Quantum Codes from Constacyclic Codes over the Ring F_q[u₁,u₂]/〈 u 1 2 -u₁, u 2 2 -u₂,u₁u₂-u₂u₁〉

Abstract

In this paper, we study the structural properties of $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic codes over $\mathcal{R}=F_q[u_1,u_2]/⟨u_1^2-u_1,u_2^2-u_2,u_1{u}_2-u_2{u}_1⟩$ where $q=p^m$ for odd prime p and $m\ge 1$. We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length $4n$. Also we have classified all possible self dual linear codes over this ring $\mathcal{R}$. We have given the applications by computing non-binary quantum codes over this ring $\mathcal{R}$.

1. Introduction

The class of constacyclic codes consists of an algebraically rich family of error-correcting codes and are the generalizations of cyclic and negacyclic codes. These codes can be easily encoded using shift registers and can be easily decoded due to their rich algebraic structure, which justify their preferred role from engineering perspective.

The study of cyclic codes over the semi-local ring $F_2+v{F}_2$, $v^2=v$ was carried by Zhu et al. [1], which was generalized to $F_3+v{F}_3$ with $v^2=1$ by Cengellenmis [2]. Later on, Zhu and Wang [3], considered $(1-2v)$-constacyclic codes over $F_p+v{F}_p$ with $v^2=v$ and proved that the image of a $(1-2v)$-constacyclic code of length n over $F_p+v{F}_p$ under the Gray map is a distance-invariant linear cyclic code of length $2n$ over $F_p$. Since then, a lot of research has been done on the study of constacyclic codes over semi-local rings.

Quantum error-correcting codes have a significant role in both quantum communication as well as in quantum computation. The investigations on quantum error-correcting codes has accomplished huge growth after it has been discovered that there exist a class of codes known as quantum error-correcting codes which preserve quantum information as classical error-correcting codes preserve classical information. Quantum error-correcting codes give an efficient technique to overcome decoherence. In [4], Shor obtained the first quantum error-correcting code. After that, a technique was given by Calderbank et al. in [5] to find quantum error-correcting codes from classical error-correcting codes. Gottesman [6] studied quantum error-correcting codes that saturate the quantum Hamming bound. Currently, the research on quantum error-correcting codes has been developed quickly. Using self-orthogonal or dual containing properties of classical cyclic codes over finite field $F_q$ (q is a power of prime number), many new quantum error-correcting codes have been obtained (for references see [7,8,9,10,11]). Further, Ouyang shown that ‘good codes’ can be attained from MDS codes using the construction given in [12].

Quantum codes over different rings is studied by many authors (for references see [9,13,14,15,16,17,18]). In these researches, researchers have used the orthogonal properties of cyclic codes to construct quantum codes over these ring. From the last few years computation of quantum codes has been going on using constacyclic codes over finite rings. Gao et al. [19] have studied non-binary quantum codes using u-constacyclic codes over $F_p+u{F}_p$ with $u^2=1$. Quantum MDS codes have been studied [7,8,11] using constacyclic codes and BCH like constacyclic codes.

Recently, Ashraf and Mohammad [20] computed quantum codes from cyclic codes over the ring $F_q[u_1,u_2]/⟨u_1^2-u_1,u_2^2-u_2,u_1{u}_2-u_2{u}_1⟩$ where $q=p^m$ for odd prime. In this paper we generalize their work over classes of constacyclic codes. Here we have studied the structural properties of $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic codes over the ring $\mathcal{R}=F_q[u_1,u_2]/⟨u_1^2-u_1,u_2^2-u_2,u_1{u}_2-u_2{u}_1⟩$ where $q=p^m$ for odd prime p and $m\ge 1$. As an application, we have constructed the quantum codes over $F_q$.

2. Preliminaries

Let $\mathcal{R}=F_q+u_1{F}_q+u_2{F}_q+u_1{u}_2{F}_q$ be the ring such that $u_1^2=u_1,u_2^2=u_2$ and $u_1{u}_2=u_{2}u-1$, here $F_q$ is the finite field with q elements, where $q=p^m$ for odd prime p and $m\ge 1$. It is commutative semi-local ring with four maximal ideals. This is a principal ideal non chain ring.

Let $\mathcal{C}$ be a linear code of length n over $\mathcal{R}$. Then, it is a $\mathcal{R}$-submodule of ${\mathcal{R}}^n$, where elements of $\mathcal{C}$ are called codeword and it is said to be $\lambda$-constacyclic code if and only if $\mathcal{C}$ is invariant under the $\lambda$-constacyclic shift operator ${\mathsf{\Psi }}_{\lambda }:{\mathcal{R}}^n\to {\mathcal{R}}^n$ defined by ${\mathsf{\Psi }}_{\lambda }(d_0,d_1,\cdots ,d_{n-1})=(\lambda d_{n-1},d_0,\cdots ,d_{n-2})$ which for $\lambda =1$ beomes the cyclic code. By identifying each codeword $d=(d_0,d_1,\cdots ,d_{n-1})\in {\mathcal{R}}^n$ to a polynomial $d\left(x\right)=d_0+d_{1}x+\cdots +d_{n-1}{x}^{n-1}$ in $\mathcal{R}\left[x\right]/⟨x^n-\lambda ⟩$, a linear code $\mathcal{C}$ is a $\lambda$-constacyclic code of length n over $\mathcal{R}$ if and only if it is an ideal of the ring $\mathcal{R}\left[x\right]/⟨x^n-\lambda ⟩$.

For any two elements $r=(r_0,r_1,\cdots ,r_{n-1})$, $s=(s_0,s_1,\cdots ,s_{n-1})\in {\mathcal{R}}^n$ the inner product is defined as $r·s={\sum }_{i=0}^{n-1}{r}_i{s}_i$. If $r·s=0$ then r and s are said to be orthogonal. For a linear code $\mathcal{C}$ of length n over $\mathcal{R}$, the dual code is defined by ${\mathcal{C}}^{\perp }=\left\{r\right|\forall s\in \mathcal{C},r·s=0\}$, which is also linear code of length n over $\mathcal{R}$. A code $\mathcal{C}$ is called self orthogonal if $\mathcal{C}\subseteq {\mathcal{C}}^{\perp }$ and self dual if ${\mathcal{C}}^{\perp }=\mathcal{C}$.

