& 1& 1& 3\end{array}\right),$$

which has the property that ${\mathbf{N}}_2{\mathbf{N}}_2^T=4I_6.$

Theorem 1.

The Gray map $\nabla :{\mathcal{R}}^l⟶{\mathbb{F}}_p^{6l}$ defined in Equation (2) is linear and isometric.

Proof. 

To prove that ∇ is a linear map, assume that $z=a_1{\kappa }_1+a_2{\kappa }_2+a_3{\kappa }_3+a_4{\kappa }_4+a_5{\kappa }_5+a_6{\kappa }_6$ and $y=b_1{\kappa }_1+b_2{\kappa }_2+b_3{\kappa }_3+b_4{\kappa }_4+b_5{\kappa }_5+b_6{\kappa }_6$ are any two elements of $\mathcal{R}$ and $\lambda$ is a non-zero scalar in ${\mathbb{F}}_p$. Then, we have

$$\begin{array}{cc}\hfill \nabla (z+y)& =(a_1+b_1,a_2+b_2,a_3+b_3,a_4+b_4,a_5+b_5,a_6+b_6)\mathbf{N}\hfill \\ \hfill & =[(a_1,a_2,a_3,a_4,a_5,a_6)+(b_1,b_2,b_3,b_4,b_5,b_6)]\mathbf{N}\hfill \\ \hfill & =(a_1,a_2,a_3,a_4,a_5,a_6)\mathbf{N}+(b_1,b_2,b_3,b_4,b_5,b_6)\mathbf{N}\hfill \\ \hfill & =\nabla \left(z\right)+\nabla \left(y\right),\hfill \\ \hfill \nabla (\lambda ·z)& =(\lambda a_1,\lambda a_2,\lambda a_3,\lambda a_4,\lambda a_5,\lambda a_6)\mathbf{N}\hfill \\ \hfill & =\lambda (a_1,a_2,a_3,a_4,a_5,a_6)\mathbf{N}\hfill \\ \hfill & =\lambda \nabla \left(z\right).\hfill \end{array}$$

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 code C are the same. As $z,y\in {\mathcal{R}}^l$, then by definition of the Lee distance, we see that

$$\begin{array}{c}\hfill d_L(z,y)=w{t}_H(\nabla (z-y))=w{t}_H(\nabla \left(z\right)-\nabla \left(y\right))=d_H(\nabla \left(z\right),\nabla \left(y\right)).\end{array}$$

(3)

Therefore, the Gray map ∇ is an isometry. □

Theorem 2.

Let C be a linear code with parameters $[l,k,d_L]$ over $\mathcal{R}$.

(i)

Then, $\nabla \left(C\right)$ is a linear code with parameters $[6l,k,d_H]$ over ${\mathbb{F}}_p$, where $d_L$ and $d_H$ are the same.

(ii)

The image $\nabla \left(C\right)$ is self-orthogonal over ${\mathbb{F}}_p$, provided C is self-orthogonal over $\mathcal{R}$.

(iii)

The image $\nabla \left(C\right)$ is a dual-containing code over ${\mathbb{F}}_p$, provided C is a dual-containing code over $\mathcal{R}$.

(iv)

C is a self-dual code over $\mathcal{R}$ if and only if $\nabla \left(C\right)$ is a self-dual code over ${\mathbb{F}}_p$.

Proof. 

(i)

The proof follows by Theorem 1.

(ii)

If C is self-orthogonal over $\mathcal{R}$. Then, for any codewords $z=(z_1,z_2,\dots ,z_l)$ and $y=(y_1,y_2,\dots ,y_l)$ in C, where $z_i=a_1^i{\kappa }_1+a_2^i{\kappa }_2+a_3^i{\kappa }_3+a_4^i{\kappa }_4+a_5^i{\kappa }_5+a_6^i{\kappa }_6$ and $y_i=b_1^i{\kappa }_1+b_2^i{\kappa }_2+b_3^i{\kappa }_3+b_4^i{\kappa }_4+b_5^i{\kappa }_5+b_6^i{\kappa }_6$ are elements of $\mathcal{R}$ for $1\le i\le l$, we have $z·y=0$. This suggests that $a_j^1{b}_j^1+a_j^2{b}_j^2+\cdots +a_j^l{b}_j^l=0$ for $1\le j\le 6$. Let $z^{\prime },y^{\prime }\in \nabla \left(C\right)$ be any two elements, then some $z,y\in C$ exists such that $z^{\prime }=\nabla \left(z\right)$ and $y^{\prime }=\nabla \left(y\right)$, i.e.,

$$\begin{array}{cc}\hfill z^{\prime }& =(\nabla \left(z_1\right),\nabla \left(z_2\right),\dots ,\nabla \left(z_l\right))\hfill \\ \hfill & =((a_1^1,a_2^1,a_3^1,a_4^1,a_5^1,a_6^1)\mathbf{N},(a_1^2,a_2^2,a_3^2,a_4^2,a_5^2,a_6^2)\mathbf{N},\dots ,(a_1^l,a_2^l,a_3^l,a_4^l,a_5^l,a_6^l)\mathbf{N}),\hfill \\ \hfill y^{\prime }& =(\nabla \left(y_1\right),\nabla \left(y_2\right),\dots ,\nabla \left(y_l\right))\hfill \\ \hfill & =((b_1^1,b_2^1,b_3^1,b_4^1,b_5^1,b_6^1)\mathbf{N},(b_1^2,b_2^2,b_3^2,b_4^2,b_5^2,b_6^2)\mathbf{N},\dots ,(b_1^l,b_2^l,b_3^l,b_4^l,b_5^l,b_6^l)\mathbf{N}),\hfill \end{array}$$

where $\mathbf{N}\in G{L}_6\left({\mathbb{F}}_p\right)$ such that $\mathbf{N}{\mathbf{N}}^T=\lambda I_6$, $\lambda \in {\mathbb{F}}_p-\left\{0\right\}$. Now, we have

$$\begin{array}{cc}\hfill z^{\prime }·y^{\prime }& =\nabla \left(z\right)·\nabla \left(y\right)=\nabla \left(z\right)·\nabla {\left(y\right)}^T\hfill \\ \hfill & =\sum _{i=1}^l(a_1^i,a_2^i,a_3^i,a_4^i,a_5^i,a_6^i)\mathbf{N}{\mathbf{N}}^T.(b_1^i,b_2^i,b_3^i,b_4^i,b_5^i,b_6^i)\hfill \\ \hfill & =\sum _{i=1}^l(a_1^i,a_2^i,a_3^i,a_4^i,a_5^i,a_6^i)\lambda I_6.(b_1^i,b_2^i,b_3^i,b_4^i,b_5^i,b_6^i)\hfill \\ \hfill & =\sum _{i=1}^l\lambda (a_1^i{b}_1^i+a_2^i{b}_2^i+a_3^i{b}_3^i+a_4^i{b}_4^i+a_5^i{b}_5^i+a_6^i{b}_6^i)\hfill \\ \hfill & =\sum _{i=j}^6\lambda (a_j^1{b}_j^1+a_j^2{b}_j^2+\cdots +a_j^l{b}_j^l)=0.\hfill \end{array}$$

Thus, we have $z^{\prime }·y^{\prime }=\nabla \left(z\right)·\nabla \left(y\right)=0$ for all $z^{\prime },y^{\prime }\in \nabla \left(C\right)$ if C is self-orthogonal over $\mathcal{R}$. Hence, $\nabla \left(C\right)$ is a self-orthogonal code of length $6l$ over ${\mathbb{F}}_p$, provided C is a self-orthogonal code over $\mathcal{R}$.

