On the Construction of Quantum and LCD Codes from Cyclic Codes over the Finite Commutative Rings

Abstract

Let ${\mathbb{F}}_q$ be a field of order q, where q is a power of an odd prime p, and $\alpha$ and $\beta$ are two non-zero elements of ${\mathbb{F}}_q$. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring $R={\mathbb{F}}_q[u_1,u_2]/⟨{u_1}^2-{\alpha }^2,u_2^2-{\beta }^2,u_1{u}_2-u_2{u}_1⟩$. We decompose the ring R by using orthogonal idempotents ${\Delta }_1,{\Delta }_2,{\Delta }_3$, and ${\Delta }_4$ as $R={\Delta }_{1}R\oplus {\Delta }_{2}R\oplus {\Delta }_{3}R\oplus {\Delta }_{4}R$, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes.

1. Introduction

Throughout this paper, unless indicated otherwise, ${\mathbb{F}}_q$ (where q is an odd prime power) denotes the field of order q, and $\alpha$ and $\beta$ are non-zero elements of ${\mathbb{F}}_q$. Next, let us consider the finite ring $R={\mathbb{F}}_q[u_1,u_2]/\langle u_1^2-{\alpha }^2,u_2^2-{\beta }^2,u_1{u}_2-u_2{u}_1\rangle$. It is straightforward to check that R is a non-chain semi-local ring of order $q^4$. Cyclic codes are very useful for the construction of quantum-error-correcting (QEC) codes. QEC codes are different from classical-error-correcting (CEC) codes. A significant breakthrough happened in 1998, when Calderbank et al. [1] solved the problem of obtaining QEC codes with the help of CEC codes over GF (4). Calderbank et al. [1] also introduced a method to construct QEC codes from CEC codes. Over finite fields, cyclic codes have been extensively investigated (see, for example, [2,3,4,5] and references therein). In 2015, from the cyclic codes over ${\mathbb{F}}_q+v{\mathbb{F}}_q+v^2{\mathbb{F}}_q+v^3{\mathbb{F}}_q$ (where $q=p^m$, p is a prime such that $3\left|\right(p-1)$, $v^4=v$, and m is a positive integer), Gao et al. [6] constructed new quantum codes over ${\mathbb{F}}_q$. Afterwards, Ozen et al. [7] constructed many ternary quantum codes from cyclic codes over ${\mathbb{F}}_3+u{\mathbb{F}}_3+v{\mathbb{F}}_3+uv{\mathbb{F}}_3$. In 2021, Ashraf et al. [8] found better quantum and LCD codes over the ring ${\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}$ with $v^2=1$, where m is a positive integer. In this article, we discuss the structural properties of cyclic codes over the ring R. On this ring R, we construct a Gray map that provides better parameters and contributes to the finding of better quantum codes over R than presented in [8,9,10,11,12,13] (and references therein).

In this paper, our main aim is to study the structural properties of cyclic codes over the finite ring R, and to construct quantum-error-correcting (QEC) codes over R. Moreover, we also study LCD codes. The major contributions of this paper are as follows:

• This paper provides superior quantum codes to those presented in recent references [8,9,10,11,12,13], see

  Table 1. Quantum codes from cyclic codes over R.

  n	${\mathit{h}}_1\left(\mathit{z}\right)$	${\mathit{h}}_2\left(\mathit{z}\right)$	${\mathit{h}}_3\left(\mathit{z}\right)$	${\mathit{h}}_4\left(\mathit{z}\right)$	$\mathit{\eta }\left(\mathcal{C}\right)$	${\left[[\mathit{n},\mathit{k},\mathit{d}]\right]}_{\mathit{q}}$	${\left[[{\mathit{n}}^{{}^{\prime }},{\mathit{k}}^{{}^{\prime }},{\mathit{d}}^{{}^{\prime }}]\right]}_{\mathit{q}}$
  10	$z+1$	$z+1$	$z+4$	$z+4$	$[40,36,2]$	${\left[[40,32,2]\right]}_5$	${\left[[40,24,2]\right]}_5$ [9]
  20	${(z+2)}^2$	${(z+2)}^2$	${(z+2)}^2$	${(z+2)}^2$	$[80,68,3]$	${\left[[80,56,3]\right]}_5$	${\left[[80,54,3]\right]}_5$ [11]
  	$(z+4)$	$(z+4)$	$(z+4)$	$(z+4)$
  22	$z+1$	$z+1$	$z+4$	$z+4$	$[88,84,2]$	${\left[[88,80,2]\right]}_5$	${\left[[88,48,2]\right]}_5$ [9]
  28	$z+2$	$z+2$	$z+3$	$z+3$	$[112,108,2]$	${\left[[112,104,2]\right]}_5$	${\left[[112,64,2]\right]}_5$ [9]
  30	${(z+1)}^2$	${(z+1)}^2$	${(z+1)}^2$	${(z+1)}^2$	$[120,104,3]$	${\left[[120,88,3]\right]}_5$	${\left[[120,32,3]\right]}_5$ [12]
  	$(z^2+z+1)$	$(z^2+z+1)$	$(z^2+z+1)$	$(z^2+z+1)$
  31	$z+4$	$z+4$	$z+4$	$(z^3+z^2+z+4)$	$[124,115,4]$	${\left[[124,106,4]\right]}_5$	${\left[[124,100,4]\right]}_5$ [8]
  				$(z^3+z^2+3z+4)$
  35	$z+4$	$z+4$	${(z+4)}^2$	$(z+4)$	$[140,129,3]$	${\left[[140,118,3]\right]}_5$	${\left[[140,112,2]\right]}_5$ [9]
  				$(z^6+z^5+z^4+$
  				$z^3+z^2+z+1)$
  42	$(z+4)$	$(z+4)$	$(z+4)$	$(z+4)$	$[168,140,4]$	${\left[[168,112,4]\right]}_5$	${\left[[168,96,2]\right]}_5$ [10]
  	$(z^6+2z^4+$	$(z^6+2z^4+$	$(z^6+2z^4+$	$(z^6+2z^4+$
  	$3z^3+2z^2+1)$	$3z^3+2z^2+1)$	$3z^3+2z^2+1)$	$3z^3+2z^2+1)$
  24	$z+3$	$z+3$	$z+3$	$(z+3)$	$[96,90,3]$	${\left[[96,84,3]\right]}_7$	${\left[[96,80,3]\right]}_7$ [8]
  				$(z^2+z+4)$
  78	${(z+3)}^2)$	${(z+3)}^2$	${(z+3)}^2$	${(z+3)}^2$	$[312,300,3]$	${\left[[312,288,3]\right]}_{13}$	${\left[[312,282,3]\right]}_{13}$ [13]
  	$(z+12$	$(z+12)$	$(z+12)$	$(z+12)$
  12	1	$z+1$	$z+1$	$(z+1)$	$[48,41,4]$	${\left[[48,34,4]\right]}_{17}$	${\left[[48,32,4]\right]}_{17}$ [8]
  				$(z^2+4z+16)$
  				$(z^2+z+1)$
  19	$z+18$	$z+18$	${(z+18)}^2$	${(z+18)}^{14}$	$[76,58,4]$	${\left[[76,40,4]\right]}_{19}$	…

  .
• This paper provides some new quantum codes, see

  Table 2. New quantum codes from cyclic codes over R.

  n	${\mathit{h}}_1\left(\mathit{z}\right)$	${\mathit{h}}_2\left(\mathit{z}\right)$	${\mathit{h}}_3\left(\mathit{z}\right)$	${\mathit{h}}_4\left(\mathit{z}\right)$	$\mathit{\eta }\left(\mathcal{C}\right)$	${\left[[\mathit{n},\mathit{k},\mathit{d}]\right]}_{\mathit{q}}$
  9	${(z+2)}^4$	${(z+2)}^2$	$z+2$	1	$[36,29,3]$	${\left[[36,22,3]\right]}_3$	New quantum code
  25	${(z+4)}^6$	$z+4$	$z+4$	1	$[100,92,3]$	${\left[[100,84,3]\right]}_5$	$Newquantumcode$
  15	${(z+4)}^2(z^2+z+1)$	$z+4$	$z^2+z+1$	1	$[60,53,3]$	${\left[[60,46,3]\right]}_5$	$Newquantumcode$
  14	$(z+1){(z+6)}^3$	$z+1$	$z+6$	1	$[56,50,4]$	${\left[[56,44,4]\right]}_7$	$Newquantumcode$
  11	${(z+10)}^5$	$z+10$	$z+10$	1	$[44,37,4]$	${\left[[44,30,4]\right]}_{11}$	$Newquantumcode$