Any element of $\mathcal{R}$ can be expressed as

$$\begin{array}{cc}\hfill & a+u_{1}b+u_{2}c+u_1{u}_{2}d=\hfill \\ \hfill & (1-u_1-u_2+u_1{u}_2)a+(u_1-u_1{u}_2)(a+b)+(u_2-u_1{u}_2)(a+c)+u_1{u}_2(a+b+c+d)\hfill \end{array}$$

(1)

Let $e_1=1-u_1-u_2+u_1{u}_2,e_2=u_1-u_1{u}_2,e_3=u_2-u_1{u}_2$ and $e_4=u_1{u}_2$. Here $e_i^2=e_i,e_i{e}_j=0$ and ${\sum }_{i=1}^4{e}_i=1,$ where $i,j=1,2,3,4$ and $i\ne j.$ It is easy to see that $e_i\mathcal{R}\cong F_q,$ for $i=1,2,3,4.$ We can express $\mathcal{R}=e_1\mathcal{R}\oplus e_2\mathcal{R}\oplus e_3\mathcal{R}\oplus e_4\mathcal{R}=e_1{F}_q\oplus e_2{F}_q\oplus e_3{F}_q\oplus e_4{F}_q.$ Therefore, for any $r\in \mathcal{R}$ can be expressed uniquely as $r=e_{1}x+e_{2}y+e_{3}w+e_{4}z$, where $x,y,w,z\in F_q$ for $i=1,2,3,4.$

We define the Gray map as follows:

$$\psi :\mathcal{R}⟶F_q^4$$

$$\psi \left(r\right)=(x,y,w,z)$$

where $x,y,w,z\in F_q.$

This is a linear map and can be extended component-wise in following way:

$$\psi :{\mathcal{R}}^n⟶F_q^{4n}$$

$$r^{\prime }=(r_0,r_1,\cdots ,r_{n-1})⟶(x_0,x_1,\cdots ,x_{n-1},y_0,\cdots ,y_{n-1},w_0,\cdots ,w_{n-1},z_0,\cdots ,z_{n-1})$$

where $r_i=e_1{x}_i+e_2{y}_i+e_3{w}_i+e_4{z}_i$, and $x_i,y_i,w_i,z_i\in F_q$ for $i=0,1,\cdots ,n-1$.

For any element $r=e_{1}x+e_{2}y+e_{3}w+e_{4}z\in \mathcal{R},$ we define the Lee weight of r as $w_L\left(r\right)=w_H\left(\psi \left(r\right)\right),$ where $w_H\left(\psi \left(r\right)\right)$ denotes the Hamming weight of $\psi \left(r\right)$ over $F_q$, where the Hamming weight of any element is defined as the number of nonzero components. The Lee distance between $r_1$ and $r_2\in \mathcal{C}$ is defined by $d_L(r_1,r_2)=w_L(r_1-r_2)=w_H\left(\psi (r_1-r_2)\right).$ The Lee distance of $\mathcal{C}$ is defined as $d_L\left(\mathcal{C}\right)$ = min $\left\{d_L(r_1,r_2)\right|r_1\ne r_2\}$.

Let $B_i,i=1,2,3,4$ be code over $\mathcal{R}.$ We denote that $B_1\oplus B_2\oplus B_3\oplus B_4=\{b_1+b_2+b_3+b_4|b_i\in A_i,i=1,2,3,4\}$ and $B_1\otimes B_2\otimes B_3\otimes B_4=\left\{(b_1,b_2,b_3,b_4)\right|b_i\in B_i,i=1,2,3,4\}.$ For a linear code $\mathcal{C}$ of length n over $\mathcal{R},$ we define

$${\mathcal{C}}_1=\{x\in F_q^n|e_{1}x+e_{2}y+e_{3}w+e_{4}z\in \mathcal{C},andy,w,z\in F_q^n\},$$

$${\mathcal{C}}_2=\{y\in F_q^n|e_{1}x+e_{2}y+e_{3}w+e_{4}z\in \mathcal{C},andx,w,z\in F_q^n\},$$

$${\mathcal{C}}_3=\{w\in F_q^n|e_{1}x+e_{2}y+e_{3}w+e_{4}z\in \mathcal{C},andx,y,z\in F_q^n\},$$

$${\mathcal{C}}_4=\{z\in F_q^n|e_{1}x+e_{2}y+e_{3}w+e_{4}z\in \mathcal{C},andx,y,w\in F_q^n\}.$$

Here ${\mathcal{C}}_i$ are linear codes over $F_q^n$ for $i=1,2,3,4$. Then ${\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3$ and ${\mathcal{C}}_4$ are q-ary linear codes of length n. So a linear code $\mathcal{C}$ of length n over $\mathcal{R}$ can be uniquely expressed as $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$ and $\left|\mathcal{C}\right|=|{\mathcal{C}}_1\left|\right|{\mathcal{C}}_2\left|\right|{\mathcal{C}}_3\left|\right|{\mathcal{C}}_4|$ and $d_L\left(\mathcal{C}\right)=min\{d_H\left({\mathcal{C}}_i\right),i=1,2,3,4\}.$

A matrix is called generator matrix of $\mathcal{C}$ if the rows of the matrix generates $\mathcal{C}$. If $G_i$ are the generator matrices of q-ary linear codes ${\mathcal{C}}_i,i=1,2,3,4$ respectively, then the generator matrix of $\mathcal{C}$ is

$$G=\left(\begin{array}{c}{e}_1{G}_1\\ e_2{G}_2\\ e_3{G}_3\\ e_4{G}_4\end{array}\right)$$

and the generator matrix of $\psi \left(\mathcal{C}\right)$ is

$$\psi \left(G\right)=\left(\begin{array}{c}\psi \left(e_1{G}_1\right)\\ \psi \left(e_2{G}_2\right)\\ \psi \left(e_3{G}_3\right)\\ \psi \left(e_4{G}_4\right)\end{array}\right).$$

3. Gray Map and Linear Codes over the Ring $\mathcal{R}$

Proposition 1.

The Gray map ψ is a distance preserving $F_q$-linear map from ${\mathcal{R}}^n$ (Lee distance) to $F_q^{4n}$ (Hamming distance).

Proof. 

Let $r_1=x_1{e}_1+y_1{e}_2+w_1{e}_3+z_1{e}_4$ and $r_2=x_2{e}_1+y_2{e}_2+w_2{e}_3+z_2{e}_4\in {\mathcal{R}}^n$, where $x_1,y_1,w_1,z_1,x_2,y_2,w_2,z_2\in F_q^n$. Then

$$\psi (r_1+r_2)=(x_1+x_2,y_1+y_2,w_1+w_2,z_1+z_2)=(x_1,y_1,w_1,z_1)+(x_2,y_2,w_2,z_2)=\psi \left(r_1\right)+\psi \left(r_2\right).$$

Also, for any $r=e_{1}x+e_{2}y+e_{3}w+e_{4}z\in {\mathcal{R}}^n$ and $\alpha \in F_q$, we have

$$\psi \left(\alpha r\right)=\psi (\alpha e_{1}x+\alpha e_{2}y+\alpha e_{3}w+\alpha e_{4}z)=\alpha \psi \left(r\right)$$

So $\psi$ is $F_q$-linear. For the other part, $d_L(r_1,r_2)=w_L(r_1-r_2)=w_H\left(\psi (r_1-r_2)\right)=w_H(\psi \left(r_1\right)-\psi \left(r_2\right))=d_H(\psi \left(r_1\right),\psi \left(r_2\right))$. Therefore the Gray map $\psi$ is a distance preserving $F_q$-linear map. □

Proposition 2.