(iii)

Suppose that $C^{\perp }\subseteq C$, then by the linearity of ∇, we have $\nabla \left(C^{\perp }\right)\subseteq \nabla \left(C\right)$. To prove that $\nabla \left(C\right)$ is dual-containing, it remains to show that $\nabla \left(C^{\perp }\right)=\nabla {\left(C\right)}^{\perp }$. For this, let $z=(z_1,z_2,\dots ,z_l)\in C$ and $y=(y_1,y_2,\dots ,y_l)\in C^{\perp }$, where $z_i=a_1^i{\kappa }_1+a_2^i{\kappa }_2+a_3^i{\kappa }_3+a_4^i{\kappa }_4+a_5^i{\kappa }_5+a_6^i{\kappa }_6$ and $y_i=b_1^i{\kappa }_1+b_2^i{\kappa }_2+b_3^i{\kappa }_3+b_4^i{\kappa }_4+b_5^i{\kappa }_5+b_6^i{\kappa }_6$ are elements of $\mathcal{R}$ for $1\le i\le l$. Now, $x·y=0$ gives that $a_j^1{b}_j^1+a_j^2{b}_j^2+\cdots +a_j^l{b}_j^l=0$ for $1\le j\le 6$. Consider

$$\begin{array}{cc}\hfill \nabla \left(z\right)& =((a_1^1,a_2^1,a_3^1,a_4^1,a_5^1,a_6^1)\mathbf{N},(a_1^2,a_2^2,a_3^2,a_4^2,a_5^2,a_6^2)\mathbf{N},\dots ,(a_1^l,a_2^l,a_3^l,a_4^l,a_5^l,a_6^l)\mathbf{N}),\hfill \\ \hfill \nabla \left(y\right)& =((b_1^1,b_2^1,b_3^1,b_4^1,b_5^1,b_6^1)\mathbf{N},(b_1^2,b_2^2,b_3^2,b_4^2,b_5^2,b_6^2)\mathbf{N},\dots ,(b_1^l,b_2^l,b_3^l,b_4^l,b_5^l,b_6^l)\mathbf{N}),\hfill \end{array}$$

Now, $\nabla \left(z\right)·\nabla \left(y\right)=0$ suggests that $\nabla \left(y\right)\in \nabla {\left(C\right)}^{\perp }$. Thus, we have $\nabla \left(C^{\perp }\right)\subseteq \nabla {\left(C\right)}^{\perp }$. Contrarily, ∇ is a bijective linear map, so the sizes of $\nabla \left(C^{\perp }\right)$ and $\nabla {\left(C\right)}^{\perp }$ are the same. Thus, $\nabla \left(C^{\perp }\right)=\nabla {\left(C\right)}^{\perp }$. Hence, $\nabla \left(C\right)$ is a dual-containing code over ${\mathbb{F}}_p$ provided C is a dual-containing code over $\mathcal{R}$.

(iv)

It follows from part (iii).

□

Theorem 3

([11]). Let $C={\kappa }_1\overline{C_1}\oplus {\kappa }_2\overline{C_2}\oplus {\kappa }_3\overline{C_3}\oplus {\kappa }_4\overline{C_4}\oplus {\kappa }_5\overline{C_5}\oplus {\kappa }_6\overline{C_6}$ be a linear code over $\mathcal{R}$. Then:

(i)

$C^{\perp }={\kappa }_1{\overline{C_1}}^{\perp }\oplus {\kappa }_2{\overline{C_2}}^{\perp }\oplus {\kappa }_3{\overline{C_3}}^{\perp }\oplus {\kappa }_4{\overline{C_4}}^{\perp }\oplus {\kappa }_5{\overline{C_5}}^{\perp }\oplus {\kappa }_6{\overline{C_6}}^{\perp }$;

(ii)

C is self-dual over $\mathcal{R}$ if $\overline{C_i}$ are self-dual codes over ${\mathbb{F}}_p$ for $1\le i\le 6$.

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

$$\begin{array}{c}\hfill D_1\oplus D_2=\{d_1+d_2|d_j\in D_j;j=1,2\},\end{array}$$

(4)

$$\begin{array}{c}\hfill D_1\otimes D_2=\left\{(d_1,d_2)\right|d_j\in D_j;j=1,2\}.\end{array}$$

(5)

Suppose that C is a linear code with length l over $\mathcal{R}$. Consider the following sets:

$$\begin{array}{cc}\hfill \overline{C_1}& =\{\overline{a}\in {\mathbb{F}}_p^l|{\kappa }_1\overline{a}+{\kappa }_2\overline{b}+{\kappa }_3\overline{c}+{\kappa }_4\overline{d}+{\kappa }_5\overline{e}+{\kappa }_6\overline{f}\in C;\mathrm{for}\mathrm{some}\overline{b},\overline{c},\overline{d},\overline{e},\overline{f}\in {\mathbb{F}}_p^l\};\hfill \\ \hfill \overline{C_2}& =\{\overline{b}\in {\mathbb{F}}_p^l|{\kappa }_1\overline{a}+{\kappa }_2\overline{b}+{\kappa }_3\overline{c}+{\kappa }_4\overline{d}+{\kappa }_5\overline{e}+{\kappa }_6\overline{f}\in C;\mathrm{for}\mathrm{some}\overline{a},\overline{c},\overline{d},\overline{e},\overline{f}\in {\mathbb{F}}_p^l\};\hfill \\ \hfill \overline{C_3}& =\{\overline{c}\in {\mathbb{F}}_p^l|{\kappa }_1\overline{a}+{\kappa }_2\overline{b}+{\kappa }_3\overline{c}+{\kappa }_4\overline{d}+{\kappa }_5\overline{e}+{\kappa }_6\overline{f}\in C;\mathrm{for}\mathrm{some}\overline{a},\overline{b},\overline{d},\overline{e},\overline{f}\in {\mathbb{F}}_p^l\};\hfill \\ \hfill \overline{C_4}& =\{\overline{d}\in {\mathbb{F}}_p^l|{\kappa }_1\overline{a}+{\kappa }_2\overline{b}+{\kappa }_3\overline{c}+{\kappa }_4\overline{d}+{\kappa }_5\overline{e}+{\kappa }_6\overline{f}\in C;\mathrm{for}\mathrm{some}\overline{a},\overline{b},\overline{c},\overline{e},\overline{f}\in {\mathbb{F}}_p^l\};\hfill \\ \hfill \overline{C_5}& =\{\overline{e}\in {\mathbb{F}}_p^l|{\kappa }_1\overline{a}+{\kappa }_2\overline{b}+{\kappa }_3\overline{c}+{\kappa }_4\overline{d}+{\kappa }_5\overline{e}+{\kappa }_6\overline{f}\in C;\mathrm{for}\mathrm{some}\overline{a},\overline{b},\overline{c},\overline{d},\overline{f}\in {\mathbb{F}}_p^l\};\hfill \\ \hfill \overline{C_6}& =\{\overline{f}\in {\mathbb{F}}_p^l|{\kappa }_1\overline{a}+{\kappa }_2\overline{b}+{\kappa }_3\overline{c}+{\kappa }_4\overline{d}+{\kappa }_5\overline{e}+{\kappa }_6\overline{f}\in C;\mathrm{for}\mathrm{some}\overline{a},\overline{b},\overline{c},\overline{d},\overline{e}\in {\mathbb{F}}_p^l\}.\hfill \end{array}$$