  .
• This paper investigates some optimal LCD codes over the ring R, see

  Table 3. Gray images of LCD codes of length n over R.

  n	${\mathit{h}}_1\left(\mathit{z}\right)$	${\mathit{h}}_2\left(\mathit{z}\right)$	${\mathit{h}}_3\left(\mathit{z}\right)$	${\mathit{h}}_4\left(\mathit{z}\right)$	$\mathit{\eta }\left(\mathcal{C}\right)$
  4	1	$z+1$	$z+1$	$(z+1)(z^2+1)$	${[16,11,4]}_3$	Optimal
  22	$z+1$	$z+1$	$z+1$	$z+1$	${[88,84,2]}_3$	$Optimal$
  6	1	$z+1$	$z+1$	$(z+1)(z^2+z+1)$	${[24,17,4]}_5$	…
  				$(z^2+4z+1)$
  8	1	$z+1$	$z+1$	$(z+1)(z^2+4z+1)$	${[32,27,4]}_7$	$Optimal$
  37	$z^6+5z^5$	$z^6+5z^5$	$z^6+4z^5$	$z^6+4z^5$	$[148,124,5]11$
  	$+5z^4+$	$+5z^4+$	$+3z^4+$	$+3z^4+$
  	$4z^3+5z^2$	$4z^3+5z^2$	$7z^3+3z^2$	$7z^3+3z^2$
  	$+5z+1$	$+5z+1$	$4z+1$	$4z+1$
  39	$(z^2+z+1)$	$(z^2+z+1)$	$(z^2+z+1)$	$(z^2+z+1)$	${[156,100,4]}_{11}$
  	$(z^{12}+z^{11}$	$(z^{12}+z^{11}$	$(z^{12}+z^{11}$	$(z^{12}+z^{11}$
  	$+z^{10}+z^9$	$+z^{10}+z^9$	$+z^{10}+z^9$	$+z^{10}+z^9$
  	$+z^8+z^7+$	$+z^8+z^7+$	$+z^8+z^7+$	$+z^8+z^7+$
  	$z^6+z^5+z^4$	$z^6+z^5+z^4$	$z^6+z^5+z^4$	$z^6+z^5+z^4$
  	$+z^3+z^2$	$+z^3+z^2$	$+z^3+z^2$	$+z^3+z^2$
  	$+z+1)$	$+z+1)$	$+z+1)$	$+z+1)$
  11	$z^{10}+z^9+z^8$	$z^{10}+z^9+z^8$	$z^{10}+z^9+z^8$	$z^{10}+z^9+z^8$	${[44,4,11]}_{19}$
  	$+z^7+z^6+$	$+z^7+z^6+$	$+z^7+z^6+$	$+z^7+z^6+$
  	$z^5+z^4+z^3$	$z^5+z^4+z^3$	$z^5+z^4+z^3$	$z^5+z^4+z^{3}1$
  	$+z^2+z+1$	$+z^2+z+1$	$+z^2+z+1$	$+z^2+z+1$
  28	$(z+1)$	$(z+1)$	$(z+1)$	$(z+1)$	${[112,60,8]}_{19}$
  	$(z^6+z^5+z^4+$	$(z^6+z^5+z^4+$	$(z^6+z^5+z^4+)$	$(z^6+z^5+z^4+$
  	$z^3+z^2+z+1)$	$z^3+z^2+z+1)$	$z^3+z^2+z+1)$	$z^3+z^2+z+1)$
  	$(z^6+11z^5+3z^4+$	$(z^6+11z^5+3z^4+$	$(z^6+8z^5+3z^4+$	$(z^6+8z^5+3z^4+$
  	$11z^3+3z^2$	$11z^3+3z^2$	$8z^3+3z^2$	$8z^3+3z^2$
  	$+11z+1)$	$+11z+1)$	$+8z+1)$	$+8z+1)$
  34	$(z+1)$	$(z+1)$	$(z+1)$	$(z+1)$	${[136,100,4]}_{19}$
  	$(z^8+13z^7+15z^6+$	$(z^8+13z^7+15z^6+$	$(z^8+13z^7+15z^6+$	$(z^8+13z^7+15z^6+$
  	$16z^5+8z^4+16z^3$	$16z^5+8z^4+16z^3$	$16z^5+8z^4+16z^3$	$16z^5+8z^4+16z^3$
  	$+15z^2+13z+1)$	$+15z^2+13z+1)$	$+15z^2+13z+1)$	$+15z^2+13z+1)$

  .

2. Preliminary Results

In this section, we deal with the study of some preliminaries and describe the Gray map over the ring R. Moreover, we establish some important results which are needed in the subsequent discussions. If a code $\mathcal{C}$ is an R-submodule of $R^n$.(where n is a positive integer), then $\mathcal{C}$ is linear. The elements of $\mathcal{C}$ are called codewords. The size of $\mathcal{C}$.refers to the total number of codewords in $\mathcal{C}$, which is indicated by $\left|\mathcal{C}\right|$. We recall some basic definitions as following:

(i)

The Hamming distance between two vectors $\mathbf{x}=(x_1,x_2,\dots ,x_n)$ and $\mathbf{y}=(y_1,y_2,\dots ,y_n)$ is the number of places where they differ, and is denoted by $d(\mathbf{x},\mathbf{y})$.

(ii)

The Hamming weight of a vector $\mathbf{x}=(x_1,x_2,\dots ,x_n)$ is the number of non-zero $x_i$ and is denoted by $wt\left(\mathbf{x}\right)$.

(iii)

The Euclidean inner product of any two vectors $\mathbf{x}=(x_1,x_2,\dots ,x_n)$ and $\mathbf{y}=(y_1,y_2,\dots ,y_n)$ is defined as $\mathbf{x}·\mathbf{y}=x_0{y}_0+x_1{y}_1+\dots +x_{n-1}{y}_{n-1}$ and the dual of linear code $\mathcal{C}$ is ${\mathcal{C}}^{\perp }=\{\mathbf{x}\in R^n|\mathbf{x}·\mathbf{y}=0\forall \mathbf{y}\in \mathcal{C}\}$.

(iv)

A code $\mathcal{C}$ is said to be self-dual if $\mathcal{C}={\mathcal{C}}^{\perp }$, self-orthogonal if $\mathcal{C}\subseteq {\mathcal{C}}^{\perp }$, and dual containing if ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$.

Clearly, the ring R can be expressed as $R={\mathbb{F}}_q+u_1{\mathbb{F}}_q+u_2{\mathbb{F}}_q+u_1{u}_2{\mathbb{F}}_q$ such that $u_1^2={\alpha }^2,u_2^2={\beta }^2$ and $u_1{u}_2=u_2{u}_1$; here ${\mathbb{F}}_q$ is the finite field of order q, where $q=p^m$ for odd prime p and $m\ge 1$. It is a commutative non-chain semi-local ring with four maximal ideals. An element z of R is of the form $z=a_1+a_2{u}_1+a_3{u}_2+a_4{u}_1{u}_2$, where $a_i\in {\mathbb{F}}_q$ and $1\le i\le 4$. With the help of a set of orthogonal idempotents, every element of this ring can be represented:

$${\Delta }_1=\frac{(\alpha +u_1)(\beta +u_2)}{4\alpha \beta },$$

$${\Delta }_2=\frac{(\alpha +u_1)(\beta -u_2)}{4\alpha \beta },$$

$${\Delta }_3=\frac{(\alpha -u_1)(\beta +u_2)}{4\alpha \beta },$$

and

$${\Delta }_4=\frac{(\alpha -u_1)(\beta -u_2)}{4\alpha \beta }.$$

It is straightforward to show that ${\Delta }_i^2={\Delta }_i,0={\Delta }_i{\Delta }_j$, and ${\Delta }_1+{\Delta }_2+{\Delta }_3+{\Delta }_4=1$, where $1\le i,j\le 4$, and $i\ne j.$ in view of the Chinese Remainder Theorem, we obtain $R={\Delta }_{1}R\oplus {\Delta }_{2}R\oplus {\Delta }_{3}R\oplus {\Delta }_{4}R={\Delta }_1{\mathbb{F}}_q\oplus {\Delta }_2{\mathbb{F}}_q\oplus {\Delta }_3{\mathbb{F}}_q\oplus {\Delta }_4{\mathbb{F}}_q$. Thus, we can express every element z of R as $z=a_1+a_2{u}_1+a_3{u}_2+a_4{u}_1{u}_2={\Delta }_1{z}_1+{\Delta }_2{z}_2+{\Delta }_3{z}_3+{\Delta }_4{z}_4$, where $a_i,z_i\in {\mathbb{F}}_q$ and $1\le i\le 4$.

