Quantum codes from constacyclic codes over $S_{k}$

Abstract

Let $S_{k}={\mathbb{F}}_{q}[u_{1},u_{2},\ldots ,u_{k}]/\langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 \rangle$, where $1\leq i,j\leq k$, $q=p^{m}$, p is an odd prime. First, we define two new Gray maps $\phi _{k}$ and $\varphi _{k}$, and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over $\mathbb{F}_{q}$ by using CSS construction, and compare the constructed codes better than the existing literature.

1 Introduction

In recent years, quantum theory and technology has become a popular research in the field of information, the research progress of some mathematical problems plays a key role in the study of quantum error correction problems. Calderbank et al. [1] gave a way to construct quantum error correcting codes from classical error correcting codes, constructing quantum error correcting codes is a systematic and effective mathematical method by using constacyclic codes. There are a lot of works about constacyclic codes over finite fields and finite rings [2–10] and many good quantum codes constructed by using cyclic codes over finite rings [11–14]. Currently, some authors have obtained quantum codes from constacyclic codes over finite non-chain ring. Wang et al. [15] studied quantum codes over ${\mathbb{F}}_{q}$ from Hermitian dual-containing constacyclic codes over ${\mathbb{F}}_{q^{2}}+v{\mathbb{F}}_{q^{2}}$. Prakash et al. [16] obtained quantum codes from skew constacyclic codes over a class of non-chain rings $R_{e,q}=\mathbb{F}_{q}[u]/\langle u^{e}-1\rangle$ by applying the CSS construction. Ashraf et al. [17] constructed quantum codes from ${\mathbb{F}}_{q} R_{1}R_{2}$-cyclic codes and introduced a Gray map to find some new and better quantum codes over ${\mathbb{F}}_{p}$. Dertli and Cengellenmis [18] studied quantum codes from constacyclic codes over the finite ring $u{\mathbb{F}}_{p}+v{\mathbb{F}}_{p}+uv{\mathbb{F}}_{p}$, Islam and Prakash [19] constructed quantum codes from $\lambda =(\lambda _{1}+u\lambda _{2}+v\lambda _{3})$-constacyclic codes over a class of finite commutative non-chain rings ${\mathbb{F}}_{q}[u,v]/\langle u^{2}-\gamma u,v^{2}-\delta v,uv=vu=0 \rangle$.

Due to the strong motivation discussed above, we construct some new quantum codes by studying the structure of constacyclic codes over a finite non-chain ring. The major two contributions of this paper are as follows.

• 1. In general, it is difficult to determine the structure of constacyclic codes over a finite non-chain ring, we study the structure of λ-constacyclic codes and their dual codes over the ring $S_{k}$, and give a necessary and sufficient conditions of dual-containing constacyclic codes.

• 2. As an application, we obtain some new quantum codes from constacyclic codes over $S_{k}$ by using CSS construction and compare these codes better than the existing codes that appeared in some recent references.

2 Preliminaries

Let $S_{k}={\mathbb{F}}_{q}[u_{1},u_{2},\ldots ,u_{k}]/\langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 \rangle$, where $q=p^{m}$ and p is an odd prime. The ring $S_{k}$ is a commutative and Frobenius ring with identity but not local, and the cardinality of $S_{k}$ is $q^{(2k+1)}$.

Let $e_{1}=\frac{u_{1}^{2}+u_{1}}{2}$, $e_{2}=\frac{u_{1}^{2}-u_{1}}{2}, \ldots , e_{2k-1}=\frac{u_{k}^{2}+u_{k}}{2}$, $e_{2k}=\frac{u_{k}^{2}-u_{k}}{2}$, $e_{2k+1}=1-u_{1}^{2}-u_{2}^{2}-\cdots -u_{k}^{2}$, where $e_{i}e_{j}=0$, when $i\neq j$, and $e_{i}^{2}=e_{i}$, when $i=1,2,\ldots ,2k+1$, and $1=e_{1}+e_{2}+\cdots +e_{2k+1}$. By the Chinese Remainder Theorem we can get that

$$ S_{k}=e_{1}S_{k}\oplus e_{2}S_{k}\oplus \cdots \oplus e_{2k+1}S_{k}. $$

$\forall r\in S_{k}$, r can be expressed uniquely as $r=r_{1}e_{1}+r_{2}e_{2}+\cdots +r_{2k+1}e_{2k+1}$, where $r_{i}\in {\mathbb{F}}_{q}$, $i=1,2,\ldots ,2k+1$.