It can be seen that $\overline{C_i}$ for $1\le i\le 6$ is a linear code with length l over ${\mathbb{F}}_p$. Therefore, we can express a linear code C with length l over $\mathcal{R}$ as $C={\kappa }_1\overline{C_1}\oplus {\kappa }_2\overline{C_2}\oplus {\kappa }_3\overline{C_3}\oplus {\kappa }_4\overline{C_4}\oplus {\kappa }_5\overline{C_5}\oplus {\kappa }_6\overline{C_6}$. If $G_i$ is the generator matrix of $\overline{C_i}$ for $1\le i\le 6$, then the generator matrix $\nabla \left(G\right)$ of the Gray image $\nabla \left(C\right)$ is given as follows:

$$\nabla \left(G\right)=\left[\begin{array}{c}\nabla \left({\kappa }_1{G}_1\right)\\ \nabla \left({\kappa }_2{G}_2\right)\\ \nabla \left({\kappa }_3{G}_3\right)\\ \nabla \left({\kappa }_4{G}_4\right)\\ \nabla \left({\kappa }_5{G}_5\right)\\ \nabla \left({\kappa }_6{G}_6\right)\end{array}\right].$$

3. $\Lambda$-Constacyclic Codes over $\mathcal{R}$

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 $\Lambda ={\kappa }_{1}a+{\kappa }_{2}b+{\kappa }_{3}c+{\kappa }_{4}d+{\kappa }_{5}e+{\kappa }_{6}f$ is a unit element in $\mathcal{R}$. Then, a linear code C with length l over $\mathcal{R}$ is called a $\Lambda$-constacyclic code if, for any codeword $c=(c_0,c_1,\dots ,c_{l-1})$ in C, it satisfies the property that ${\omega }_{\Lambda }\left(c\right)=(\Lambda c_{l-1},c_0,\dots ,c_{l-2})$ is again a member of C. In particular, if $\Lambda =1$, then $\Lambda$-constacyclic code C becomes a cyclic code, and if $\Lambda =-1$, then C becomes a negacyclic code.

Lemma 1.

Let $\Lambda ={\kappa }_{1}a+{\kappa }_{2}b+{\kappa }_{3}c+{\kappa }_{4}d+{\kappa }_{5}e+{\kappa }_{6}f\in \mathcal{R}$ be a non-zero element. Then, the element $\Lambda \in \mathcal{R}$ is a unit element in $\mathcal{R}$ if $a,b,c,d,e,f$ are unit elements in ${\mathbb{F}}_p$. Moreover, when $\Lambda \in \mathcal{R}$ is a unit element, then its inverse is given by ${\Lambda }^{-1}={\kappa }_1{a}^{-1}+{\kappa }_2{b}^{-1}+{\kappa }_3{c}^{-1}+{\kappa }_4{d}^{-1}+{\kappa }_5{e}^{-1}+{\kappa }_6{f}^{-1}$.

Proof. 

Suppose that $\Lambda ={\kappa }_{1}a+{\kappa }_{2}b+{\kappa }_{3}c+{\kappa }_{4}d+{\kappa }_{5}e+{\kappa }_{6}f\in \mathcal{R}$ is a unit element. Then, an element ${\Lambda }_1={\kappa }_1{a}_1+{\kappa }_2{b}_1+{\kappa }_3{c}_1+{\kappa }_4{d}_1+{\kappa }_5{e}_1+{\kappa }_6{f}_1\in \mathcal{R}$ exists such that $\Lambda {\Lambda }_1=1$. Using the idempotent orthogonality of ${\kappa }_i$ for $1\le i\le 6$, we have ${\kappa }_{1}a{a}_1+{\kappa }_{2}b{b}_1+{\kappa }_{3}c{c}_1+{\kappa }_{4}d{d}_1+{\kappa }_{5}e{e}_1+{\kappa }_{6}f{f}_1=1$. Putting the values of ${\kappa }_i$ for $1\le i\le 6$ 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

$$\begin{array}{cc}\hfill a{a}_1+b{b}_1& =2,\hfill \\ \hfill a{a}_1-b{b}_1& =0,\hfill \\ \hfill -c{c}_1-d{d}_1+e{e}_1+f{f}_1& =0,\hfill \\ \hfill c{c}_1+d{d}_1+e{e}_1+f{f}_1& =4,\hfill \\ \hfill -c{c}_1+d{d}_1+e{e}_1-f{f}_1& =0,\hfill \\ \hfill c{c}_1-d{d}_1+e{e}_1-f{f}_1& =0.\hfill \end{array}$$

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 ${\Lambda }^{-1}={\kappa }_1{a}^{-1}+{\kappa }_2{b}^{-1}+{\kappa }_3{c}^{-1}+{\kappa }_4{d}^{-1}+{\kappa }_5{e}^{-1}+{\kappa }_6{f}^{-1}$.

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

Theorem 4.

Let $C={\kappa }_1\overline{C_1}\oplus {\kappa }_2\overline{C_2}\oplus {\kappa }_3\overline{C_3}\oplus {\kappa }_4\overline{C_4}\oplus {\kappa }_5\overline{C_5}\oplus {\kappa }_6\overline{C_6}$ be a linear code over $\mathcal{R}$ and $\Lambda ={\kappa }_1{\Lambda }_1+{\kappa }_2{\Lambda }_2+{\kappa }_3{\Lambda }_3+{\kappa }_4{\Lambda }_4+{\kappa }_5{\Lambda }_5+{\kappa }_6{\Lambda }_6\in \mathcal{R}$ be a unit element. Then, C is a Λ-constacyclic code over $\mathcal{R}$ if $\overline{C_i}$ is a ${\Lambda }_i$-constacyclic code for $1\le i\le 6$ over ${\mathbb{F}}_p$.

Proof. 

Suppose that C is a $\Lambda$-constacyclic code with length l over $\mathcal{R}$. If $c=(r_0,r_1,\dots ,r_{l-1})\in C$, where $r_j={\kappa }_1{a}_{1,j}+{\kappa }_2{a}_{2,j}+{\kappa }_3{a}_{3,j}+{\kappa }_4{a}_{4,j}+{\kappa }_5{a}_{5,j}+{\kappa }_6{a}_{6,j}$ such that $a_{i,j}\in {\mathbb{F}}_p$ for $1\le i\le 6$ and $0\le j\le l-1$, then we have $(a_{i,0},a_{i,1},\dots ,a_{i,l-1})\in \overline{C_i}$. Thus, the $\Lambda$-constacyclic shift of c is ${\omega }_{\Lambda }\left(c\right)=(\Lambda r_{l-1},r_0,\dots ,r_{l-2})\in C$, where

$$\Lambda r_{l-1}={\kappa }_1{\Lambda }_1{a}_{1,l-1}+{\kappa }_2{\Lambda }_2{a}_{2,l-1}+{\kappa }_3{\Lambda }_3{a}_{3,l-1}+{\kappa }_4{\Lambda }_4{a}_{4,l-1}+{\kappa }_5{\Lambda }_5{a}_{5,l-1}+{\kappa }_6{\Lambda }_6{a}_{6,l-1}.$$

Therefore, we obtain ${\omega }_{\Lambda }\left(c\right)={\sum }_{i=1}^6{\kappa }_i({\Lambda }_i{a}_{i,l-1},a_{i,0},a_{i,1},\dots ,a_{i,l-2})\in C$, which leads to $({\Lambda }_i{a}_{i,l-1},a_{i,0},a_{i,1},\dots ,a_{i,l-2})\in \overline{C_i}.$ Therefore, $\overline{C_i}$ is a ${\Lambda }_i$-constacyclic code for $1\le i\le 6$ of length l over ${\mathbb{F}}_p$.