The Gray map $\eta :R⟶{\mathbb{F}}_q^4$ is defined by

$$\eta ({\Delta }_1{z}_1+{\Delta }_2{z}_2+{\Delta }_3{z}_3+{\Delta }_4{z}_4)=(z_1,z_2,z_3,z_4)A,$$

(1)

where $A\in G{L}_4\left({\mathbb{F}}_q\right)$ is a fixed matrix and $G{L}_4\left({\mathbb{F}}_q\right)$ is the linear group of all $4\times 4$ invertible matrices over the field ${\mathbb{F}}_q$ such that $A{A}^T=ϵI_{4\times 4},$ where $A^T$ is the transpose of A and $ϵ\in {\mathbb{F}}_q\setminus \left\{0\right\}$.

The above Gray map is linear, and we can extend it component-wise from $R^n$ to ${\mathbb{F}}_q^{4n}$, where n is a positive integer. For any element $z={\Delta }_1{z}_1+{\Delta }_2{z}_2+{\Delta }_3{z}_3+{\Delta }_4{z}_4\in R$, the Lee weight of z is defined as $w_L\left(z\right)=w_H\left(\eta \left(z\right)\right)$, where $w_H$ represents the Hamming weight over ${\mathbb{F}}_q$. We begin our discussion with the following result related to the Gray map (1):

Proposition 1.

The map $\eta :R⟶{\mathbb{F}}_q^4$ defined in (1) is an ${\mathbb{F}}_q$-linear and distance-preserving map from $(R^n,d_L)$ to $({\mathbb{F}}_q^{4n},d_H)$, where $d_L=d_H$.

Proof. 

Let $z,z^{{}^{\prime }}\in R^n$ such that

$$z={\Delta }_1{z}_1+{\Delta }_2{z}_2+{\Delta }_3{z}_3+{\Delta }_4{z}_4$$

$$z^{{}^{\prime }}={\Delta }_1{z}_1^{{}^{\prime }}+{\Delta }_2{z}_2^{{}^{\prime }}+{\Delta }_3{z}_3^{{}^{\prime }}+{\Delta }_4{z}_4^{{}^{\prime }}$$

and $z_i,z_i^{{}^{\prime }}\in {\mathbb{F}}_q^n$ for $1\le i\le 4$. Then, we have

$$\begin{array}{ccc}\hfill \eta (z+z^{{}^{\prime }})& =& \eta ({\Delta }_1{z}_1+{\Delta }_1{z}_1^{{}^{\prime }}+{\Delta }_2{z}_2+{\Delta }_2{z}_2^{{}^{\prime }}+{\Delta }_3{z}_3+{\Delta }_3{z}_3^{{}^{\prime }}+{\Delta }_4{z}_4+{\Delta }_4{z}_4^{{}^{\prime }})\hfill \\ & =& \eta ({\Delta }_1(z_1+z_1^{{}^{\prime }})+{\Delta }_2(z_2+z_2^{{}^{\prime }})+{\Delta }_3(z_3+z_3^{{}^{\prime }})+{\Delta }_4(z_4+z_4^{{}^{\prime }}))\hfill \\ & =& (z_1+z_1^{{}^{\prime }},z_2+z_2^{{}^{\prime }},z_3+z_3^{{}^{\prime }},z_4+z_4^{{}^{\prime }})A\hfill \\ & =& (z_1,z_2,z_3,z_4)A+(z_1^{{}^{\prime }},z_2^{{}^{\prime }},z_3^{{}^{\prime }},z_4^{{}^{\prime }})A\hfill \\ & =& \eta \left(z\right)+\eta \left(z^{{}^{\prime }}\right)forallz,z^{{}^{\prime }}\in R^n.\hfill \end{array}$$

Furthermore, for any $\alpha \in {\mathbb{F}}_q$, we have

$$\begin{array}{ccc}\hfill \eta \left(\alpha z\right)& =& \eta ({\Delta }_1\alpha z_1+{\Delta }_2\alpha z_2+{\Delta }_3\alpha z_3+{\Delta }_4\alpha z_4)\hfill \\ & =& (\alpha z_1,\alpha z_2,\alpha z_3,\alpha z_4)A\hfill \\ & =& \alpha (z_1,z_2,z_3,z_4)A\hfill \\ & =& \alpha \eta \left(z\right)forallz\in R^n.\hfill \end{array}$$

Hence, η is an ${\mathbb{F}}_q$-linear. As for the second part, we know that

$$\begin{array}{ccc}\hfill d_L(z,z^{{}^{\prime }})& =& w_L(z-z^{{}^{\prime }})\hfill \\ & =& w_H\left(\eta (z-z^{{}^{\prime }})\right)\hfill \\ & =& w_H(\eta \left(z\right)-\eta \left(z^{{}^{\prime }}\right))\hfill \\ & =& d_H(\eta \left(z\right),\eta \left(z^{{}^{\prime }}\right)).\hfill \end{array}$$

Therefore, η is a distance-preserving map. □

Define ${\Theta }_1\otimes {\Theta }_2\otimes {\Theta }_3\otimes {\Theta }_4=\left\{({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4)\right|{\theta }_i\in {\Theta }_i:1\le i\le 4\}$ and ${\Theta }_1\oplus {\Theta }_2\oplus {\Theta }_3\oplus {\Theta }_4=\left\{({\theta }_1+{\theta }_2+{\theta }_3+{\theta }_4)\right|{\theta }_i\in {\Theta }_i:1\le i\le 4\}$. Let C be a linear code of length n over R. We define that

$${\mathcal{C}}_1=\{z_1\in {\mathbb{F}}_q^n|{\Delta }_1{z}_1+{\Delta }_2{z}_2+{\Delta }_3{z}_3+{\Delta }_4{z}_4\in \mathcal{C},where{z}_2,z_3,z_4\in {\mathbb{F}}_q^n\},$$

$${\mathcal{C}}_2=\{z_2\in {\mathbb{F}}_q^n|{\Delta }_1{z}_1+{\Delta }_2{z}_2+{\Delta }_3{z}_3+{\Delta }_4{z}_4\in \mathcal{C},where{z}_1,z_3,z_4\in {\mathbb{F}}_q^n\},$$

$${\mathcal{C}}_3=\{z_3\in {\mathbb{F}}_q^n|{\Delta }_1{z}_1+{\Delta }_2{z}_2+{\Delta }_3{z}_3+{\Delta }_4{z}_4\in \mathcal{C},where{z}_1,z_2,z_4\in {\mathbb{F}}_q^n\},$$

and

$${\mathcal{C}}_4=\{z_4\in {\mathbb{F}}_q^n|{\Delta }_1{z}_1+{\Delta }_2{z}_2+{\Delta }_3{z}_3+{\Delta }_4{z}_4\in \mathcal{C},where{z}_1,z_2,z_3\in {\mathbb{F}}_q^n\}.$$

Now, each ${\mathcal{C}}_i$ is a linear code of length n over ${\mathbb{F}}_q$ for $1\le i\le 4$. Hence, any linear code of length n can be represented as $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ and $\left|\mathcal{C}\right|=|{\mathcal{C}}_1\left|\right|{\mathcal{C}}_2\left|\right|{\mathcal{C}}_3\left|\right|{\mathcal{C}}_4|$ over R. A matrix is called a generator matrix of $\mathcal{C}$ if the rows of the matrix generate $\mathcal{C}$. If $M_i$ are the generator matrices of the linear code ${\mathcal{C}}_i$, for $i=1,2,3,4$, respectively, then a generator matrix of $\mathcal{C}$ is

$$M=\left(\begin{array}{c}{\Delta }_1{M}_1\\ {\Delta }_2{M}_2\\ {\Delta }_3{M}_3\\ {\Delta }_4{M}_4\end{array}\right)$$

and a generator matrix of $\eta \left(\mathcal{C}\right)$ is

$$\eta \left(M\right)=\left(\begin{array}{c}\eta \left({\Delta }_1{M}_1\right)\\ \eta \left({\Delta }_2{M}_2\right)\\ \eta \left({\Delta }_3{M}_3\right)\\ \eta \left({\Delta }_4{M}_4\right)\end{array}\right).$$

Proposition 2.

Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a linear code of length n over R. Then, $\eta \left(\mathcal{C}\right)$ is a $[4n,{\displaystyle \sum _{i=1}^4}{k}_i,d]$ linear code over ${\mathbb{F}}_q$ for $1\le i\le 4$, where each ${\mathcal{C}}_i$ is an $[n,k_i,d]$ code.

Proof. 

The proof is obvious with the help of the Gray map. □

Proposition 3.

If $\mathcal{C}$ is a linear code of length n over R, then $\eta \left(\mathcal{C}\right)={\mathcal{C}}_1\otimes {\mathcal{C}}_2\otimes {\mathcal{C}}_3\otimes {\mathcal{C}}_4$.

Proof. 

The proof is similar to the one in [14]. □

Theorem 1.

Let $\mathcal{C}$ be a self-orthogonal linear code of length n over R and A be a $4\times 4$ non-singular matrix over ${\mathbb{F}}_q$ which has the property $A{A}^T=ϵI_4$, where $I_4$ is the identity matrix, $0\ne ϵ\in {\mathbb{F}}_q$, and $A^T$ is the transpose of matrix A. Then, the Gray image $\eta \left(\mathcal{C}\right)$ is a self-orthogonal linear code of length $4n$ over ${\mathbb{F}}_q$.

Proof. 

Suppose $\mathcal{C}$ is a self-orthogonal linear code of length n over R, i.e., $\mathcal{C}\subseteq {\mathcal{C}}^{\perp }$ and let $P,Q\in \eta \left(\mathcal{C}\right)$ such that $P=\eta \left(p\right)=(p_{0}A,p_{1}A,\dots ,p_{n-1}A)$ and $Q=\eta \left(q\right)=(q_{0}A,q_{1}A,$.$\dots ,q_{n-1}A)$. We have to show that $\eta \left(\mathcal{C}\right)$ is self-orthogonal, that is, $P·Q=0$. Since $\mathcal{C}$ is self-orthogonal, $p·q={\displaystyle \sum _{j=0}^{n-1}}{p}_j.q_j=0$. Therefore, $P·Q=P{Q}^{\perp }={\displaystyle \sum _{j=0}^{n-1}}{p}_{j}A{A}^T{q}_j^{\perp }=m{\displaystyle \sum _{j=0}^{n-1}}{p}_j.q_j=0$. Suppose P and Q are arbitrary, then $\eta \left(\mathcal{C}\right)\subseteq \eta \left({\mathcal{C}}^{\perp }\right)$. Thus, $\eta \left(\mathcal{C}\right)$ is a self-orthogonal linear code of length $4n$ over ${\mathbb{F}}_q.$ □

3. Structural Properties of Cyclic Codes over R

On ring R, as described, we shall explore various structural properties of cyclic codes and prove some results. We begin with the following definition:

Definition 1.

A linear code $\mathcal{C}$ of length n over R is said to be a cyclic code if every cyclic shift of a codeword c in $\mathcal{C}$ is again a codeword in $\mathcal{C}$, i.e., if $c=(c_0,c_1,c_2,\dots ,c_{n-1})\in \mathcal{C}$, then its cyclic shift $\zeta \left(c\right)=(c_{n-1},c_0,\dots ,c_{n-2})\in \mathcal{C}$, where the operator ζ is known as cyclic shift.

Theorem 2.

Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a linear code of length n over R. Then, $\mathcal{C}$ is a cyclic code over R if, and only if, each ${\mathcal{C}}_i$ is a cyclic code over ${\mathbb{F}}_q$, where $1\le i\le 4$.

Proof. 

Suppose s is any codeword in $\mathcal{C}$ such that $s=(s_0,s_1,\dots ,s_{n-1})$. We can write its components as $s_i={\Delta }_1{z}_{1,i}+{\Delta }_2{z}_{2,i}+{\Delta }_3{z}_{3,i}+{\Delta }_4{z}_{4,i}$, where $z_{1,i},z_{2,i},z_{3,i},z_{4,i}\in {\mathbb{F}}_q$ and $1\le i\le n-1$. Let

$$z_1=(z_{0,1},z_{1,1},\dots ,z_{n-1,1}),$$

$$z_2=(z_{0,2},z_{1,2},\dots ,z_{n-1,2}),$$

$$z_3=(z_{0,3},z_{1,3},\dots ,z_{n-1,3}),$$

$$z_4=(z_{0,4},z_{1,4},\dots ,z_{n-1,4}),$$

where $z_i\in {\mathcal{C}}_i$ and $1\le i\le 4$. Now, let us assume that every ${\mathcal{C}}_i$ is a cyclic code over ${\mathbb{F}}_q$, where $1\le i\le 4$. This implies that

$$\zeta \left(z_1\right)=(z_{n-1,1},z_{0,1},\dots ,z_{n-2,1})\in {\mathcal{C}}_1,$$

$$\zeta \left(z_2\right)=(z_{n-1,2},z_{0,2},\dots ,z_{n-2,2})\in {\mathcal{C}}_2,$$

$$\zeta \left(z_3\right)=(z_{n-1,3},z_{0,3},\dots ,z_{n-2,3})\in {\mathcal{C}}_3,$$

$$\zeta \left(z_4\right)=(z_{n-1,4},z_{0,4},\dots ,z_{n-2,4})\in {\mathcal{C}}_4,$$

Thus, ${\Delta }_1\zeta \left(z_1\right)+{\Delta }_2\zeta \left(z_2\right)+{\Delta }_3\zeta \left(z_3\right)+{\Delta }_4\zeta \left(z_4\right)\in \mathcal{C}$. It can easily be seen that ${\Delta }_1\zeta \left(z_1\right)+{\Delta }_2\zeta \left(z_2\right)+{\Delta }_3\zeta \left(z_3\right)+{\Delta }_4\zeta \left(z_4\right)=\zeta \left(s\right)$. Hence, $\zeta \left(s\right)\in \mathcal{C}.$ We can conclude that $\mathcal{C}$ is a cyclic code over R.

On the other hand, let us assume that $\mathcal{C}$ is a cyclic code over R. Next, let us consider $s_i={\Delta }_1{z}_{1,i}+{\Delta }_2{z}_{2,i}+{\Delta }_3{z}_{3,i}+{\Delta }_4{z}_{4,i}$, where $z_1=(z_{0,1},z_{1,1},\dots ,z_{n-1,1})$, $z_2=(z_{0,2},z_{1,2},$.$\dots ,z_{n-1,2})$, $z_3=(z_{0,3},z_{1,3},\dots ,z_{n-1,3})$, and $z_4=(z_{0,4},z_{1,4},\dots ,z_{n-1,4})$. Then, $z_1\in {\mathcal{C}}_1,z_2\in {\mathcal{C}}_2,z_3\in {\mathcal{C}}_3,and{z}_4\in {\mathcal{C}}_4$. Again, $s=(s_0,s_1,\dots ,s_{n-1})\in \mathcal{C}$, and by this hypothesis $\zeta \left(s\right)\in \mathcal{C}$. We have ${\Delta }_1\zeta \left(z_1\right)+{\Delta }_2\zeta \left(z_2\right)+{\Delta }_3\zeta \left(z_3\right)+{\Delta }_4\zeta \left(z_4\right)\in \mathcal{C}$. Here, $\zeta \left(z_i\right)\in {\mathcal{C}}_i$, where $1\le i\le 4$. Consequently, every ${\mathcal{C}}_i$ is a cyclic code of length n over ${\mathbb{F}}_q$, where $1\le i\le 4$. □

Theorem 3.

Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a cyclic code of length n over R and $h_i\left(z\right)$ be the standard generator polynomial of ${\mathcal{C}}_i$. Then, $\mathcal{C}=\langle h\left(z\right)\rangle$ and $\left|\mathcal{C}\right|=q^{4n-{\displaystyle \sum _{i=0}^4}{h}_i\left(z\right)}$, where $h\left(z\right)={\Delta }_1{h}_1\left(z\right)+{\Delta }_2{h}_2\left(z\right)+{\Delta }_3{h}_3\left(z\right)+{\Delta }_4{h}_4\left(z\right)$ and $1\le i\le 4$.

Proof. 

Given ${\mathcal{C}}_i=\langle h_i\left(z\right)\rangle$, where $1\le i\le 4$ and $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$. Let $c\in \mathcal{C}$ be such that $c=\left\{c\left(z\right)\right|{\Delta }_1{h}_1\left(z\right)+{\Delta }_2{h}_2\left(z\right)+{\Delta }_3{h}_3\left(z\right)+{\Delta }_4{h}_4\left(z\right)$ for $h_i\left(z\right)\in {\mathcal{C}}_i\}$. Therefore, $\mathcal{C}\subseteq \langle {\Delta }_1{h}_1\left(z\right),{\Delta }_2{h}_2\left(z\right),{\Delta }_3{h}_3\left(z\right),{\Delta }_4{h}_4\left(z\right)\rangle \subseteq R\left[z\right]/\langle z^n-1\rangle$. For any ${\Delta }_1{t}_1\left(z\right)h_1\left(z\right)+{\Delta }_2{t}_2\left(z\right)h_2\left(z\right)+{\Delta }_3{t}_3\left(z\right)h_3\left(z\right)+{\Delta }_4{t}_4\left(z\right)h_4\left(z\right)\in \langle {\Delta }_1{h}_1\left(z\right)+{\Delta }_2{h}_2\left(z\right)+{\Delta }_3{h}_3\left(z\right)+{\Delta }_4{h}_4\left(z\right)\rangle \subseteq R\left[z\right]/\langle z^n-1\rangle$, where $t_1\left(z\right),t_2\left(z\right),t_3\left(z\right)$, and $t_4\left(z\right)\in R\left[z\right]/\langle z^n-1\rangle$, there exist $s_1\left(z\right),s_2\left(z\right)$, $s_3\left(z\right)$, and $s_4\left(z\right)\in {\mathbb{F}}_q\left[z\right]$ such that

$${\Delta }_i{t}_i\left(z\right)={\Delta }_i{s}_i\left(z\right),$$

where $1\le i\le 4$. Hence, $\langle {\Delta }_1{h}_1\left(z\right),{\Delta }_2{h}_2\left(z\right),{\Delta }_3{h}_3\left(z\right),{\Delta }_4{h}_4\left(z\right)\rangle \subseteq \mathcal{C}$. This implies $\langle {\Delta }_1{h}_1\left(z\right),{\Delta }_2{h}_2\left(z\right),{\Delta }_3{h}_3\left(z\right),{\Delta }_4{h}_4\left(z\right)\rangle =\mathcal{C}$. Since $\left|\mathcal{C}\right|=|{\mathcal{C}}_1\left|\right|{\mathcal{C}}_2\left|\right|{\mathcal{C}}_3\left|\right|{\mathcal{C}}_4|$, we have

$$\left|\mathcal{C}\right|=q^{4n-{\displaystyle \sum _{i=0}^4}{h}_i\left(z\right)}.$$

□

Theorem 4.

Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a cyclic code of length n over R. Then, there exists a unique monic polynomial $h\left(z\right)\in R\left[z\right]$ such that $\mathcal{C}=\langle h\left(z\right)\rangle$ and $h\left(z\right)$.divides $(z^n-1)$. If $h_i\left(z\right)$ is the standard generator polynomial of ${\mathcal{C}}_i$, $1\le i\le 4$, then $h\left(z\right)={\Delta }_1{h}_1\left(z\right)+{\Delta }_2{h}_2\left(z\right)+{\Delta }_3{h}_3\left(z\right)+{\Delta }_4{h}_4\left(z\right)$.

Proof. 

By Theorem 3, $\mathcal{C}=\langle {\Delta }_1{h}_1\left(z\right),{\Delta }_2{h}_2\left(z\right),{\Delta }_3{h}_3\left(z\right),{\Delta }_4{h}_4\left(z\right)\rangle$, where $h_i\left(z\right)$ is the generator polynomial of ${\mathcal{C}}_i$ and $1\le i\le 4$. Let $h\left(z\right)={\Delta }_1{h}_1\left(z\right)+{\Delta }_2{h}_2\left(z\right)+{\Delta }_3{h}_3\left(z\right)+{\Delta }_4{h}_4\left(z\right)$. From here, $\langle h\left(z\right)\rangle \subseteq \mathcal{C}$. Now, ${\Delta }_i{h}_i\left(z\right)={\Delta }_{i}h\left(z\right)$ and $1\le i\le 4$, so $\mathcal{C}\subseteq \langle h\left(z\right)\rangle$, hence $\mathcal{C}=\langle h\left(z\right)\rangle$. Since $h_i\left(z\right)$ is a monic right divisor of $(z^n-1)$, there are $s_i\left(z\right)\in {\mathbb{F}}_q\left[z\right]/\langle z^n-1\rangle$, where $1\le i\le 4$, such that $z^n-1=s_1\left(z\right)h_1\left(z\right)=s_2\left(z\right)h_2\left(z\right)=s_3\left(z\right)h_3\left(z\right)=s_4\left(z\right)h_4\left(z\right)$. This shows that $z^n-1=[{\Delta }_1{s}_1\left(z\right)+{\Delta }_2{s}_2\left(z\right)+{\Delta }_3{s}_3\left(z\right)+{\Delta }_4{s}_4\left(z\right)]h\left(z\right)$, i.e., $h\left(z\right)|(z^n-1)$. Here, each $h_i\left(z\right)$ is unique, and hence $h\left(z\right)$ is unique. □

Theorem 5.

Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a cyclic code of length n over R. Then, ${\mathcal{C}}^{\perp }={\Delta }_1{\mathcal{C}}_1^{\perp }\oplus {\Delta }_2{\mathcal{C}}_2^{\perp }\oplus {\Delta }_3{\mathcal{C}}_3^{\perp }\oplus {\Delta }_4{\mathcal{C}}_4^{\perp }$ is also a cyclic code of length n over R.

Proof. 

${\mathcal{C}}^{\perp }$ is a cyclic code of length n over R, since $\mathcal{C}$ is a cyclic code of length n over R. Now, we will show that ${\mathcal{C}}^{\perp }={\Delta }_1{\mathcal{C}}_1^{\perp }\oplus {\Delta }_2{\mathcal{C}}_2^{\perp }\oplus {\Delta }_3{\mathcal{C}}_3^{\perp }\oplus {\Delta }_4{\mathcal{C}}_4^{\perp }$. Here, $\mathcal{C}$ is a cyclic code of of length n over R. This implies $\mathcal{C}$ is a linear code of length n over R. Let $T_1=\{t_1\in {\mathbb{F}}_q^n|\exists t_2,t_3,t_4$ such that ${\displaystyle \sum _{i=1}^4}{t}_i{\Delta }_i\in {\mathcal{C}}^{\perp }$., for $1\le i\le 4.$ hence, ${\mathcal{C}}^{\perp }$ is uniquely expressed as ${\mathbb{C}}^{\perp }={\oplus }_{i=1}^4{\Delta }_i{T}_i$. Therefore, $T_1\subseteq {\mathcal{C}}_1^{\perp }$. Conversely, let $q\in {\mathcal{C}}_1^{\perp }$. This implies $q·s_1=0\forall s_1\in {\mathcal{C}}_1$. Consider $y={\displaystyle \sum _{i=1}^4}{\Delta }_i{s}_i\in \mathcal{C}.$ Now, ${\Delta }_{1}q·y={\Delta }_1{s}_1·q=0$. This shows that ${\Delta }_{1}q\in {\mathcal{C}}_1^{\perp }$. From the specific expression of ${\mathcal{C}}^{\perp }$, we obtain $q\in T_1$. From here, ${\mathcal{C}}^{\perp }\subseteq T_1$. Therefore, ${\mathcal{C}}_1^{\perp }=T_1$. In the same manner, ${\mathcal{C}}_i^{\perp }=T_i$ for $1\le i\le 4.$ hence, ${\mathcal{C}}^{\perp }={\Delta }_1{\mathcal{C}}_1^{\perp }\oplus {\Delta }_2{\mathcal{C}}_2^{\perp }\oplus {\Delta }_3{\mathcal{C}}_3^{\perp }\oplus {\Delta }_4{\mathcal{C}}_4^{\perp }$. □

Lemma 1

([1]). Let $\mathcal{C}=\langle h\left(z\right)\rangle$ be a cyclic code of length n over ${\mathbb{F}}_q$. $h\left(z\right)$ be the generator polynomial of $\mathcal{C}$. Then ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ if, and only if,

$$z^n-1\equiv 0\left(modh\left(z\right)h^{*}\left(z\right)\right),$$

where the reciprocal polynomial of $h\left(z\right)$ is denoted by $h^{*}\left(z\right)$.