Let $\mathcal{C}$ be a $[n,k,d_L]$ linear code over $\mathcal{R}$. Then $\psi \left(\mathcal{C}\right)$ is a $[4n,k,d_H]$ linear code over $F_q^{4n}$, where $d_L=d_H$.

Proof. 

By the above result, $\psi$ is $F_q$-linear distance preserving map, therefore $d_L=d_H$. As $\psi$ is bijection therefore $\left|\mathcal{C}\right|=\left|\psi \left(\mathcal{C}\right)\right|=q^k$ also $\psi \left(\mathcal{C}\right)$ has length $4n$. Therefore, $\psi \left(\mathcal{C}\right)$ is a $[4n,k,d_H]$ linear code over $F_q^{4n}$, where $d_L=d_H$. □

Theorem 1.

If $\mathcal{C}$ is a linear code of length n over $\mathcal{R}$, then $\psi \left({\mathcal{C}}^{\perp }\right)=\psi {\left(\mathcal{C}\right)}^{\perp }$. Moreover, if $\mathcal{C}$ is self orthogonal, then $\psi \left(\mathcal{C}\right)$ is also self orthogonal.

Proof. 

Let $r_1=x_1{e}_1+y_1{e}_2+w_1{e}_3+z_1{e}_4\in \mathcal{C}$ and $r_2=x_2{e}_1+y_2{e}_2+w_2{e}_3+z_2{e}_4\in {\mathcal{C}}^{\perp }$. Then by inner product of $r_1$ and $r_2$, we have

$$r_1·r_2=e_1{x}_1{x}_2+e_2{y}_1{y}_2+e_3{w}_1{w}_2+e_4{z}_1{z}_2=0.$$

which implies, $x_1{x}_2=y_1{y}_2=w_1{w}_2=z_1{z}_2=0.$ Also

$$\psi \left(r_1\right)·\psi \left(r_2\right)=(x_1,y_1,w_1,z_1)·(x_2,y_2,w_2,z_2)=x_1{x}_2+y_1{y}_2+w_1{w}_2+z_1{z}_2=0$$

As $\psi \left(r_1\right)\in \psi \left(\mathcal{C}\right)$ so $\psi \left(r_2\right)\in \psi {\left(\mathcal{C}\right)}^{\perp }$. Thus, $\psi \left({\mathcal{C}}^{\perp }\right)\subseteq \psi {\left(\mathcal{C}\right)}^{\perp }$. Since $|\psi \left({\mathcal{C}}^{\perp }\right)|=|\psi {\left(\mathcal{C}\right)}^{\perp }|$, we get $\psi \left({\mathcal{C}}^{\perp }\right)=\psi {\left(\mathcal{C}\right)}^{\perp }$. Similarly we can show the other part. □

Proposition 3.

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

Proof. 

Since $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$ is a code of length n over R, where $C_i,i=1\dots 4$ are codes of length n over $F_q$, each $r\in C$ can be written as $r=e_{1}x+e_{2}y+e_{3}z+e_{4}w$, where $x\in C_1,y\in C_2,z\in C_3$ and $w\in C_4$. Now by definition of the Gray map, we have $\psi \left(r\right)=(x,y,z,w)\in {\mathcal{C}}_1\otimes {\mathcal{C}}_2\otimes {\mathcal{C}}_3\otimes {\mathcal{C}}_4$. Hence $\psi \left(\mathcal{C}\right)={\mathcal{C}}_1\otimes {\mathcal{C}}_2\otimes {\mathcal{C}}_3\otimes {\mathcal{C}}_4$. □

Proposition 4.

Let $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$ be linear code over $\mathcal{R}$ of length n. Then ${\mathcal{C}}^{\perp }=e_1{\mathcal{C}}_1^{\perp }\oplus e_2{\mathcal{C}}_2^{\perp }\oplus e_3{\mathcal{C}}_3^{\perp }\oplus e_4{\mathcal{C}}_4^{\perp }$ is also a linear code over $\mathcal{R}$ of length n.

4. Properties of $(\mathit{\alpha }+{\mathit{u}}_{\mathbf{1}}\mathit{\beta }+{\mathit{u}}_{\mathbf{2}}\mathit{\gamma }+{\mathit{u}}_{\mathbf{1}}{\mathit{u}}_{\mathbf{2}}\mathit{\delta })$t-Constacyclic Codes over $\mathcal{R}$

Theorem 2.

If $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$ is a unit in $\mathcal{R}$ and $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$ is a linear code of length n over $\mathcal{R}$, then $\mathcal{C}$ is a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$ if and only if ${\mathcal{C}}_1$ is α-constacyclic, ${\mathcal{C}}_2$ is $(\alpha +\beta )$-constacyclic, ${\mathcal{C}}_3$ is $(\alpha +\gamma )$-constacyclic and ${\mathcal{C}}_4$ is $(\alpha +\beta +\gamma +\delta )$-constacyclic codes of length n over $F_q$ respectively.

Proof. 

Suppose $\mathcal{C}$ is a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$ and $x=(x_0,x_1,\cdots ,x_{n-1})\in {\mathcal{C}}_1$, $y=(y_0,y_1,\cdots ,y_{n-1})\in {\mathcal{C}}_2$, $w=(w_0,w_1,\cdots ,w_{n-1})\in {\mathcal{C}}_3$ and $z=(z_0,z_1,\cdots ,z_{n-1})\in {\mathcal{C}}_4$ where $x_i,y_i,w_i,z_i\in F_q$ for $i=0,1\cdots ,n-1$. Let $d=(d_0,d_1,\cdots ,d_{n-1})\in \mathcal{C}$, where $d_i=e_1{x}_i+e_2{y}_i+e_3{w}_i+e_4{z}_i$ for $i=0,1\cdots ,n-1$. Since $\mathcal{C}$ is a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$, so

$${\mathsf{\Psi }}_{(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )}\left(c\right)=((\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )c_{n-1},c_0,\cdots ,c_{n-2})\in \mathcal{C}$$

As, $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )c_{n-1}=e_1\alpha x_{n-1}+e_2(\alpha +\beta )y_{n-1}+e_3(\alpha +\gamma )w_{n-1}+e_4(\alpha +\beta +\gamma +\delta )z_{n-1}$

Therefore,

$${\mathsf{\Psi }}_{(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )}\left(c\right)=e_1(\alpha x_{n-1},x_0,\cdots ,x_{n-2})+e_2((\alpha +\beta )y_{n-1},y_0,\cdots ,y_{n-2})$$

$$+e_3((\alpha +\gamma )w_{n-1},w_0,\cdots ,w_{n-2})+e_4((\alpha +\beta +\gamma +\delta )z_{n-1},z_0,\cdots ,z_{n-2})\in \mathcal{C}$$