Conversely, assume that $\overline{C_i}$ is a ${\Lambda }_i$-constacyclic code of length l over ${\mathbb{F}}_p$ for $1\le i\le 6$. Then, for a vector $\overline{a_i}=(a_{i,0},a_{i,1},\dots ,a_{i,l-1})\in \overline{C_i}$, we have ${\omega }_{{\Lambda }_i}\overline{\left(a_i\right)}=({\Lambda }_i{a}_{i,l-1},a_{i,0},a_{i,1},\dots ,a_{i,l-2})\in \overline{C_i}$. Thus, we have

$$\sum _{i=1}^6{\kappa }_i{\omega }_{{\Lambda }_i}\overline{\left(a_i\right)}=\sum _{i=1}^6{\kappa }_i({\Lambda }_i{a}_{i,l-1},a_{i,0},a_{i,1},\dots ,a_{i,l-2})=(\Lambda r_{l-1},r_0,\dots ,r_{l-2})={\omega }_{\Lambda }\left(c\right).$$

Therefore, if $\overline{C_i}$ is a ${\Lambda }_i$-constacyclic code for $1\le i\le 6$ of length l over ${\mathbb{F}}_p$, then C is a $\Lambda$-constacyclic code over $\mathcal{R}$. □

Theorem 5.

Let $C={\kappa }_1\overline{C_1}\oplus {\kappa }_2\overline{C_2}\oplus {\kappa }_3\overline{C_3}\oplus {\kappa }_4\overline{C_4}\oplus {\kappa }_5\overline{C_5}\oplus {\kappa }_6\overline{C_6}$ be a Λ-constacyclic code over $\mathcal{R}$ and $p_i\left(z\right)\in {\displaystyle \frac{{\mathbb{F}}_p\left[z\right]}{⟨z^l-{\Lambda }_i⟩}}$ a unique monic polynomial of the lowest degree such that $\overline{C_i}=⟨p_i\left(z\right)⟩$ and $p_i\left(z\right)|(z^l-{\Lambda }_i)$ for $1\le i\le 6$. Then, $C=⟨p\left(z\right)⟩$, where $p\left(z\right)={\kappa }_1{p}_1\left(z\right)+{\kappa }_2{p}_2\left(z\right)+{\kappa }_3{p}_3\left(z\right)+{\kappa }_4{p}_4\left(z\right)+{\kappa }_5{p}_5\left(z\right)+{\kappa }_6{p}_6\left(z\right)$ and $p\left(z\right)|(z^l-\Lambda )$.

Proof. 

Suppose that $C={\kappa }_1\overline{C_1}\oplus {\kappa }_2\overline{C_2}\oplus {\kappa }_3\overline{C_3}\oplus {\kappa }_4\overline{C_4}\oplus {\kappa }_5\overline{C_5}\oplus {\kappa }_6\overline{C_6}$ is a $\Lambda$-constacyclic code with length l over $\mathcal{R}$, then each $\overline{C_i}$ is a ${\Lambda }_i$-constacyclic code over ${\mathbb{F}}_p$ for $1\le i\le 6$. Therefore, $\overline{C_i}\subseteq {\displaystyle \frac{{\mathbb{F}}_p\left[z\right]}{⟨z^l-{\Lambda }_i⟩}}$ is a principal ideal generated by a monic polynomial $p_i\left(z\right)\in {\displaystyle \frac{{\mathbb{F}}_p\left[z\right]}{⟨z^l-{\Lambda }_i⟩}}$ of lowest degree such that $p_i\left(z\right)|(z^l-{\Lambda }_i)$ for $1\le i\le 6$. Thus, ${\kappa }_i{p}_i\left(z\right)$ are the generator polynomials of C.

If we take $p\left(z\right)={\kappa }_1{p}_1\left(z\right)+{\kappa }_2{p}_2\left(z\right)+{\kappa }_3{p}_3\left(z\right)+{\kappa }_4{p}_4\left(z\right)+{\kappa }_5{p}_5\left(z\right)+{\kappa }_6{p}_6\left(z\right)$, then $⟨p\left(z\right)⟩\subseteq C$. Furthermore, we see that ${\kappa }_{i}p\left(z\right)={\kappa }_i{p}_i\left(z\right)\in ⟨p\left(z\right)⟩$ implies that $C\subseteq ⟨p\left(z\right)⟩$. Thus, we conclude that $C=⟨p\left(z\right)⟩$.

Moreover, we have $p_i\left(z\right)\in {\displaystyle \frac{{\mathbb{F}}_p\left[z\right]}{⟨z^l-{\Lambda }_i⟩}}$ such that $p_i\left(z\right)|(z^l-{\Lambda }_i)$. Thus, polynomials $q_i\left(z\right)\in {\mathbb{F}}_p\left[z\right]$ exist such that $(z^l-{\Lambda }_i)=p_i\left(z\right)q_i\left(z\right)$ for $1\le i\le 6$. Thus, we have

$$p\left(z\right)(\sum _{i=1}^6{\kappa }_i{q}_i\left(z\right))=\sum _{i=1}^6{\kappa }_i{p}_i\left(z\right)q_i\left(z\right)={\kappa }_i(z^l-{\Lambda }_i)=z^l-\Lambda .$$

Thus, we conclude that $p\left(z\right)|(z^l-\Lambda )$. □

Corollary 1.

Let $C={\oplus }_{i=1}^6{\kappa }_i\overline{C_i}$ be a Λ-constacyclic code over $\mathcal{R}$, and $\overline{C_i}=⟨p_i\left(z\right)⟩$ such that $z^l-{\Lambda }_i=p_i\left(z\right)q_i\left(z\right)$ for $1\le i\le 6$. Then:

(i)

$C^{\perp }={\oplus }_{i=1}^6{\kappa }_i{\overline{C_i}}^{\perp }$ is a ${\Lambda }^{-1}$-constacyclic code over $\mathcal{R}$;

(ii)

$C^{\perp }=⟨{\sum }_{i=1}^6{\kappa }_i{q}_i^{*}\left(z\right)⟩$, where $q_i^{*}\left(z\right)$ is the reciprocal polynomial of $q_i\left(z\right)$, which is defined as $q_i^{*}\left(z\right)=z^{deg\left(q_i\left(z\right)\right)}{q}_i\left(z^{-1}\right)$ for $1\le i\le 6$;

(iii)

$|C^{\perp }|=p^{{\sum }_{i=1}^{6}deg\left(p_i\left(z\right)\right)}$.

4. Dual-Containing $\Lambda$-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 a Λ-constacyclic code of length l over $\mathcal{R}$, where Λ is a unit element of $\mathcal{R}$. Then, C is said to be dual-containing if $C^{\perp }\subseteq C$.

Proposition 1.

Let C be a Λ-constacyclic code over $\mathcal{R}$, where $\Lambda ={\kappa }_1{\Lambda }_1+{\kappa }_2{\Lambda }_2+{\kappa }_3{\Lambda }_3+{\kappa }_4{\Lambda }_4+{\kappa }_5{\Lambda }_5+{\kappa }_6{\Lambda }_6\in \mathcal{R}$. If C is a non-trivial dual-containing code, then ${\Lambda }_i=\pm 1$ for $1\le i\le 6$, i.e., $\Lambda \in \{\pm {\kappa }_1\pm {\kappa }_2\pm {\kappa }_3\pm {\kappa }_4\pm {\kappa }_5\pm {\kappa }_6\}\in \mathcal{R}$.