Theorem 6.

Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a cyclic code of length n over R and $\mathcal{C}=\langle h\left(z\right)\rangle =\langle {\displaystyle \sum _{i=1}^4}{\Delta }_i{h}_i\left(z\right)\rangle$, where $h_i\left(z\right)$ be the generator polynomial of ${\mathcal{C}}_i$. Then, ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ if, and only, if

$$z^n-1\equiv 0\left(mod{h}_i\left(z\right)h_i^{*}\left(z\right)\right),$$

where the reciprocal polynomial of $h_i\left(z\right)$ is denoted by $h_i^{*}\left(z\right)$ and $1\le i\le 4.$

Proof. 

Suppose $z^n-1\equiv 0\left(mod{h}_i\left(z\right)h_i^{*}\left(z\right)\right)$ for $1\le i\le 4$. Hence, by Lemma 1, we have ${\mathcal{C}}_i^{\perp }\subseteq {\mathcal{C}}_i$. From here, we can write ${\Delta }_i{\mathcal{C}}^{\perp }\subseteq {\Delta }_i{\mathcal{C}}_i$ for $1\le i\le 4$. Similarly, ${\mathcal{C}}^{\perp }={\displaystyle \sum _{i=0}^4}{\Delta }_i{C}_i^{\perp }\subseteq {\displaystyle \sum _{i=0}^4}{\Delta }_i{\mathcal{C}}_i=\mathcal{C}$. Conversely, assume ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ and ${\displaystyle \sum _{i=0}^4}{\Delta }_i{\mathcal{C}}_i^{\perp }\subseteq {\displaystyle \sum _{i=0}^4}{\Delta }_i{\mathcal{C}}_i$, but each ${\mathcal{C}}_i$ is a cyclic code over ${\mathbb{F}}_q$, such that ${\Delta }_i{\mathcal{C}}_i\equiv \mathcal{C}\left(mod{\Delta }_i\right)$. This implies that ${\mathcal{C}}_i^{\perp }\subseteq {\mathcal{C}}_i$, where $1\le i\le 4$. By Lemma 1, we obtain

$$z^n-1\equiv 0\left(mod{h}_i\left(z\right)h_i^{*}\left(z\right)\right),$$

where the reciprocal polynomial of $h_i\left(z\right)$ is denoted by $h_i^{*}\left(z\right)$ for $1\le i\le 4$. □

Corollary 1.

Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a cyclic code of length n over R Then, ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ if, and only if, ${\mathcal{C}}_i^{\perp }\subseteq {\mathcal{C}}_i$ and $1\le i\le 4$.

4. Quantum and LCD Codes

This section deals with the study of quantum and LCD codes over the ring R. We begin with the following definition: Let p be a prime and $q=p^m$ for a positive integer m. Let $H\left(\mathbb{C}\right)$ be a q-dimensional Hilbert space over the complex field $\mathbb{C}$. Then, the set of n-folded tensor products $H{\left(\mathbb{C}\right)}^n=\underset{n-times}{{\displaystyle \underbrace{H\otimes H\otimes \dots \otimes H}}}$ is also a $q^n$.dimensional Hilbert space.

Definition 2

([15]). A quantum code represented by ${\left[[n,k,d]\right]}_q$ is defined as a subspace of $H{\left(\mathbb{C}\right)}^n$ with dimension $q^k$ and minimum distance d. Moreover, we consider ${\left[[n,k,d]\right]}_q$ to be better than ${\left[[n^{{}^{\prime }},k^{{}^{\prime }},d^{{}^{\prime }}]\right]}_q$ if either or both of the following conditions hold:

(i) $d>d^{{}^{\prime }}$.whenever the code rate $\frac{k}{n}=\frac{k^{{}^{\prime }}}{n^{{}^{\prime }}}$.(larger distance).

(ii) $\frac{k}{n}>\frac{k^{{}^{\prime }}}{n^{{}^{\prime }}}$, whenever the distance $d=d^{{}^{\prime }}$.(larger code rate).

Lemma 2

([2]). (Theorem 3) (CSS Construction) Let ${\mathcal{C}}_1={[n,k_1,d_1]}_q$ and ${\mathcal{C}}_2={[n,k_2,d_2]}_q$ be two linear codes over GF(q) with ${\mathcal{C}}_2^{\perp }\subseteq {\mathcal{C}}_1$. Furthermore, let $d=min\{wgt\left(v\right):v\in ({\mathcal{C}}_1\setminus {\mathcal{C}}_2^{\perp })\cup ({\mathcal{C}}_2\setminus {\mathcal{C}}_1^{\perp })\}\ge min(d_1,d_2)$. Then, there exists a QEC code with the parameters ${\left[[n,k_1+k_2-n,d]\right]}_q$. In particular, if ${\mathcal{C}}_1^{\perp }\subseteq {\mathcal{C}}_1$, then there exists a QEC code with the parameters ${\left[[n,2k_1-n,d_1]\right]}_q$, where $d_1=min\{wgt\left(v\right):v\in ({\mathcal{C}}_1\setminus {\mathcal{C}}_1^{\perp })\}$.

Theorem 7.

Let $\mathcal{C}$ be a cyclic code of length n over R and let the parameters of its Gray image be $[4n,k,d_H]$. If ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$, then there exists a QECC ${\left[[4n,2k-4n,d_H]\right]}_q$ over ${\mathbb{F}}_q$.

Proof. 

Let us consider $\mathbf{x}=(x_0,x_1,\dots ,x_{n-1})\in \mathcal{C},\mathbf{y}=(y_0,y_1,\dots ,y_{n-1})\in {\mathcal{C}}^{\perp },$ where $x_i=a_i+u_1{b}_i+u_2{c}_i+u_1{u}_2{d}_i$ and $y_i=a_i^{{}^{\prime }}+u_1{b}_i^{{}^{\prime }}+u_2{c}_i^{{}^{\prime }}+u_1{u}_2{d}_i^{{}^{\prime }}$, $a_i,b_i,c_i,d_i,a_i^{{}^{\prime }},b_i^{{}^{\prime }},c_i^{{}^{\prime }},d_i^{{}^{\prime }}\in {\mathbb{F}}_q$ for $0\le i\le n-1$. Since $\mathbf{x}·\mathbf{y}=0$. This gives