Therefore,

$${\mathsf{\Psi }}_{\left(\alpha \right)}x=(\alpha x_{n-1},x_0,\cdots ,x_{n-2})\in {\mathcal{C}}_1$$

$${\mathsf{\Psi }}_{(\alpha +\beta )}y=((\alpha +\beta )y_{n-1},y_0,\cdots ,y_{n-2})\in {\mathcal{C}}_2$$

$${\mathsf{\Psi }}_{(\alpha +\gamma )}w=((\alpha +\gamma )w_{n-1},w_0,\cdots ,w_{n-2})\in {\mathcal{C}}_3$$

and

$${\mathsf{\Psi }}_{(\alpha +\beta +\gamma +\delta )}z=((\alpha +\beta +\gamma +\delta )z_{n-1},z_0,\cdots ,z_{n-2})\in {\mathcal{C}}_4$$

Hence ${\mathcal{C}}_1$ is $\alpha$-constacyclic, ${\mathcal{C}}_2$ is $(\alpha +\beta )$-constacyclic, ${\mathcal{C}}_3$ is $(\alpha +\gamma )$-constacyclic and ${\mathcal{C}}_4$ is $(\alpha +\beta +\gamma +\delta )$-constacyclic codes of length n over $F_q$ respectively.

Conversely, considering the above notations let ${\mathcal{C}}_1$ is $\alpha$-constacyclic, ${\mathcal{C}}_2$ is $(\alpha +\beta )$-constacyclic, ${\mathcal{C}}_3$ is $(\alpha +\gamma )$-constacyclic and ${\mathcal{C}}_4$ is $(\alpha +\beta +\gamma +\delta )$-constacyclic codes of length n over $F_q$. Therefore,

$${\mathsf{\Psi }}_{\left(\alpha \right)}x=(\alpha x_{n-1},x_0,\cdots ,x_{n-2})\in {\mathcal{C}}_1$$

$${\mathsf{\Psi }}_{(\alpha +\beta )}y=((\alpha +\beta )y_{n-1},y_0,\cdots ,y_{n-2})\in {\mathcal{C}}_2$$

$${\mathsf{\Psi }}_{(\alpha +\gamma )}w=((\alpha +\gamma )w_{n-1},w_0,\cdots ,w_{n-2})\in {\mathcal{C}}_3$$

and

$${\mathsf{\Psi }}_{(\alpha +\beta +\gamma +\delta )}z=((\alpha +\beta +\gamma +\delta )z_{n-1},z_0,\cdots ,z_{n-2})\in {\mathcal{C}}_4$$

Now as

$$e_1(\alpha x_{n-1},x_0,\cdots ,x_{n-2})+e_2((\alpha +\beta )y_{n-1},y_0,\cdots ,y_{n-2})$$

$$+e_3((\alpha +\gamma )w_{n-1},w_0,\cdots ,w_{n-2})+e_4((\alpha +\beta +\gamma +\delta )z_{n-1},z_0,\cdots ,z_{n-2})$$

$$=((\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )c_{n-1},c_0,\cdots ,c_{n-2})={\mathsf{\Psi }}_{(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )}\left(c\right)$$

Therefore, $\mathcal{C}$ is a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$. □

Definition 1.

Let $a\in F_q^{mn}$ with $a=\left(a^1\right|a^2|\cdots |a^{r-1}\left|a^m\right)$, where $a^i\in F_q^n$ for $i=1,\cdots ,m$. Let ${\eta }_m^{\lambda }$ be a map from $F_q^{mn}$ to $F_q^{mn}$ defined ${\eta }_m^{\lambda }\left(a\right)=({\mathsf{\Psi }}_{{\lambda }_1}\left(a^1\right)|\cdots |{\mathsf{\Psi }}_{{\lambda }_m}\left(a^m\right))$, where ${\mathsf{\Psi }}_{{\lambda }_i}$ is the ${\lambda }_i$-constacyclic shift from $F_q^n$ to $F_q^n$. A code of length $mn$ over $F_q$ is called a quasi constacyclic code of index m if ${\eta }_m^{\lambda }\left(\mathcal{C}\right)=\mathcal{C}.$

Theorem 3.

Let $\mathcal{C}$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$. Then $\psi \left(\mathcal{C}\right)$ is a quasi constacyclic code of index 4 and of length $4n$.

Proof. 

Let $\mathcal{C}$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$. Then by Theorem 2, ${\mathcal{C}}_1$ is $\alpha$-constacyclic, ${\mathcal{C}}_2$ is $(\alpha +\beta )$-constacyclic, ${\mathcal{C}}_3$ is $(\alpha +\gamma )$-constacyclic and ${\mathcal{C}}_4$ is $(\alpha +\beta +\gamma +\delta )$-constacyclic codes of length n over $F_q$. Then by the above definition, ${\mathcal{C}}_1\otimes {\mathcal{C}}_2\otimes {\mathcal{C}}_3\otimes {\mathcal{C}}_4$ is quasi constacyclic code of index 4 and of length $4n$. As by the Proposition 3, $\psi \left(\mathcal{C}\right)={\mathcal{C}}_1\otimes {\mathcal{C}}_2\otimes {\mathcal{C}}_3\otimes {\mathcal{C}}_4$. Hence, $\psi \left(\mathcal{C}\right)$ is a quasi constacyclic code of index 4 and of length $4n$. □

Corollary 1.

Let $\mathcal{C}$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$. Then its dual ${\mathcal{C}}^{\perp }=e_1{\mathcal{C}}_1^{\perp }\oplus e_2{\mathcal{C}}_2^{\perp }\oplus e_3{\mathcal{C}}_3^{\perp }\oplus e_4{\mathcal{C}}_4^{\perp }$ is a ${(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )}^{-1}$-constacyclic code of length n over $\mathcal{R}$ if and only if ${\mathcal{C}}_1^{\perp }$ is ${\alpha }^{-1}$-constacyclic, ${\mathcal{C}}_2^{\perp }$ is ${(\alpha +\beta )}^{-1}$-constacyclic, ${\mathcal{C}}_3^{\perp }$ is ${(\alpha +\gamma )}^{-1}$-constacyclic and ${\mathcal{C}}_4^{\perp }$ is ${(\alpha +\beta +\gamma +\delta )}^{-1}$-constacyclic codes of length n over $F_q$ respectively.

Proof. 

Using Proposition 4 and arguing similar like Theorem 2, we can prove this result. □

Corollary 2.

If $\mathcal{C}$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$, then $\mathcal{C}$ is self dual if and only if $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )=1,-1,1-2u_1,1-2u_2,1-2u_1{u}_2,-1+2u_1,-1+2u_2,-1+2u_1{u}_2,1-2u_1+2u_1{u}_2,1-2u_2+2u_1{u}_2,-1+2u_1-2u_1{u}_2,-1+2u_2-2u_1{u}_2,1-2u_1-2u_2+2u_1{u}_2,-1+2u_1+2u_2-2u_1{u}_2,1-2u_1-2u_2+4u_1{u}_2,-1+2u_1+2u_2-4u_1{u}_2.$

Proof. 