By the definition above, it can be easily seen that $S_{k}$ is a principal ideal ring but not a chain ring, which has $2k+1$ maximal ideals. For any element $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$ of $S_{k}$, $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$ is a unit if and only if $\lambda _{1},\lambda _{2},\ldots ,\lambda _{2k+1}$ are units over ${\mathbb{F}}_{q}$.

If C is a code of length n over $S_{k}$, then C is a subset of $S_{k}^{n}$. C is a linear code of length n over $S_{k}$ if and only if C is an $S_{k}$-submodule of $S_{k}^{n}$.

For any unit $\lambda \in S_{k}$, a code C is called a λ-constacyclic code of length n over $S_{k}$ if and only if C is invariant under constacyclic shift operator $\sigma _{\lambda}:S_{k}^{n}\rightarrow S_{k}^{n}$ by

$$ \sigma _{\lambda}(c_{0},c_{1},\ldots ,c_{n-1})=(\lambda c_{n-1},c_{0}, \ldots ,c_{n-2}). $$

When $\lambda =1$, C is a cyclic code, when $\lambda =-1$, C is a negacyclic code.

If C is a linear code of length n over $S_{k}$, the dual code of C is defined as

$$ C^{\perp}=\{x \mid \forall y\in C,x\cdot y=0\}, $$

where $x\cdot y=\sum_{i=0}^{n-1}x_{i}y_{i}$, $x=(x_{0},x_{1},\ldots ,x_{n-1})\in S_{k}^{n}$, $y=(y_{0},y_{1},\ldots ,y_{n-1})\in S_{k}^{n}$.

3 Gray maps

Let A be an $n\times n$ matrix, such that $AA^{T}=\lambda E_{n}$, where $A^{T}$ denotes the transpose of the matrix A, $E_{n}$ is the identity matrix of order n, $\lambda \in {\mathbb{F}}_{q}$ and $\lambda \neq 0$.

Definition 1

We define a Gray map $\phi _{k}: S_{k}\rightarrow {\mathbb{F}}_{q}^{2k+1}$ by $r\mapsto (r_{1},r_{2},\ldots ,r_{2k+1})$, where $r=r_{1}e_{1}+r_{2}e_{2}+\cdots +r_{2k+1}e_{2k+1}$.

And $\phi _{k}$ can be expanded as:

$$\begin{aligned} \phi _{k}:{}& S_{k}^{n}\rightarrow { \mathbb{F}}_{q}^{(2k+1)n}\\ &(a_{0},a_{1},\ldots ,a_{n-1})\mapsto \bigl(a^{(1)}A,a^{(2)}A,\ldots ,a^{(2k+1)}A \bigr), \end{aligned}$$

where

$$ a_{j}=a_{1,j}e_{1}+a_{2,j}e_{2}+ \cdots +a_{2k+1,j}e_{2k+1}\in S_{k},\quad j=0,1,2, \ldots ,n-1, $$

and

$$ a^{(i)}=(a_{i,0},a_{i,1},\ldots ,a_{i,n-1}), \quad i=1,2,\ldots ,2k+1. $$

When the Gray map is defined as $\phi _{k}$, the Gray weight of $a\in S_{k}$ is defined as $w_{G}(a)=w_{H}(\phi _{k}(a))$, where $w_{H}(\phi _{k}(a))$ denotes the Hamming weight of $\phi _{k}(a)$.