$$\begin{array}{ccc}\hfill \sum _{i=0}^{n-1}(a_i+u_1{b}_i+u_2{c}_i+u_1{u}_2{d}_i)(a_i^{{}^{\prime }}+u_1{b}_i^{{}^{\prime }}+u_2{c}_i^{{}^{\prime }}+u_1{u}_2{d}_i^{{}^{\prime }})& =& 0\hfill \\ \hfill \sum _{i=0}^{n-1}(a_i{a}_i^{{}^{\prime }}+u_1{a}_i{b}_i^{{}^{\prime }}+u_2{a}_i{c}_i^{{}^{\prime }}+u_1{u}_2{a}_i{d}_i^{{}^{\prime }}+u_1{b}_i{a}_i^{{}^{\prime }}+{\alpha }^2{b}_i{b}_i^{{}^{\prime }}+u_1{u}_2{b}_i{c}_i^{{}^{\prime }}+{\alpha }^2{u}_2{b}_i{d}_i^{{}^{\prime }}+u_2{c}_i{a}_i^{{}^{\prime }}+& & \\ \hfill u_1{u}_2{c}_i{b}_i^{{}^{\prime }}+{\beta }^2{c}_i{c}_i^{{}^{\prime }}+{\beta }^2{u}_1{c}_i{d}_i^{{}^{\prime }}+u_1{u}_2{d}_i{a}_i^{{}^{\prime }}+{\alpha }^2{u}_2{d}_i{b}_i^{{}^{\prime }}+{\beta }^2{u}_1{d}_i{c}_i^{{}^{\prime }}+{\alpha }^2{\beta }^2{d}_i{d}_i^{{}^{\prime }})& =& 0\hfill \\ \hfill \sum _{i=0}^{n-1}[(a_i{a}_i^{{}^{\prime }}+{\alpha }^2{b}_i{b}_i^{{}^{\prime }}+{\beta }^2{c}_i{c}_i^{{}^{\prime }}+{\alpha }^2{\beta }^2{d}_i{d}_i^{{}^{\prime }})+u_1(a_i{b}_i^{{}^{\prime }}+b_i{a}_i^{{}^{\prime }}+{\beta }^2{c}_i{d}_i^{{}^{\prime }}+{\beta }^2{d}_i{c}_i^{{}^{\prime }})+& & \\ \hfill u_2(a_i{c}_i^{{}^{\prime }}+{\alpha }^2{b}_i{d}_i^{{}^{\prime }}+c_i{a}_i^{{}^{\prime }}+{\alpha }^2{d}_i{b}_i^{{}^{\prime }})+u_1{u}_2(a_i{d}_i^{{}^{\prime }}+b_i{c}_i^{{}^{\prime }}+c_i{b}_i^{{}^{\prime }}+d_i{a}_i^{{}^{\prime }})]& =& 0.\hfill \end{array}$$

The above relation yields

$$\begin{array}{ccc}\hfill \sum _{i=0}^{n-1}(a_i{a}_i^{{}^{\prime }}+{\alpha }^2{b}_i{b}_i^{{}^{\prime }}+{\beta }^2{c}_i{c}_i^{{}^{\prime }}+{\alpha }^2{\beta }^2{d}_i{d}_i^{{}^{\prime }})& =& 0\hfill \\ \hfill \sum _{i=0}^{n-1}(a_i{b}_i^{{}^{\prime }}+b_i{a}_i^{{}^{\prime }}+{\beta }^2{c}_i{d}_i^{{}^{\prime }}+{\beta }^2{d}_i{c}_i^{{}^{\prime }})& =& 0\hfill \\ \hfill \sum _{i=0}^{n-1}(a_i{c}_i^{{}^{\prime }}+{\alpha }^2{b}_i{d}_i^{{}^{\prime }}+c_i{a}_i^{{}^{\prime }}+{\alpha }^2{d}_i{b}_i^{{}^{\prime }})& =& 0\hfill \\ \hfill \sum _{i=0}^{n-1}(a_i{d}_i^{{}^{\prime }}+b_i{c}_i^{{}^{\prime }}+c_i{b}_i^{{}^{\prime }}+d_i{a}_i^{{}^{\prime }})& =& 0.\hfill \end{array}$$

Additionally, $\eta \left(\mathbf{x}\right)·\eta \left(\mathbf{y}\right)=0.$.Therefore, $\eta \left({\mathcal{C}}^{\perp }\right)\subseteq \eta {\left(\mathcal{C}\right)}^{\perp }$. Since $\eta$ is bijective, $|\eta \left({\mathcal{C}}^{\perp }\right)|=|\eta {\left(\mathcal{C}\right)}^{\perp }|$. Hence, $\eta \left({\mathcal{C}}^{\perp }\right)=\eta {\left(\mathcal{C}\right)}^{\perp }$. Moreover, ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ implies $\eta {\left(\mathcal{C}\right)}^{\perp }\subseteq \eta \left(\mathcal{C}\right)$. Hence, $\eta \left(\mathcal{C}\right)$ is a dual-containing linear code with parameters $[4n,k,d_H]$. Thus by Lemma 2, there exists a quantum-error-correcting code with the parameters ${\left[[4n,2k-4n,d_H]\right]}_q$ over ${\mathbb{F}}_q$. □

Definition 3

([16]). A linear code $\mathcal{C}$ of length n over R is said to be linear complementary dual (LCD) if $\mathcal{C}\cap {\mathcal{C}}^{\perp }=\left\{0\right\}.$

Lemma 3

([17]). Let $\mathcal{C}$ be a cyclic code of length n over ${\mathbb{F}}_q$ generated by a polynomial $h\left(z\right)$ such that $n=p^{k_1}t$, where p and t are relatively prime and $k_1\ge 0$. Then, $\mathcal{C}$ is an LCD code if, and only if, $h\left(z\right)$ is a self-reciprocal and all the monic irreducible factors of $h\left(z\right)$ have the same multiplicity in $h\left(z\right)$ and in $z^n-1$.

Definition 4.

A linear code $\mathcal{C}$ of length n over R is said to be reversible if $(c_{n-1},c_{n-2},\dots ,c_1,c_0)\in \mathcal{C}$, for all $(c_0,c_1,c_2,\dots ,c_{n-1})\in \mathcal{C}$.

Lemma 4

([17]). Let $\mathcal{C}$ be a cyclic code of length n over ${\mathbb{F}}_q$ such that $gcd(n,p)=1$. Then, $\mathcal{C}$ is a reversible code if, and only if, $\mathcal{C}$ is an LCD code.

The proofs of Theorems 8–10, Corollaries 2 and 3, and Lemma 5 are similar to those in [18].

Theorem 8.

Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a cyclic code of length n over R. Then, $\mathcal{C}$ is an LCD code if, and only if, ${\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3$, and ${\mathcal{C}}_4$ are LCD codes of length n over ${\mathbb{F}}_q$.

Corollary 2.

Let $n=p^{t}m$ and $gcd(m,p)=1$. Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a cyclic code of length n over R, where ${\mathcal{C}}_i=\langle h_i\left(z\right)\rangle$ such that $h_i\left(z\right)\in {\mathbb{F}}_q$ and $h_i\left(z\right)|(z^n-1)$ for $i=1,2,3,4$. Then, $\mathcal{C}$ is an LCD code if, and only if, $h_i\left(z\right)$ is self-reciprocal and each monic irreducible factor of $h_i\left(z\right)$ has the same multiplicity in $h_i\left(z\right)$ and in $z^n-1$ for $i=1,2,3,4$.

Theorem 9.

Let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a cyclic code of length n over Rwith $gcd(n,p)=1$. Then, $\mathcal{C}$ is an LCD code if, and only if, ${\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3$, and ${\mathcal{C}}_4$ are reversible codes of length n over ${\mathbb{F}}_q$.

Corollary 3.

For $gcd(n,p)=1$, let $\mathcal{C}={\Delta }_1{\mathcal{C}}_1\oplus {\Delta }_2{\mathcal{C}}_2\oplus {\Delta }_3{\mathcal{C}}_3\oplus {\Delta }_4{\mathcal{C}}_4$ be a cyclic code of length n over R, where ${\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3$, and ${\mathcal{C}}_4$ are cyclic codes of length n over ${\mathbb{F}}_q$. Then, $\mathcal{C}$ is an LCD code if, and only if, $h_i\left(z\right)$ is a self-reciprocal polynomial in ${\mathbb{F}}_q$ for $i=1,2,3,4$.

Lemma 5.

Let $\mathcal{C}$ be a linear code of length n over R. Then, $\eta (\mathcal{C}\cap {\mathcal{C}}^{\perp })=\eta \left(\mathcal{C}\right)\cap \eta {\left(\mathcal{C}\right)}^{\perp }.$

Theorem 10.

Let $\mathcal{C}$ be a linear code of length n over R Then, $\mathcal{C}$ is an LCD code if, and only if, its Gray image $\eta \left(\mathcal{C}\right)$ is an LCD code of length $4n$ over ${\mathbb{F}}_q$.

5. Applications

In this section, we present some applications of the results proven in the previous sections. Example 3 and Table 1 demonstrate that our results provide several quantum codes which are better than the existing quantum codes that have been reported [8,9,10,11,12,13]. Moreover, we obtained new quantum codes in Example 1 and in Table 2. All of the computations involved in these examples were accomplished by using the Magma computation system [19]. We begin our discussion with the following example:

Example 1.