Proof follows from the fact that, $\alpha =\pm 1$, $(\alpha +\beta )=\pm 1$, $(\alpha +\gamma )=\pm 1$ and $(\alpha +\beta +\gamma +\delta )$ = ±1. □

Now using the values $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$ from Corollary 2, here we classify all possible types of linear codes over $F_q$ depending upon the values of units. Using the Theorem 2 and Corollary 2 we give the classification of codes over $F_q$ in Table below.

Each row of the Table 1, demonstrates the structure of codes for certain value of a unit. For example, if we take the last row of the above Table 1, then $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )=-1+2u_1+2u_2-4u_1{u}_2$. As in this case $\alpha =-1,\beta =2,\gamma =2,\delta =-4$. Then by Theorem 2, we have the following result.

Table 1. Behaviour of ${\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3$ and ${\mathcal{C}}_4$ depending on $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$.

$(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$	${\mathcal{C}}_1$	${\mathcal{C}}_2$	${\mathcal{C}}_3$	${\mathcal{C}}_4$
$(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$	${\mathcal{C}}_1$	${\mathcal{C}}_2$	${\mathcal{C}}_3$	${\mathcal{C}}_4$
1	Cyclic	Cyclic	Cyclic	Cyclic
−1	Negacyclic	Negacyclic	Negacyclic	Negacyclic
$1-2u_1$	Cyclic	Negacyclic	Cyclic	Cyclic
$-1+2u_1$	Negacyclic	Cyclic	Negacyclic	Cyclic
$1-2u_2$	Cyclic	Negacyclic	Cyclic	Negacyclic
$-1+2u_1$	Negacyclic	Cyclic	Negacyclic	Cyclic
$1-2u_1{u}_2$	Cyclic	Cyclic	Cyclic	Negacyclic
$-1+2u_1{u}_2$	Negacyclic	Negacyclic	Negacyclic	Cyclic
$1-2u_1+2u_1{u}_2$	Cyclic	Negacyclic	Cyclic	Cyclic
$-1+2u_1-2u_1{u}_2$	Negacyclic	Cyclic	Negacyclic	Negacyclic
$1-2u_2+2u_1{u}_2$	Cyclic	Cyclic	Negacyclic	Cyclic
$-1+2u_2-2u_1{u}_2$	Negacyclic	Negacyclic	Cyclic	Negacyclic
$1-2u_1-2u_2+2u_1{u}_2$	Cyclic	Negacyclic	Negacyclic	Negacyclic
$-1+2u_1+2u_2-2u_1{u}_2$	Negacyclic	Cyclic	Cyclic	Cyclic
$1-2u_1-2u_2+4u_1{u}_2$	Cyclic	Negacyclic	Negacyclic	Cyclic
$-1+2u_1+2u_2-4u_1{u}_2$	Negacyclic	Cyclic	Cyclic	Negacyclic

Proposition 5.

$\mathcal{C}$ is a $(-1+2u_1+2u_2-4u_1{u}_2)$-constacyclic code of length n over $\mathcal{R}$ if and only if ${\mathcal{C}}_1,{\mathcal{C}}_4$ are negacyclic and ${\mathcal{C}}_2,{\mathcal{C}}_3$ are cyclic codes of length n over the $F_q$.

Remark 1.

For each different values of $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$ from the first column of the Table 1, we can show similar type of results as in Proposition 5.

Proposition 6.

Let $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$. Then

 1 

$\mathcal{C}=⟨e_1{f}_1,e_2{f}_2,e_3{f}_3,e_4{f}_4⟩and\mid \mathcal{C}\mid =q^{4n-(deg\left(f_1\right)+deg\left(f_2\right)+deg\left(f_3\right)+deg\left(f_4\right))}$.

 2 

$\mathcal{C}=⟨f\left(x\right)⟩$ and $f\left(x\right)\left|\right(x^n-(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta ))$, such that $f\left(x\right)=e_1{f}_1\left(x\right)+e_2{f}_2\left(x\right)+e_3{f}_3\left(x\right)+e_4{f}_4\left(x\right).$

where $f_1,f_2,f_3$ and $f_4$ are the generator polynomials of ${\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3$ and ${\mathcal{C}}_4$ respectively.

Proof. 

$(1.)$ Let $\mathcal{C}$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$, then by Theorem 2 ${\mathcal{C}}_1$ is $\alpha$-constacyclic, ${\mathcal{C}}_2$ is $(\alpha +\beta )$-constacyclic, ${\mathcal{C}}_3$ is $(\alpha +\gamma )$-constacyclic code and ${\mathcal{C}}_4$ is $(\alpha +\beta +\gamma +\delta )$-constacyclic codes of length n over $F_q$ respectively. Then we can write, ${\mathcal{C}}_1=\left(f_1\left(x\right)\right)\subseteq F_q\left[x\right]/(x^n-\alpha )$, ${\mathcal{C}}_2=\left(f_2\left(x\right)\right)\subseteq F_q\left[x\right]/(x^n-(\alpha +\beta ))$, ${\mathcal{C}}_3=\left(f_3\left(x\right)\right)\subseteq F_q\left[x\right]/(x^n-(\alpha +\gamma ))$ and ${\mathcal{C}}_4=\left(f_4\left(x\right)\right)\subseteq F_q\left[x\right]/(x^n-(\alpha +\beta +\gamma +\delta ))$. Also as $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$, we can write $\mathcal{C}$ as,

$$\mathcal{C}=\left\{f\left(x\right)\right|f\left(x\right)=e_1{f}_1\left(x\right)+e_2{f}_2\left(x\right)+e_3{f}_3\left(x\right)+e_4{f}_4\left(x\right)where{f}_i\left(x\right)\in {\mathcal{C}}_{i}fori=1,2,3,4\}$$

This implies $\mathcal{C}\subseteq ⟨e_1{f}_1,e_2{f}_2,e_3{f}_3,e_4{f}_4⟩\subseteq \mathcal{R}\left[x\right]/(x^n-(\alpha +u\beta +v\gamma +uv\delta ))$.

On the other hand, let $e_1{f}_1\left(x\right)g_1\left(x\right)+e_2{f}_2\left(x\right)g_2\left(x\right)+e_3{f}_3\left(x\right)g_3\left(x\right)+e_4{f}_4\left(x\right)g_4\left(x\right)\in ⟨e_1{f}_1,e_2{f}_2,e_3{f}_3,e_4{f}_4⟩$, where $g_1\left(x\right),g_2\left(x\right),g_3\left(x\right)$ and $g_4\left(x\right)$ are elements of $\mathcal{R}\left[x\right]/(x^n-(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta ))$. Then there exists $k_1\left(x\right)\in F_q\left[x\right]/(x^n-\alpha )$, $k_2\left(x\right)\in F_q\left[x\right]/(x^n-(\alpha +\beta ))$, $k_3\left(x\right)\in F_q\left[x\right]/(x^n-(\alpha +\gamma ))$ and $k_4\left(x\right)\in F_q\left[x\right]/(x^n-(\alpha +\beta +\gamma +\delta ))$ such that $e_1{g}_1\left(x\right)=e_1{k}_1\left(x\right)$, $e_2{g}_2\left(x\right)=e_2{k}_2\left(x\right)$, $e_3{g}_3\left(x\right)=e_3{k}_3\left(x\right)$ and $e_4{g}_4\left(x\right)=e_4{k}_4\left(x\right)$. Therefore, $⟨e_1{f}_1,e_2{f}_2,e_3{f}_3,e_4{f}_4⟩\subseteq \mathcal{C}$. Hence, $\mathcal{C}=⟨e_1{f}_1,e_2{f}_2,e_3{f}_3,e_4{f}_4⟩$.