The Gray weight of a vector $r=(x_{1},x_{2},\ldots ,x_{n})\in S_{k}^{n}$ is defined as $w_{G}(r)=\sum_{i=1}^{n}w_{G}(x_{i})$, the Gray distance of $x,y \in S_{k}^{n}$ is given by $d_{G}(x,y)=w_{G}(x-y)$, and the minimum Gray distance of C is defined as

$$ d_{G}(C)=\min \bigl\{ d_{G}(x-y), x,y\in C,x\neq y \bigr\} . $$

Lemma 1

$\phi _{k}$ is both a bijection and a distance preserving linear map from $S_{k}^{n}$ to ${\mathbb{F}}^{(2k+1)n}_{q}$.

Proof

Let $a=(a_{0},a_{1},\ldots ,a_{n-1})\in S_{k}^{n}$, $b=(b_{0},b_{1},\ldots ,b_{n-1}) \in S_{k}^{n}$, $l\in {\mathbb{F}}_{q}$, where $a_{j}=a_{1,j}e_{1}+a_{2,j}e_{2}+\cdots +a_{2k+1,j}e_{2k+1}\in S_{k}$, $b_{j}=b_{1,j}e_{1}+b_{2,j}e_{2}+ \cdots +b_{2k+1,j}e_{2k+1}\in S_{k}$, $j=0,1,2,\ldots ,n-1$, $a^{(i)}=(a_{i,0},a_{i,1},\ldots ,a_{i,n-1})$, $b^{(i)}=(b_{i,0},b_{i,1},\ldots ,b_{i,n-1})$, $i=1,2,\ldots ,2k+1$.

Then

$$\begin{aligned}& \begin{aligned} \phi _{k}(a+b)&=\phi _{k}(a_{0}+b_{0},a_{1}+b_{1}, \ldots ,a_{n-1}+b_{n-1}) \\ &= \bigl( \bigl(a^{(1)}+b^{(1)} \bigr)A, \bigl(a^{(2)}+b^{(2)} \bigr)A,\ldots , \bigl(a^{(2k+1)}+b^{(2k+1)} \bigr)A \bigr) \\ &= \bigl(a^{(1)}A,a^{(2)}A,\ldots ,a^{(2k+1)}A \bigr)+ \bigl(b^{(1)}A,b^{(2)}A,\ldots ,b^{(2k+1)}A \bigr) \\ &=\phi _{k}(a)+\phi _{k}(b), \end{aligned} \\& \begin{aligned} \phi _{k}(la)&=\phi _{k}(la_{0},la_{1}, \ldots ,la_{n-1}) \\ &=(la_{0}A,la_{1}A,\ldots ,la_{n-1}A) \\ &=l \bigl(a^{(1)}A,a^{(2)}A,\ldots ,a^{(2k+1)}A \bigr) \\ &=l\phi _{k}(a). \end{aligned} \end{aligned}$$

So $\phi _{k}$ is linear.

$\forall a,b\in S_{k}^{n}$, suppose $\phi _{k}(a)=\phi _{k}(b)$, then

$$ \bigl(a^{(1)}A,a^{(2)}A,\ldots ,a^{(2k+1)}A \bigr)= \bigl(b^{(1)}A,b^{(2)}A,\ldots ,b^{(2k+1)}A \bigr). $$

Because A is an invertible matrix, we have

$$ \bigl(a^{(1)},a^{(2)},\ldots ,a^{(2k+1)} \bigr)= \bigl(b^{(1)},b^{(2)},\ldots ,b^{(2k+1)} \bigr), $$

so $a=b$, $\phi _{k}$ is an injection.

As

$$ \bigl\vert S_{k}^{n} \bigr\vert = \bigl\vert { \mathbb{F}}_{q}^{(2k+1)n} \bigr\vert =q^{(2k+1)n}, $$

so $\phi _{k}$ is a bijection.