Let $R={\mathbb{F}}_3[u_1,u_2]/\langle u_1^2-1,u_2^2-1,u_1{u}_2-u_2{u}_1\rangle$ be a finite commutative ring, $n=9$, and $\alpha =\beta =1$. Then,

$$z^9-1={(z+2)}^9\in {\mathbb{F}}_3\left[x\right].$$

Take

$$h_1\left(z\right)={(z+2)}^4$$

$$h_2\left(z\right)={(z+2)}^2$$

$$h_3\left(z\right)=(z+2)$$

$$h_4\left(z\right)=1$$

and

$$A=\left[\begin{array}{cccc}-1& 1& 1& 1\\ 1& 1& 1& -1\\ 1& -1& 1& 1\\ 1& 1& -1& 1\end{array}\right]$$

Here, matrix A satisfies the condition AA^T = I_4×4, where A∈GL₄$\left({\mathbb{F}}_3\right)$ and I_4×4 an identity matrix. The cyclic code $\mathcal{C}=\langle {\displaystyle \sum _{i=1}^4}{\Delta }_i{h}_i\left(z\right)\rangle$ is of length 9 over R and its Gray image is of length 36, dimension 29, and distance 3 over ${\mathbb{F}}_3$, i.e., [36, 29, 3]₃. Moreover

$$z^9-1\equiv 0\left(mod{h}_i\left(z\right)h_i^{*}\left(z\right)\right),$$

for 1 ≤ i ≤ 4. thus, ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ by Theorem 6. In view of Theorem 7, we conclude that there exists a quantum code [[36, 22, 3]]₃. This quantum code is a new quantum code (see [20] for details).

Example 2.

Let $R={\mathbb{F}}_{19}[u_1,u_2]/\langle u_1^2-4,u_2^2-1,u_1{u}_2-u_2{u}_1\rangle$ be a finite commutative ring, $n=19$, and $\alpha =2,\beta =1$. Then,

$$z^{19}-1={(z+18)}^{19}\in {\mathbb{F}}_{19}\left[x\right].$$

Take

$$h_1\left(z\right)=(z+18)$$

$$h_2\left(z\right)=(z+18)$$

$$h_3\left(z\right)={(z+18)}^2$$

$$h_4\left(z\right)={(z+18)}^{14}$$

and

$$A=\left[\begin{array}{cccc}-1& 1& 1& 1\\ 1& 1& 1& -1\\ 1& -1& 1& 1\\ 1& 1& -1& 1\end{array}\right]$$

Here, matrix A satisfies the condition AA^T = 5I_4×4, where A∈GL₄$\left({\mathbb{F}}_{19}\right)$ and I_4×4 is an identity matrix. The cyclic code $\mathcal{C}=\langle {\displaystyle \sum _{i=1}^4}{\Delta }_i{h}_i\left(z\right)\rangle$ is of length 19 over R and its Gray image is of length 76, dimension 58, and distance 4 over ${\mathbb{F}}_{19}$, i.e., [76, 58, 4]₁₉. Moreover

$$z^{19}-1\equiv 0\left(mod{h}_i\left(z\right)h_i^{*}\left(z\right)\right),$$

for 1 ≤ i ≤ 4. Application of Theorem 6 yields ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ by Theorem 7, we conclude that there exists a quantum code [[76, 40, 4]]₁₉.

Example 3.

Let $R={\mathbb{F}}_5[u_1,u_2]/\langle u_1^2-1,u_2^2-1,u_1{u}_2-u_2{u}_1\rangle$ be a finite commutative ring, $n=30$, and $\alpha =\beta =1$. Then,

$$z^{30}-1={(z+1)}^5{(z+4)}^5{(z^2+z+1)}^5{(z^2+4z+1)}^5\in {\mathbb{F}}_5\left[x\right].$$

Take

$$h_1\left(z\right)=h_2\left(z\right)=h_3\left(z\right)=h_4\left(z\right)={(z+1)}^2(z^2+z+1)$$

and

$$A=\left[\begin{array}{cccc}-1& 1& 1& 1\\ 1& 1& 1& -1\\ 1& -1& 1& 1\\ 1& 1& -1& 1\end{array}\right]$$

Here, matrix A satisfies the condition AA^T = 4I_4×4, where A∈GL₄$\left({\mathbb{F}}_5\right)$ and I_4×4 is an identity matrix. The cyclic code $\mathcal{C}=\langle {\displaystyle \sum _{i=1}^4}{\Delta }_i{h}_i\left(z\right)\rangle$ is of length 30 over R and its Gray image is of length 120, dimension 104, and distance 3 over ${\mathbb{F}}_5$, i.e., [120, 104, 3]₅. Moreover

$$z^{30}-1\equiv 0\left(mod{h}_i\left(z\right)h_i^{*}\left(z\right)\right),$$

for 1 ≤ i ≤ 4. This implies that ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ by Theorem 6. In view of Theorem 7, we conclude that there exists a quantum code [[120, 88, 3]]₅. which has the same minimum distance but a larger code rate than the best previously known quantum code [[120, 32, 3]]₅ (see [12] for details). Therefore, our quantum code [[120, 88, 3]]₅ is better than the best previously known quantum code [[120, 32, 3]]₅ reported in [12].

Example 4.

Let $R={\mathbb{F}}_5[u_1,u_2]/\langle u_1^2-1,u_2^2-4,u_1{u}_2-u_2{u}_1\rangle$ be a finite commutative ring, $n=6$, $\alpha =1,and\beta =2$. Then,

$$z^6-1=(z+1)(z+4)(z^2+z+1)(z^2+4z+1)\in {\mathbb{F}}_5\left[x\right].$$

Take

$$h_1\left(z\right)=1$$

$$h_2\left(z\right)=(z+1)$$

$$h_3\left(z\right)=(z+1)$$

$$h_4\left(z\right)=(z+1)(z^2+z+1)(z^2+4z+1)$$

and

$$A=\left[\begin{array}{cccc}-1& 1& 1& 1\\ 1& 1& 1& -1\\ 1& -1& 1& 1\\ 1& 1& -1& 1\end{array}\right]$$

Matrix A satisfies the condition AA^T = 4I_4×4, where A∈ GL₄ $\left({\mathbb{F}}_5\right)$ and I_4×4 is an identity matrix. Here, h₁(z), h₂(z), h₃(z), and h₄(z) are self-reciprocal polynomials. By Corollary 3, $\mathcal{C}=\langle {\displaystyle \sum _{i=1}^4}{\Delta }_i{h}_i\left(z\right)\rangle$ is an LCD code of length 6 over the ring R. Hence, by Theorem 10, its Gray image $\eta \left(\mathcal{C}\right)$ is also an LCD code with the parameters [24, 17, 4]₅ over ${\mathbb{F}}_5$.

In Table 1 we present QEC codes obtained from cyclic codes $\mathcal{C}=\langle {\displaystyle \sum _{i=0}^4}{\Delta }_i{h}_i\left(z\right)\rangle$ of length n over R, where ${\mathcal{C}}_i=\langle h_i\left(z\right)\rangle$, such that $z^n-1\equiv 0\left(mod{h}_i\left(z\right)h_i^{*}\left(z\right)\right)$ for $i=1,2,3,4$. It is noted that our QEC codes ${\left[[n,k,d]\right]}_q$ are better than the existing quantum codes ${\left[[n^{{}^{\prime }},k^{{}^{\prime }},d^{{}^{\prime }}]\right]}_q$ collected from the different references mentioned in this article. In Table 2, we obtain new quantum codes, and in Table 3 we construct LCD codes $\mathcal{C}=\langle {\displaystyle \sum _{i=0}^4}{\Delta }_i{h}_i\left(z\right)\rangle$ of length n over R, where $gcd(n,p)=1$, ${\mathcal{C}}_i=\langle h_i\left(z\right)\rangle$, and $h_i\left(z\right)$ is the self-reciprocal divisor of $z^n-1$ in ${\mathbb{F}}_q\left[z\right]$ for $i=1,2,3,4.$

6. Conclusions

In this article, we discuss some of the structural properties of cyclic codes over the ring $R={\mathbb{F}}_q[u_1,u_2]/\langle u_1^2-{\alpha }^2,u_2^2-{\beta }^2,u_1{u}_2-u_2{u}_1\rangle$, where α and β are non-zero elements of ${\mathbb{F}}_q$. Furthermore, we obtain better quantum codes than presented in [8,9,10,11,12,13]. As an application, we obtain LCD codes over the ring R. This study can be generalized to a product of finite rings. We hope that this study will encourage readers to investigate these codes over other finite rings to explore new and better quantum codes in the future.