Remark 1.

Suppose that C is a Λ-constacyclic code over $\mathcal{R}$, then from Proposition 1 we conclude that:

(i)

If $\Lambda =1$, then ${\Lambda }_i=1$ and $C_i$ is a cyclic code over ${\mathbb{F}}_p$ for $1\le i\le 6$.

(ii)

If $\Lambda =-1$, then ${\Lambda }_i=-1$ and $C_i$ is a negacyclic code over ${\mathbb{F}}_p$ for $1\le i\le 6$.

(iii)

If ${\Lambda }_i=1$ and ${\Lambda }_j=-1$, then $C_i$ is a cyclic code, and $C_j$ is a negacyclic code over ${\mathbb{F}}_p$ for $1\le i\ne j\le 6$.

Example 3.

Let C be a $(w_1-w_2^2-w_1{w}_2^2)$-constacyclic code over $\mathcal{R}$, then ${\Lambda }_1=1$ implies that $C_1$ is a cyclic code, and ${\Lambda }_j=-1$ further implies that $C_j$ is a negacyclic code for $2\le j\le 6$ over ${\mathbb{F}}_p$.

Example 4.

Let C be a $(1-2w_2^2)$-constacyclic code over $\mathcal{R}$, then ${\Lambda }_i=1$ implies that $C_i$ is a cyclic code for $i=1,2$, and ${\Lambda }_j=-1$ further implies that $C_j$ is a negacyclic code for $3\le j\le 6$ over ${\mathbb{F}}_p$.

Example 5.

Let C be a $(1-{\displaystyle \frac{1}{2}}{w}_2-{\displaystyle \frac{3}{2}}{w}_2^2-w_1{w}_2+w_1{w}_2^2)$-constacyclic code over $\mathcal{R}$, then ${\Lambda }_i=1$ implies that $C_i$ is a cyclic code for $i=1,2,3$, and ${\Lambda }_j=-1$ further implies that $C_j$ is a negacyclic code for $j=4,5,6$ over ${\mathbb{F}}_p$.

Lemma 2

([6]). Let $C_j$ be a ${\Lambda }_j$-constacyclic code with generator polynomial $p_j\left(z\right)$ over ${\mathbb{F}}_p$. Then, $C_j$ is a dual-containing code if $z^n-{\Lambda }_j\equiv 0mod\left(p_j\left(z\right)p_j^{*}\left(z\right)\right)$, where ${\Lambda }_j=\pm 1$ and $p_j^{*}\left(z\right)$ is the reciprocal polynomial of $p_j\left(z\right)$, for $j=1,2,\dots ,6$.

Lemma 3.

Let C be a linear code over $\mathcal{R}$ and $C^{\perp }$ be the dual of C. If ∇ is a Gray map as defined in Equation (2), then $\nabla \left(C^{\perp }\right)=\nabla {\left(C\right)}^{\perp }$. Moreover, if C is a self-orthogonal (self-dual) code over $\mathcal{R}$, then $\nabla \left(C\right)$ is a self-orthogonal (resp. self-dual) code over ${\mathbb{F}}_p$.

Proof. 

The set $K=\{{\kappa }_1,{\kappa }_2,\dots ,{\kappa }_6\}$ forms a basis for 6-dimensional vector space $\mathcal{R}$ over ${\mathbb{F}}_p$. An element $r\in \mathcal{R}$ can be uniquely expressed as $r={\kappa }_1{a}_1+{\kappa }_2{a}_2+{\kappa }_3{a}_3+{\kappa }_4{a}_4+{\kappa }_5{a}_5+{\kappa }_6{a}_6$, where $a_i\in {\mathbb{F}}_p$ for $1\le i\le 6$. Then, we have

$$\nabla \left(r\right)=(a_1,a_2,a_3,a_4,a_5,a_6)M,$$

where $M\in G{L}_6\left({\mathbb{F}}_p\right)$ such that $M{M}^T=\alpha I_6$ and $\alpha \in {\mathbb{F}}_p-\left\{0\right\}$. Let $z=(z_0,z_1,\dots ,z_{l-1})\in C\subseteq {\mathcal{R}}^l$, where $z_j={\kappa }_1{a}_1^j+{\kappa }_2{a}_2^j+{\kappa }_3{a}_3^j+{\kappa }_4{a}_4^j+{\kappa }_5{a}_5^j+{\kappa }_6{a}_6^j\in \mathcal{R}$ for $0\le j\le l-1$. Then, we have $z={\kappa }_1{a}_1+{\kappa }_2{a}_2+{\kappa }_3{a}_3+{\kappa }_4{a}_4+{\kappa }_5{a}_5+{\kappa }_6{a}_6$, where $a_i=(a_i^0,a_i^1,\dots ,a_i^{l-1})\in {\mathbb{F}}_p^l$. Suppose that $y={\kappa }_1{b}_1+{\kappa }_2{b}_2+{\kappa }_3{b}_3+{\kappa }_4{b}_4+{\kappa }_5{b}_5+{\kappa }_6{b}_6\in C^{\perp }$. Then, we obtain that $z·y=0$ implies that

$${\kappa }_1{a}_1{b}_1+{\kappa }_2{a}_2{b}_2+{\kappa }_3{a}_3{b}_3+{\kappa }_4{a}_4{b}_4+{\kappa }_5{a}_5{b}_5+{\kappa }_6{a}_6{b}_6=0.$$

Since K is linearly independent, we obtain $a_i{b}_i=0$ for $1\le i\le 6$. Also, we have $\nabla \left(z\right)=(a_1,a_2,a_3,a_4,a_5,a_6)M\in \nabla \left(C\right)$ and $\nabla \left(y\right)=(b_1,b_2,b_3,b_4,b_5,b_6)M\in \nabla \left(C^{\perp }\right)$. Consider

$$\begin{array}{cc}\hfill \nabla \left(z\right)·\nabla \left(y\right)=\nabla \left(z\right)·\nabla {\left(y\right)}^T=& (a_1,a_2,a_3,a_4,a_5,a_6)M{M}^T(b_1,b_2,b_3,b_4,b_5,b_6)\hfill \\ \hfill =& \alpha (a_1{b}_1+a_2{b}_2+a_3{b}_3+a_4{b}_4+a_5{b}_5+a_6{b}_6)\hfill \\ \hfill =& 0.\hfill \end{array}$$

Therefore, $\nabla \left(y\right)\in \nabla {\left(C\right)}^{\perp }$, i.e., $\nabla \left(C^{\perp }\right)\subseteq \nabla {\left(C\right)}^{\perp }$. Since the Gray map ∇ is bijective, $|\nabla \left(C^{\perp }\right)|=|\nabla {\left(C\right)}^{\perp }|$ suggests that $\nabla \left(C^{\perp }\right)=\nabla {\left(C\right)}^{\perp }$. If C is a self-orthogonal code, then $C\subseteq C^{\perp }$, and hence, $\nabla \left(C\right)\subseteq \nabla \left(C^{\perp }\right)=\nabla {\left(C\right)}^{\perp }$. Therefore, $\nabla \left(C\right)$ 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 $GF\left(q\right)$ with $C_2^{\perp }\subseteq C_1$. Then, a QECC exists with parameters ${\left[[n,k_1+k_2-n,d]\right]}_q$, where $d=min\{wt\left(v\right):v\in (C_1\setminus C_2^{\perp })\cup (C_2\setminus C_1^{\perp })\}\ge min\{d_1,d_2\}$. Moreover, if $C_1$ is a dual-containing code, then a QECC with parameters ${\left[[n,2k_1-n,d_1]\right]}_q$ exists, where $d_1=min\{wt\left(v\right):v\in C_1\setminus C_1^{\perp }\}$.