$\forall a,b\in S_{k}^{n}$, then

$$\begin{aligned}& a-b=(a_{0}-b_{0},a_{1}-b_{1}, \ldots ,a_{n-1}-b_{n-1}),\\& \phi _{k}(a-b)= \bigl( \bigl(a^{(1)}-b^{(1)} \bigr)A, \bigl(a^{(2)}-b^{(2)} \bigr)A,\ldots , \bigl(a^{(2k+1)}-b^{(2k+1)} \bigr)A \bigr) =\phi _{k}(a)-\phi _{k}(b),\\& d_{G}(a,b)=w_{G}(a-b)=w_{H} \bigl(\phi _{k}(a-b) \bigr)=w_{H} \bigl(\phi _{k}(a)- \phi _{k}(b) \bigr)=d_{H} \bigl( \phi _{k}(a), \phi _{k}(b) \bigr). \end{aligned}$$

So $\phi _{k}$ is a distance preserving map from $S_{k}^{n}$ to ${\mathbb{F}}^{(2k+1)n}_{q}$. □

By Lemma 1 and the definition of $\phi _{k}$, we can have the following lemma.

Lemma 2

Let C be a linear code of length n over $S_{k}^{n}$ and the minimal Gray distance of C is d, then $\phi _{k}(C)$ is a $[(2k+1)n, l,d]$ linear code over ${\mathbb{F}}_{q}$, where $l=\log_{q}{| C |}$.

Let B be a $(2k+1)\times (2k+1)$ matrix, such that $BB^{T}=\lambda E_{2k+1}$, where $B^{T}$ denotes the transpose of the matrix B, $E_{2k+1}$ is the identity matrix of order $2k+1$, $\lambda \in {\mathbb{F}}_{q}$ and $\lambda \neq 0$. $\forall r=r_{1}e_{1}+r_{2}e_{2}+\cdots +r_{2k+1}e_{2k+1}\in S_{k}$, the vector form of r is written as $r=(r_{1},r_{2},\ldots ,r_{2k+1})$.

Definition 2

We define a Gray map $\varphi _{k}: S_{k}\rightarrow {\mathbb{F}}_{q}^{2k+1}$ by $r\mapsto rB$.

And $\varphi _{k}$ can be expanded as

$$\begin{aligned} \varphi _{k}:{}& S_{k}^{n}\rightarrow { \mathbb{F}}_{q}^{(2k+1)n}\\ &(a_{0},a_{1},\ldots ,a_{n-1})\mapsto (a_{0}B,a_{1}B,\ldots ,a_{n-1}B), \end{aligned}$$

where $a_{i}=a_{1,i}e_{1}+a_{2,i}e_{2}+\cdots +a_{2k+1,i}e_{2k+1}\in S_{k}$, $i=0,1,2,\ldots ,n-1$.

When the Gray map is defined as $\varphi _{k}$, the Gray weight of $a\in S_{k}$ is defined as $w_{G}(a)=w_{H}(\varphi _{k}(a))$, where $w_{H}(\varphi _{k}(a))$ denotes the Hamming weight of $\varphi _{k}(a)$.

The Gray weight of a vector $r=(x_{1},x_{2},\ldots ,x_{n})\in S_{k}^{n}$ is defined as $w_{G}(r)=\sum_{i=1}^{n}w_{G}(x_{i})$, the Gray distance of $x,y \in S_{k}^{n}$ is given by $d_{G}(x,y)=w_{G}(x-y)$, and the minimum Gray distance of C is defined as

$$ d_{G}(C)=\min \bigl\{ d_{G}(x-y), x,y\in C,x\neq y \bigr\} . $$

Lemma 3

$\varphi _{k}$ is both a bijection and a distance preserving linear map from $S_{k}^{n}$ to ${\mathbb{F}}^{(2k+1)n}_{q}$.

Proof

Let $a,b\in S_{k}^{n}$, where $a=(a_{0},a_{1},\ldots ,a_{n-1})$, $b=(b_{0},b_{1},\ldots ,b_{n-1})$, $l\in {\mathbb{F}}_{q}$. Then

$$\begin{aligned}& \begin{aligned} \varphi _{k}(a+b)&=\varphi _{k}(a_{0}+b_{0},a_{1}+b_{1}, \ldots ,a_{n-1}+b_{n-1}) \\ &= \bigl((a_{0}+b_{0})B,(a_{1}+b_{1})B, \ldots ,(a_{n-1}+b_{n-1})B \bigr) \\ &=(a_{0}B,a_{1}B,\ldots ,a_{n-1}B)+(b_{0}B,b_{1}B, \ldots ,b_{n-1}B) \\ &=\varphi _{k}(a)+\varphi _{k}(b), \end{aligned} \\& \begin{aligned} \varphi _{k}(la)&=\phi _{k}(la_{0},la_{1}, \ldots ,la_{n-1})=(la_{0}B,la_{1}B, \ldots ,la_{n-1}B) \\ &=l(a_{0}B,a_{1}B,\ldots ,a_{n-1}B) \\ &=l\phi _{k}(a). \end{aligned} \end{aligned}$$