As $\mid \psi \left(\mathcal{C}\right)\mid =|{\mathcal{C}}_1\left|\right|{\mathcal{C}}_2\left|\right|{\mathcal{C}}_3\left|\right|{\mathcal{C}}_4|$ and $\left|\psi \right(\mathcal{C}\left)\right|=\mid \mathcal{C}\mid$, Therefore,

$$\mid \mathcal{C}\mid =|{\mathcal{C}}_1\left|\right|{\mathcal{C}}_2\left|\right|{\mathcal{C}}_3\left|\right|{\mathcal{C}}_4|=q^{n-deg\left(f_1\right)}{q}^{n-deg\left(f_2\right)}{q}^{n-deg\left(f_3\right)}{q}^{n-deg\left(f_4\right)}=q^{4n-\left({\sum }_{i=1}^{4}deg\left(f_i\right)\right)}$$

$(2.)$ Let $\mathcal{C}$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$ and $f_i$ be the generator polynomial of ${\mathcal{C}}_i$ for $i=1,2,3,4$. Then by the above result $\mathcal{C}=⟨e_1{f}_1,e_2{f}_2,e_3{f}_3,e_4{f}_4⟩.$ Suppose $\mathcal{C}=⟨e_1{f}_1+e_2{f}_2+e_3{f}_3+e_4{f}_4⟩$, then it is obvious that, $\mathcal{C}\subseteq \mathcal{C}$. As $e_i^2=e_i,e_i{e}_j=0$ where $i,j=1,2,3,4$ and $i\ne j$ then, $e_i(e_1{f}_1+e_2{f}_2+e_3{f}_3+e_4{f}_4)=e_i{f}_i$ for $i=1,2,3,4$, this implies $\mathcal{C}\subseteq \mathcal{C}$. Therefore $\mathcal{C}=\mathcal{C},$ and $\mathcal{C}=⟨f\left(x\right)⟩$, where $f\left(x\right)=e_1{f}_1+e_2{f}_2+e_3{f}_3+e_4{f}_4$.

Now suppose $f_i$ are the generator polynomials of ${\mathcal{C}}_i,$ for $i=1,2,3,4$. Then $f_1$ divides $x^n-\alpha$, $f_2$ divides $x^n-(\alpha +\beta )$, $f_3$ divides $x^n-(\alpha +\gamma )$ and $f_4$ divides $x^n-(\alpha +\beta +\gamma +\delta )$ such that, $x^n-\alpha =g_1{f}_1$, $x^n-(\alpha +\beta )=g_2{f}_2$, $x^n-(\alpha +\gamma )=g_3{f}_3$ and $x^n-(\alpha +\beta +\gamma +\delta )=g_4{f}_4$, for some $g_i\in {\mathcal{C}}_i$. Then

$$x^n-(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$$

$$=x^n-(e_1\alpha +e_2(\alpha +\beta )+e_3(\alpha +\gamma )+e_4(\alpha +\beta +\gamma +\delta ))$$

$$=(e_1{g}_1+e_2{g}_2+e_3{g}_3+e_4{g}_4)(e_1{f}_1+e_2{f}_2+e_3{f}_3+e_4{f}_4).$$

Therefore, $x^n-(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )=f\left(x\right)(e_1{g}_1+e_2{g}_2+e_3{g}_3+e_4{g}_4)$. Hence, $f\left(x\right)\left|\right(x^n-(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta ))$. □

Corollary 3.

Let $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$ and $f_i$ are the generator polynomials of ${\mathcal{C}}_i$ for $i=1,2,3,4$. Then

 1 

${\mathcal{C}}^{\perp }=⟨e_1{h}_1^{\ast }\left(x\right)+e_2{h}_2^{\ast }\left(x\right)+e_3{h}_3^{\ast }\left(x\right)+e_4{h}_4^{\ast }\left(x\right)⟩$ and $|{\mathcal{C}}^{\perp }|=q^{{\sum }_{i=1}^4\left(deg\left(f_i\left(x\right)\right)\right)}$.

 2 

${\mathcal{C}}^{\perp }=⟨h^{\ast }\left(x\right)⟩where{h}^{\ast }\left(x\right)=e_1{h}_1^{\ast }\left(x\right)+e_2{h}_2^{\ast }\left(x\right)+e_3{h}_3^{\ast }\left(x\right)+e_4{h}_4^{\ast }\left(x\right)$

where $h_i^{\ast }\left(x\right)$ are the reciprocal polynomials of $h_i\left(x\right)$ for $i=1,2,3,4$, such that $h_1{f}_1=x^n-\alpha$, $h_2{f}_2=x^n-(\alpha +\beta )$, $h_3{f}_3=x^n-(\alpha +\gamma )$, $h_4{f}_4=x^n-(\alpha +\beta +\gamma +\delta )$.

5. Quantum Codes from $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-Constacyclic Codes over $\mathcal{R}$

Quantum error-correcting codes (QECC) are a key ingredient to implement information processing based on quantum mechanics. For quantum systems composed of n subsystems of dimension $q\ge 2$, so-called qudits, a quantum code $\mathcal{C}={\left((N;K)\right)}_q$ is a k-dimensional subspace of the Hilbert space ${\left({\mathbb{C}}^q\right)}^{\otimes }n$. If the dimension of the code $\mathcal{C}$ is $q^k$, it will be denoted by $\mathcal{C}={\left[[n;k;d]\right]}_q$, where d is the minimum distance.

The following theorem is an important construction of quantum error-correcting codes which is used to construct quantum codes and is called CSS construction:

Theorem 4

([5]). (CSS Construction) Suppose $C_1$ and ${\mathcal{C}}_2$ are ${[n,k_1,d_1]}_q$ and ${[n,k_2,d_2]}_q$ linear codes over $GF\left(q\right)$ respectively such that ${\mathcal{C}}_2^{\perp }\subseteq {\mathcal{C}}_1$. Also, if $d=min\{d_1,d_2\}$, then there exists a quantum error-correcting code $\mathcal{C}$ with 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 quantum error-correcting code $\mathcal{C}$ with parameters ${\left[[n,2k_1-n,d_1]\right]}_q$.