Theorem 7

([6]). Let C be a Λ-constacyclic code over ${\mathbb{F}}_p$ having a generator polynomial $p\left(z\right)$. Then, C is dual-containing if $(z^l-\Lambda )\equiv 0mod\left(p\left(z\right)p^{*}\left(z\right)\right)$, where $\Lambda =\pm 1$.

The dual-containing cyclic and negacyclic codes over ${\mathbb{F}}_p$ are provided by Theorem 7. Using this outcome, we can now ascertain the prerequisites and requirements for $\Lambda$-constacyclic codes over $\mathcal{R}$ to have their duals, as demonstrated in the following theorem.

Theorem 8.

Let $C={\oplus }_{i=1}^6{\kappa }_i\overline{C_i}$ be a Λ-constacyclic code of length l over $\mathcal{R}$, where $\Lambda ={\kappa }_1{\Lambda }_1+{\kappa }_2{\Lambda }_2+\cdots +{\kappa }_6{\Lambda }_6\in \mathcal{R}$ and $C=⟨p\left(z\right)⟩=⟨{\kappa }_1{p}_1\left(z\right)+{\kappa }_2{p}_2\left(z\right)+\cdots +{\kappa }_6{p}_6\left(z\right)⟩$, where $p_i\left(z\right)$ is the generating polynomial of code $\overline{C_i}$ over ${\mathbb{F}}_p$ for $1\le i\le 6$. Then, C is a dual-containing code if and only if $(z^l-{\Lambda }_i)\equiv 0mod\left(p_i\left(z\right)p_i^{*}\left(z\right)\right)$, where ${\Lambda }_i=\pm 1$ for $1\le i\le 6$.

Proof. 

Suppose that $C={\oplus }_{i=1}^6{\kappa }_i\overline{C_i}$ is a $\Lambda$-constacyclic code over $\mathcal{R}$, where $\Lambda ={\kappa }_1{\Lambda }_1+{\kappa }_2{\Lambda }_2+\cdots +{\kappa }_6{\Lambda }_6\in \mathcal{R}$. Then, by Theorem 4, the code $\overline{C_i}$ is a ${\Lambda }_i$-constacyclic code with generating polynomial $p_i\left(z\right)$ over ${\mathbb{F}}_p$. If C is a dual-containing code, then we have ${\oplus }_{i=1}^6{\kappa }_i{\overline{C_i}}^{\perp }\subseteq {\oplus }_{i=1}^6{\kappa }_i\overline{C_i}$. Since this expression is unique, we have ${\overline{C_i}}^{\perp }\subseteq \overline{C_i}$. Therefore, by Lemma 2 we have $(z^l-{\Lambda }_i)\equiv 0mod(p_i\left(z\right)·p_i^{*}\left(z\right))$. □

Corollary 2.

Let $C={\oplus }_{i=1}^6{\kappa }_i\overline{C_i}$ be a Λ-constacyclic code over $\mathcal{R}$. Then, C is a dual-containing code over $\mathcal{R}$ if and only if $C_i$ is a dual-containing code over ${\mathbb{F}}_p$ for $1\le i\le 6$.

Theorem 9.

Let $C={\oplus }_{i=1}^6{\kappa }_i\overline{C_i}$ be a Λ-constacyclic code of length l over $\mathcal{R}$, and ∇ be the Gray map. If $\nabla \left(C\right)$ has parameters $[6l,k,d_H]$, where $k=k_1+k_2+\cdots +k_6$ is the dimension of $\nabla \left(C\right)$ and $d_L$ is the Lee distance of C, if C is a dual-containing code, then a QECC exists with parameters $[6l,2k-6l,d_H]$ over ${\mathbb{F}}_p$.

Proof. 

Suppose that C is a dual-containing code over $\mathcal{R}$ and ∇ is a Gray map. Then, $\nabla \left(C\right)$ is also a dual-containing code with parameters $[6l,k,d_H]$ over ${\mathbb{F}}_p$. Therefore, by Theorem 6, a QECC with parameters ${\left[[6l,2k-6l,d_H]\right]}_p$ exists over ${\mathbb{F}}_p$. □

Example 6.

Let ${\mathcal{R}}_3={\displaystyle \frac{{\mathbb{F}}_3[w_1,w_2]}{⟨w_1^2-1,w_2^3-w_2,w_1{w}_2-w_2{w}_1⟩}}$ be a finite non-chain ring. Suppose that $\Lambda =2w_2^2-1$ is a unit element in ${\mathcal{R}}_3$. Then, ${\Lambda }_1={\Lambda }_2=-1$ and ${\Lambda }_3={\Lambda }_4={\Lambda }_5={\Lambda }_6=1$. Thus, in ${\mathbb{F}}_3\left[z\right]$, we have

$$\begin{array}{cc}\hfill z^9-1& ={(z+2)}^9\hfill \\ \hfill z^9+1& ={(z+1)}^9.\hfill \end{array}$$

Let $p_1\left(z\right)=p_2\left(z\right)={(z+1)}^4$, $p_3\left(z\right)=p_4\left(z\right)=(z+2)$, and $p_5\left(z\right)=p_6\left(z\right)=1$ be the generator polynomials of $\overline{C_i}$ for $1\le i\le 6$, respectively. Then, $C=⟨{\kappa }_1{p}_1\left(z\right)+{\kappa }_2{p}_2\left(z\right)+\cdots +{\kappa }_6{p}_6\left(z\right)⟩$ is a $(2w_2^2-1)$-constacyclic code of length 9 over ${\mathcal{R}}_3$. Let ${\mathbf{N}}_1\in G{L}_6\left({\mathbb{F}}_3\right)$, as given in Example 1, then ${\mathbf{N}}_1{\mathbf{N}}_1^T=2I_6$ and the Gray image $\nabla \left(C\right)$ has the parameters $[54,44,4]$. Moreover, $(x^9-{\Lambda }_i)\equiv 0mod\left(p_i\left(z\right)p_i^{*}\left(z\right)\right)$ for $1\le i\le 6$; thus, by Theorem 8, we find that C is a dual-containing code; so, by Theorem 9, we have a QECC ${\left[[54,34,4]\right]}_3$, which is a new QECC with this parameter.

Remark 2.

In the previous example, we have seen that the Gray image $\nabla \left(C\right)$ is a linear code with parameters $[54,44,4]$ over the field ${\mathbb{F}}_3$. Specifically, for a code with a length 9, the Gray image’s length is 54, and its dimension is equal to the rational sum of the dimensions of the individual codes, yielding 44 as a result. Let $G_i$ denote the generator matrix of $\overline{C_i}=⟨p_i\left(z\right)⟩$ for $i\in \{1,2,\dots ,6\}$. Then, the generator matrix for $\nabla \left(C\right)$ is given in Section 2.

After providing the generator matrix $\nabla \left(G\right)$ as input to the Magma Computation System [22], it was determined that the minimum distance of $\nabla \left(C\right)$ is 4. 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, $d_H\left(C_1\right)=d_H\left(C_2\right)=3$, $d_H\left(C_3\right)=d_H\left(C_4\right)=2$, and $d_H\left(C_5\right)=d_H\left(C_6\right)=1$, while the Lee distance is 4. Notably, employing the canonical Gray map rather than the Gray map ∇ would result in a Lee distance of 1 instead of 4. Which underlines one of the primary advantages of using the Gray map ∇.

Example 7.

Let ${\mathcal{R}}_5={\displaystyle \frac{{\mathbb{F}}_5[w_1,w_2]}{⟨w_1^2-4,w_2^3-4w_2,w_1{w}_2-w_2{w}_1⟩}}$ be a non-chain ring. Suppose that $\Lambda =1+w_2(1+3w_1)(1+3w_2)$ is a unit element in ${\mathcal{R}}_5$. Then, ${\Lambda }_1={\Lambda }_2={\Lambda }_3=1$ and ${\Lambda }_4={\Lambda }_5={\Lambda }_6=-1$. Thus, in ${\mathbb{F}}_5\left[z\right]$, we have

$$\begin{array}{cc}\hfill z^{15}-1& ={(z+4)}^5{(z^2+z+1)}^5\hfill \\ \hfill z^{15}+1& ={(z+1)}^5{(z^2+4z+1)}^5.\hfill \end{array}$$

Let $p_1\left(z\right)=(z^2+z+1)$, $p_2\left(z\right)={(z+4)}^2$, $p_3\left(z\right)=1$, and $p_4\left(z\right)=p_5\left(z\right)=p_6\left(z\right)=(z+1)$ be the generator polynomials of $\overline{C_i}$ for $1\le i\le 6$, respectively. Then, $C=⟨{\kappa }_1{p}_1\left(z\right)+{\kappa }_2{p}_2\left(z\right)+\cdots +{\kappa }_6{p}_6\left(z\right)⟩$ is a $(1+{\displaystyle \frac{1}{2}}(1-w_1)(w_2-w_2^2))$-constacyclic code with length 15 over ${\mathcal{R}}_5$. Let ${\mathbf{N}}_2\in G{L}_6\left({\mathbb{F}}_5\right)$, as given in Example 2, then ${\mathbf{N}}_2{\mathbf{N}}_2^T=4I_6$ and the Gray image $\nabla \left(C\right)$ has the parameters $[90,83,3]$. Moreover, $(z^{15}-{\Lambda }_i)\equiv 0mod\left(p_i\left(z\right)p_i^{*}\left(z\right)\right)$ for $1\le i\le 6$; thus, by Theorem 8, we find that C is a dual-containing code; hence, by Theorem 9 we have a new QECC ${\left[[90,76,3]\right]}_5$, with this parameter. Again, here we can see that the distance of $\nabla \left(C\right)\ge d_H\left(\overline{C_i}\right)$ for $1\le i\le 6$.

Example 8.

Let ${\mathcal{R}}_5={\displaystyle \frac{{\mathbb{F}}_5[w_1,w_2]}{⟨w_1^2+1,w_2^3-4w_2,w_1{w}_2-w_2{w}_1⟩}}$ be a non-chain ring. Suppose that $\Lambda =4$ is a unit element in ${\mathcal{R}}_5$. Then, ${\Lambda }_1=-1={\Lambda }_2$ and ${\Lambda }_3=1={\Lambda }_4={\Lambda }_5={\Lambda }_6$. Thus, in ${\mathbb{F}}_5\left[z\right]$, we have

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

Let $p_1\left(z\right)=(z^6+4z^5+z^4+4z^3+z^2+4z+1)$, $p_2\left(z\right)={(z+1)}^2$, $p_3\left(z\right)=1$, and $p_4\left(z\right)=p_5\left(z\right)=p_6\left(z\right)=(z+4)$ be the generator polynomials of $\overline{C_i}$ for $1\le i\le 6$, respectively. Then, $C=⟨{\kappa }_1{p}_1\left(z\right)+{\kappa }_2{p}_2\left(z\right)+\cdots +{\kappa }_6{p}_6\left(z\right)⟩$ is a Λ-constacyclic code with length 35 over ${\mathcal{R}}_5$. Let ${\mathbf{N}}_2\in G{L}_6\left({\mathbb{F}}_5\right)$, as given in Example 2, then ${\mathbf{N}}_2{\mathbf{N}}_2^T=4I_6$ and the Gray image $\nabla \left(C\right)$ has the parameters $[210,199,3]$. Moreover, $(z^{35}-{\Lambda }_i)\equiv 0mod\left(p_i\left(z\right)p_i^{*}\left(z\right)\right)$ for $1\le i\le 6$; thus, by Theorem 8, we find that C is a dual-containing code; so, by Theorem 9 we have an improved QECC ${\left[[210,188,3]\right]}_5$ against the existing code ${\left[[210,186,3]\right]}_5$ [14]. Here, we can see that the distance of $\nabla \left(C\right)\ge d_H\left(\overline{C_i}\right)$ for $1\le i\le 6$.

Example 9.

Let ${\mathcal{R}}_5$ be a non-chain ring, as in Example 7. Suppose that $\Lambda =4+w_2(2+w_1)(2+w_2)$ is a unit element in ${\mathcal{R}}_5$. Then, ${\Lambda }_1=-1={\Lambda }_2={\Lambda }_3$ and ${\Lambda }_4=1={\Lambda }_5={\Lambda }_6$. Thus, in ${\mathbb{F}}_5\left[z\right]$, we have

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

Let $p_1\left(z\right)=(z^6+4z^3+1)$, $p_2\left(z\right)=(z+1)$, $p_3\left(z\right)={(z+1)}^2$, $p_4\left(z\right)=1$, and $p_5\left(z\right)=p_6\left(z\right)=(z+4)$ be the generator polynomials of $\overline{C_i}$ for $1\le i\le 6$, respectively. Then, $C=⟨{\kappa }_1{p}_1\left(z\right)+{\kappa }_2{p}_2\left(z\right)+\cdots +{\kappa }_6{p}_6\left(z\right)⟩$ is a $(4+w_2(2+w_1)(2+w_2))$-constacyclic code with length 45 over ${\mathcal{R}}_5$. Let ${\mathbf{N}}_2\in G{L}_6\left({\mathbb{F}}_5\right)$, as given in Example 2, then ${\mathbf{N}}_2{\mathbf{N}}_2^T=4I_6$ and the Gray image $\nabla \left(C\right)$ has the parameters $[270,259,3]$. Moreover, $(z^{45}-{\Lambda }_i)\equiv 0mod\left(p_i\left(z\right)p_i^{*}\left(z\right)\right)$ for $1\le i\le 6$; thus, by Theorem 8, we find that C is a dual-containing code; thus, by Theorem 9 we have an improved QECC with parameters ${\left[[270,248,3]\right]}_5$ against the existing code ${\left[[270,246,3]\right]}_5$ [28]. Here, we can see that the distance of $\nabla \left(C\right)\ge d_H\left(\overline{C_i}\right)$ for $1\le i\le 6$.

Example 10.