Lemma 1

([5]). If $\mathcal{C}$ is a q-ary linear cyclic or negacyclic code with generator polynomial $f\left(x\right)$, then $\mathcal{C}$ contains its dual code if and only if

$$x^n-\kappa \equiv 0\left(modf{f}^{\ast }\right)$$

where $f^{\ast }$ is the reciprocal polynomial of f and $\kappa =\pm 1$

Theorem 5.

Let $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$. Then ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ if and only if $x^n-\alpha \equiv 0\left(mod{f}_1{f}_1^{\ast }\right)$, $x^n-(\alpha +\beta )\equiv 0\left(mod{f}_2{f}_2^{\ast }\right)$, $x^n-(\alpha +\gamma )\equiv 0\left(mod{f}_3{f}_3^{\ast }\right)$ and $x^n-(\alpha +\beta +\gamma +\delta )\equiv 0\left(mod{f}_4{f}_4^{\ast }\right)$, where $f_i^{\ast }$ is the reciprocal polynomial of $f_i$ for $i=1,2,3,4$ and $\alpha =\alpha +\beta =\alpha +\gamma =\alpha +\beta +\gamma +\delta =\pm 1.$

Proof. 

Let $x^n-\alpha \equiv 0\left(mod{f}_1{f}_1^{\ast }\right)$, $x^n-(\alpha +\beta )\equiv 0\left(mod{f}_2{f}_2^{\ast }\right)$, $x^n-(\alpha +\gamma )\equiv 0\left(mod{f}_3{f}_3^{\ast }\right)$ and $x^n-(\alpha +\beta +\gamma +\delta )\equiv 0\left(mod{f}_4{f}_4^{\ast }\right)$. Then by Lemma 1, we have ${\mathcal{C}}_i^{\perp }\subseteq {\mathcal{C}}_i,i=1,2,3,4.$ This implies that $e_i{\mathcal{C}}_i^{\perp }\subseteq e_i{\mathcal{C}}_i,i=1,2,3,4.$ Therefore, $e_1{\mathcal{C}}_1^{\perp }\oplus e_2{\mathcal{C}}_2^{\perp }\oplus e_3{\mathcal{C}}_3^{\perp }\oplus e_4{\mathcal{C}}_4^{\perp }\subseteq e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$. Hence, ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$.

Conversely, if ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$, then $e_1{\mathcal{C}}_1^{\perp }\oplus e_2{\mathcal{C}}_2^{\perp }\oplus e_3{\mathcal{C}}_3^{\perp }\oplus e_4{\mathcal{C}}_4^{\perp }\subseteq e_4{\mathcal{C}}_4\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4.$ Since ${\mathcal{C}}_i$ are the linear codes over $F_q$ such that $e_i{\mathcal{C}}_i=\mathcal{C}\left(mod{e}_i\right)$, for $i=1,2,3,4$. So ${\mathcal{C}}_i^{\perp }\subseteq {\mathcal{C}}_i,i=1,2,3,4$. Therefore, $x^n-\alpha \equiv 0\left(mod{f}_1{f}_1^{\ast }\right)$, $x^n-(\alpha +\beta )\equiv 0\left(mod{f}_2{f}_2^{\ast }\right)$, $x^n-(\alpha +\gamma )\equiv 0\left(mod{f}_3{f}_3^{\ast }\right)$ and $x^n-(\alpha +\beta +\gamma +\delta )\equiv 0\left(mod{f}_4{f}_4^{\ast }\right)$. □

Corollary 4.

Let $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$. Then ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ if and only if ${\mathcal{C}}_i^{\perp }\subseteq {\mathcal{C}}_i,i=1,2,3,4.$

Theorem 6.

Let $\mathcal{C}=e_1{\mathcal{C}}_1\oplus e_2{\mathcal{C}}_2\oplus e_3{\mathcal{C}}_3\oplus e_4{\mathcal{C}}_4$ be a $(\alpha +u_1\beta +u_2\gamma +u_1{u}_2\delta )$-constacyclic code of length n over $\mathcal{R}$. If ${\mathcal{C}}_i^{\perp }\subseteq {\mathcal{C}}_i,i=1,2,3,4$, then ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ and there exists a quantum error-correcting code with parameters ${\left[[4n,2k-4n,d_L]\right]}_q$, where $d_L$ denotes the minimum Lee weight of the code $\mathcal{C}$ and k denotes the dimension of the code $\psi \left(\mathcal{C}\right)$.

Proof. 

Let ${\mathcal{C}}_i^{\perp }\subseteq {\mathcal{C}}_i$ for $i=1,2,3,4$. Then by the Corollary 4, ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$. Now let $x\in \psi \left({\mathcal{C}}^{\perp }\right)=\psi {\left(\mathcal{C}\right)}^{\perp }$ then there exists $y\in {\mathcal{C}}^{\perp }$ such that $x=\psi \left(y\right)$ where $y·y^{\prime }=0$ for all $y^{\prime }\in \mathcal{C}$. As ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}$ and $y\in {\mathcal{C}}^{\perp }$ therefore $y\in \mathcal{C}$. Hence, $x=\psi \left(y\right)\in \psi \left(\mathcal{C}\right)$. Therefore, $\psi {\left(\mathcal{C}\right)}^{\perp }\subseteq \psi \left(\mathcal{C}\right)$. As $\psi \left(\mathcal{C}\right)$ is a $[4n,k,d_L]$ linear code over $F_q^{4n}$. Then by Theorem 4, there exists a quantum error-correcting code with parameters ${\left[[4n,2k-4n,d_L]\right]}_q$. □

6. Example

Example 1.

Let $\mathcal{R}=F_3[u_1,u_2]/⟨u_1^2-u_1,u_2^2-u_2,u_1{u}_2-u_2{u}_1⟩$ and $n=12.$ We have

$$x^{12}-1={(1+x)}^3{(2+x)}^3{(1+x^2)}^3\in F_3\left[x\right],$$

$$x^{12}+1={(2+x+x^2)}^3{(2+2x+x^2)}^3\in F_3\left[x\right]$$

Let $f_1\left(x\right)=f_2\left(x\right)=f_3\left(x\right)=(1+x^2)$ and $f_4\left(x\right)=(2+x+x^2)$. Then ${\mathcal{C}}_1={\mathcal{C}}_2={\mathcal{C}}_3=⟨1+x^2⟩$ are the cyclic codes over $F_3$ having the same parameters $[12,10,2]$ and ${\mathcal{C}}_4=⟨2+x+x^2⟩$ is a negacyclic codes over $F_3$ with parameters $[12,10,3]$. Thus,

$$\mathcal{C}=⟨e_1{f}_1\left(x\right),e_2{f}_2\left(x\right),e_3{f}_3\left(x\right),e_4{f}_4\left(x\right)⟩$$

is a $(1-2u_1{u}_2)$-constacyclic code of length 12 over $\mathcal{R}$. Since all $f_i\left(x\right)f_i^{\ast }\left(x\right)$ divide $x^{12}-1$ for $i=1,2,3$ and $f_4\left(x\right)f_4^{\ast }\left(x\right)$ divides $x^{12}+1$, ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}.$ Also, $\psi \left(\mathcal{C}\right)$ is a linear quasi constacyclic code over $F_3$ of index 4 with parameters $[48,40,2]$. Now, using Theorem 6, we get a quantum code with parameters ${\left[[48,32,2]\right]}_3.$

Example 2.

Let $\mathcal{R}=F_5[u_1,u_2]/⟨u_1^2-u_1,u_2^2-u_2,u_1{u}_2-u_2{u}_1⟩$ and $n=15.$ We have

$$x^{15}-1={(4+x)}^5{(1+x+x^2)}^5\in F_5\left[x\right],$$

$$x^{15}+1={(1+x)}^5{(1+4x+x^2)}^5\in F_5\left[x\right]$$

Let $f_1\left(x\right)=f_3\left(x\right)=f_4\left(x\right)=(1+x+x^2)$ and $f_2\left(x\right)=(1+4x+x^2)$. Then ${\mathcal{C}}_1={\mathcal{C}}_3={\mathcal{C}}_4=⟨1+x+x^2⟩$ are the cyclic codes over $F_5$ having the same parameters $[15,13,3]$ and ${\mathcal{C}}_2=⟨1+4x+x^2⟩$ is a negacyclic codes over $F_5$ with parameters $[15,13,3]$. Thus,

$$\mathcal{C}=⟨e_1{f}_1\left(x\right),e_2{f}_2\left(x\right),e_3{f}_3\left(x\right),e_4{f}_4\left(x\right)⟩$$

is a $(1-2u_1+2u_1{u}_2)$-constacyclic code of length 15 over $\mathcal{R}$. Since all $f_i\left(x\right)f_i^{\ast }\left(x\right)$ divide $x^{15}-1$ for $i=1,3,4$ and $f_2\left(x\right)f_2^{\ast }\left(x\right)$ divides $x^{15}+1$, ${\mathcal{C}}^{\perp }\subseteq \mathcal{C}.$ Also, $\psi \left(\mathcal{C}\right)$ is a linear quasi constacyclic code over $F_5$ of index 4 with parameters $[60,52,3]$. Now, using Theorem 6, we get a quantum code with parameters ${\left[[60,44,3]\right]}_5.$

Now, we find some new quantum error correcting codes over the field $F_5$ using the Gray images of $(1-2u_1{u}_2)$-constacyclic codes over the ring $\mathcal{R}=F_5[u_1,u_2]/⟨u_1^2-u_1,u_2^2-u_2,u_1{u}_2-u_2{u}_1⟩$. First column of the following table denotes the length of cyclic codes over $\mathcal{R}$, $f_i\left(x\right)$ are generator polynomials of codes ${\mathcal{C}}_i$ for $i=1,2,3,4$, parameters of the Gray images of $(1-2u_1{u}_2)$-constacyclic codes over $\mathcal{R}$ are represented by column four and the parameters of the corresponding quantum codes are denoted by the last column.

Comparison Compared to previously known quantum error-correcting codes, some of our quantum error-correcting codes are new, e.g., the codes ${\left[[60,52,2]\right]}_5$ and ${\left[[180,164,3]\right]}_5$ have better parameters than the known quantum codes ${\left[[60,48,2]\right]}_5$ and ${\left[[180,156,3]\right]}_5$, respectively in [21]. Also, the codes ${\left[[88,68,6]\right]}_5$ and ${\left[[208,180,9]\right]}_5$ have better parameters than the known quantum codes ${\left[[93,69,6]\right]}_5$ and ${\left[[208,178,9]\right]}_5$, respectively in (https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHTab5.html).

In [22], minimum Hamming distance is restricted to $d\le q+1$, but in our case we can find the codes that have distances greater than $q+1$, for example, the codes ${\left[[200,172,7]\right]}_5$ and ${\left[[208,180,9]\right]}_5$ have the minimum Hamming distances greater than 6.

7. Conclusions

In this paper, we have studied properties of constacyclic codes over the ring $F_q[u_1,u_2]/⟨u_1^2-u_1,u_2^2-u_2,u_1{u}_2-u_2{u}_1⟩$. We have classified all self dual linear codes over $F_q$ using self duality condition of the units. As an application, we have obtained non binary quantum codes from these classes of codes using self duality condition. For future work, it would be interesting to compute quantum codes from other classes of constacyclic codes given in Table 2.

Table 2. Parameters of Quantum Codes.

n	${\mathit{f}}_1\left(\mathit{x}\right)={\mathit{f}}_2\left(\mathit{x}\right)={\mathit{f}}_3\left(\mathit{x}\right)$	${\mathit{f}}_4\left(\mathit{x}\right)$	$\mathit{\psi }\left(\mathcal{C}\right)$	$\left[\right[\mathit{n},\mathit{k},\mathit{d}\left]\right]$
15	$4+x$	$1+x$	$[60,56,2]$	${\left[[60,52,2]\right]}_5$
22	$4+x+x^2+4x^3+2x^4+x^5$	$2+3x+2x^2+x^3+2x^4+x^5$	$[88,68,6]$	${\left[[88,48,6]\right]}_5$
25	$4+x$	$1+x$	$[100,96,2]$	${\left[[100,92,2]\right]}_5$
25	$4+x$	$4+2x+x^2$	$[100,95,2]$	${\left[[100,90,2]\right]}_5$
30	$1+x+x^2$	$1+x$	$[120,113,2]$	${\left[[120,106,2]\right]}_5$
35	$4+x$	$1+4x+x^2+4x^3+x^4+4x^5+x^6$	$[140,131,2]$	${\left[[140,122,2]\right]}_5$
40	$1+x$	$2+x^4$	$[160,153,2]$	${\left[[160,146,2]\right]}_5$
45	$1+x+x^2$	$1+4x+x^2$	$[180,172,3]$	${\left[[180,164,3]\right]}_5$
45	$1+x+x^2$	$1+x$	$[180,173,2]$	${\left[[180,166,2]\right]}_5$
50	$1+x$	$2+x$	$[200,196,2]$	${\left[[200,192,2]\right]}_5$
50	${(1+x)}^4{(4+x)}^2$	${(2+x)}^4{(3+x)}^2$	$[200,186,7]$	${\left[[200,172,7]\right]}_5$
52	$(1+4x+x^3+x^4)$	$(4+2x+x^3+x^4)$	$[208,194,9]$	${\left[[208,180,9]\right]}_5$
	$\times (1+2x+2x^3+x^4)$	$\times (4+x+2x^3+x^4)$
55	$(4+x+x^2+4x^3+2x^4+x^5)$	$(1+3x+4x^2+4x^3+x^4+x^5)$	$[220,210,6]$	${\left[[220,200,6]\right]}_5$