Let ${\mathcal{R}}_7={\displaystyle \frac{{\mathbb{F}}_7[w_1,w_2]}{⟨w_1^2-1,w_2^3-w_2,w_1{w}_2-w_2{w}_1⟩}}$ be a non-chain ring. Suppose that $\Lambda =2w_2^2-1$ is a unit element in ${\mathcal{R}}_7$. Then, ${\Lambda }_1=-1={\Lambda }_2$ and ${\Lambda }_3={\Lambda }_4=1={\Lambda }_5={\Lambda }_6$. Thus, in ${\mathbb{F}}_7\left[z\right]$, we have

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

Let $p_1\left(z\right)=(z^3+2)$, $p_2\left(z\right)=1$, and $p_3\left(z\right)=p_4\left(z\right)=(z+3)=p_5\left(z\right)=p_6\left(z\right)$ be the generator polynomials of $\overline{C_i}$ for $1\le i\le 6$, respectively. Then, $C=⟨{\kappa }_1{p}_1\left(z\right)+{\kappa }_2{p}_2\left(z\right)+\cdots +{\kappa }_6{p}_6\left(z\right)⟩$ is a $(2w_2^2-1)$-constacyclic code of length 9 over ${\mathcal{R}}_7$. Let ${\mathbf{N}}_3\in G{L}_6\left({\mathbb{F}}_7\right)$ such that

$$N_3=\left[\begin{array}{ccccccc}3& 2& 2& 2& 2& 2& \\ 2& 3& 2& 2& 2& 2& \\ 2& 2& 3& 2& 2& 2& \\ 2& 2& 2& 3& 2& 2& \\ 2& 2& 2& 2& 3& 2& \\ 2& 2& 2& 2& 2& 3\end{array}\right],$$

then ${\mathbf{N}}_3{\mathbf{N}}_3^T=I_6$ and the Gray image $\nabla \left(C\right)$ has the parameters $[54,47,3]$. Moreover, $(z^9-{\Lambda }_i)\equiv 0mod\left(p_i\left(z\right)p_i^{*}\left(z\right)\right)$ for $1\le i\le 6$; thus, by Theorem 8, we find that C is a dual-containing code; so, by Theorem 9, we have a new QECC with parameters ${\left[[54,40,3]\right]}_7$. Here, one can see that the distance of $\nabla \left(C\right)\ge d_H\left(\overline{C_i}\right)$ for $1\le i\le 6$.

Note: In Table 1, $q,n$, and $\Lambda$ represent the order of the field, the length of the code defined over $\mathcal{R}$, and the unit element in $\mathcal{R}$, respectively. $p_i\left(z\right)$ 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 ${\mathbb{F}}_3$, ${\mathbb{F}}_5$, ${\mathbb{F}}_7$, respectively, used to define the Gray map ∇. The parameters of the corresponding Gray image (dual-containing code) are denoted by $\nabla \left(C\right)$. $\left[\right[n,k,d\left]\right]$ and $\left[[n,k^{\prime },d^{\prime }]\right]$ represent the parameters of the new QECC and existing QECC, respectively.

Table 1. Some new and improved QECCs over ${\mathbb{F}}_p$ from constacyclic codes over ${\mathcal{R}}_p$ for $(p=3,5,7)$.

q	n	$\Lambda$	${\mathit{p}}_1\left(\mathit{z}\right)$	${\mathit{p}}_2\left(\mathit{z}\right)$	${\mathit{p}}_3\left(\mathit{z}\right)={\mathit{p}}_4\left(\mathit{z}\right)$	${\mathit{p}}_5\left(\mathit{x}\right)={\mathit{p}}_6\left(\mathit{x}\right)$	N	$\mathit{\nabla }\left(\mathit{C}\right)$	$\left[\right[\mathit{n},\mathit{k},\mathit{d}\left]\right]$	$\left[[\mathit{n},{\mathit{k}}^{\prime },{\mathit{d}}^{\prime }]\right]$
3	9	$-w_1+w_2^2+w_1{w}_2^2$	11	12	1	1	$N_1$	$[54,52,2]$	${\left[[54,50,2]\right]}_3$	${\left[[54,46,2]\right]}_3$ [12]
3	9	$2w_2^2-1$	11011	11011	12	1	$N_1$	$[54,44,4]$	${\left[[54,34,4]\right]}_3$	New QECC
3	11	$-w_1+w_2^2+w_1{w}_2^2$	102221	1	102122	1	$N_1$	$[66,51,3]$	${\left[[66,36,3]\right]}_3$	New QECC
3	11	$-w_1+w_2^2+w_1{w}_2^2$	102221	102122	102122	1	$N_1$	$[66,46,4]$	${\left[[66,26,4]\right]}_3$	New QECC
3	11	$-w_1+w_2^2+w_1{w}_2^2$	102221	1	102122	1022122	$N_1$	$[66,41,5]$	${\left[[66,16,5]\right]}_3$	New QECC
3	13	$2w_2^2-1$	1021	1021	1102	1	$N_1$	$[78,66,4]$	${\left[[78,54,4]\right]}_3$	New QECC
3	18	1	11011	11011	12	1	$N_1$	$[108,98,3]$	${\left[[108,88,3]\right]}_3$	New QECC
5	6	$-1$	134	13	12	12	$N_2$	$[36,29,4]$	${\left[[36,22,4]\right]}_5$	New QECC
5	18	$-1$	1003004	13	12	12	$N_2$	$[108,97,4]$	${\left[[108,86,4]\right]}_5$	New QECC
5	19	$-1$	1	1033234341	1033234341	1033234341	$N_2$	$[114,69,6]$	${\left[[114,24,6]\right]}_5$	New QECC
5	19	$-1$	1033234341	1033234341	1033234341	1033234341	$N_2$	$[114,60,7]$	${\left[[114,6,7]\right]}_5$	New QECC
5	20	$-w_1+w_2^2+w_1{w}_2^2$	10404	11	11	11	$N_2$	$[120,111,3]$	${\left[[120,102,3]\right]}_5$	${\left[[120,96,3]\right]}_5$ [14]
5	22	$-w_1-w_2^2+w_1{w}_2^2$	111212	1	13	12	$N_2$	$[132,123,4]$	${\left[[132,114,4]\right]}_5$	${\left[[132,110,4]\right]}_5$ [21]
5	25	$1-2w_2^2$	1400041	1	11	11	$N_2$	$[150,140,3]$	${\left[[150,130,3]\right]}_5$	New QECC
5	25	$1-2w_2^2$	140000000014	1	11	11	$N_2$	$[150,135,4]$	${\left[[150,120,4]\right]}_5$	New QECC
5	40	1	12342	1	12	13	$N_2$	$[240,232,3]$	${\left[[240,224,3]\right]}_5$	New QECC
7	7	$1-2w_2^2$	1436	1	11	11	$N_3$	$[42,35,4]$	${\left[[42,28,4]\right]}_7$	New QECC
7	9	$2w_2^2-1$	1002	12	13	13	$N_3$	$[54,46,4]$	${\left[[54,38,4]\right]}_7$	New QECC
7	14	$-w_1+w_2^2+w_1{w}_2^2$	10201	1331	11	11	$N_3$	$[84,73,4]$	${\left[[84,62,4]\right]}_7$	New QECC
7	15	$1-2w_2^2$	153356	1	12	14	$N_3$	$[90,81,4]$	${\left[[90,72,4]\right]}_7$	New QECC
7	18	$-w_1+w_2^2+w_1{w}_2^2$	102	13026	12	12	$N_3$	$[108,98,4]$	${\left[[108,88,4]\right]}_7$	New QECC

5. Conclusions

This article focuses on the exploration of constacyclic codes in the context of non-chain rings $\mathcal{R}={\displaystyle \frac{{\mathbb{F}}_p[u,v]}{⟨w_1^2-{\alpha }^2,w_2^3-{\beta }^2{w}_2,w_1{w}_2-w_2{w}_1⟩}}$, where $\alpha ,\beta \in {\mathbb{F}}_p-\left\{0\right\}$ for a prime p. From this investigation, numerous new and improved quantum codes have been derived. Substantial potential exists for discovering additional quantum codes within the finite field ${\mathbb{F}}_p$ by considering prime powers instead of primes. Applying the Gray map ∇ harnesses this potential. In a more general context, substituting the ring $\mathcal{R}$ with alternative commutative finite rings offers the prospect of developing many fresh quantum code constructions.