So $\varphi _{k}$ is linear.

$\forall a,b\in S_{k}^{n}$, suppose $\varphi _{k}(a)=\varphi _{k}(b)$, then

$$ (a_{0}B,a_{1}B,\ldots ,a_{n-1}B)=(b_{0}B,b_{1}B, \ldots ,b_{n-1}B). $$

Because B is an invertible matrix, we have $a=(a_{0},a_{1},\ldots ,a_{n-1})=(b_{0},b_{1},\ldots ,b_{n-1})=b$, $\varphi _{k}$ is an injection.

As

$$ \bigl\vert S_{k}^{n} \bigr\vert = \bigl\vert { \mathbb{F}}_{q}^{(2k+1)n} \bigr\vert =q^{(2k+1)n}, $$

so $\varphi _{k}$ is a bijection.

$\forall a,b\in S_{k}^{n}$, then

$$\begin{aligned}& a-b=(a_{0}-b_{0},a_{1}-b_{1}, \ldots ,a_{n-1}-b_{n-1}),\\& \varphi _{k}(a-b)= \bigl((a_{0}-b_{0})B,(a_{1}-b_{1})B, \ldots ,(a_{n-1}-b_{n-1})B \bigr)= \varphi _{k}(a)-\varphi _{k}(b),\\& d_{G}(a,b)=w_{G}(a-b)=w_{H} \bigl(\varphi _{k}(a-b) \bigr)=w_{H} \bigl(\varphi _{k}(a)- \varphi _{k}(b) \bigr)=d_{H} \bigl(\varphi _{k}(a),\varphi _{k}(b) \bigr). \end{aligned}$$

So $\varphi _{k}$ is a distance preserving map from $S_{k}^{n}$ to ${\mathbb{F}}^{(2k+1)n}_{q}$. □

By Lemma 3 and the definition of $\varphi _{k}$, we can have the following lemma.

Lemma 4

Let C be a linear code of length n over $S_{k}^{n}$ and the minimal Gray distance of C is d, then $\varphi _{k}(C)$ is a $[(2k+1)n, l,d]$ linear code over ${\mathbb{F}}_{q}$, where $l=\log_{q}{| C|}$.

4 Constacyclic codes over $S_{k}$

Let C be a linear code of length n over $S_{k}$ and define

$$ C_{j}= \Biggl\{ x_{j}\in {\mathbb{F}}_{q}^{n} \biggm| \sum_{i=1}^{2k+1}x_{i}e_{i} \in C, x_{i}\in {\mathbb{F}}_{q}^{n} \Biggr\} , \quad j=1,2,\ldots ,2k+1, $$

then, $C_{1},C_{2},\ldots ,C_{2k+1}$ are linear codes of length n over ${\mathbb{F}}_{q}$.

Moreover, the linear code C of length n over $S_{k}$ can be represented as

$$ C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}. $$

Let $G_{j}$ be the Generator matrices of $C_{j}$, then the Generator matrix of C is

$$ G= \begin{bmatrix} e_{1}G_{1} \\ e_{2}G_{2} \\ \cdots \\ e_{2k+1}G_{2k+1}\end{bmatrix} . $$

Definition 3

We define a quasi-cyclic shift on $({\mathbb{F}}_{q}^{n})^{2k+1}$,

$$\begin{aligned} &\psi _{2k+1}(a_{1,0},a_{1,1}\cdots ,a_{1,n-1},a_{2,0},a_{2,1}\cdots ,a_{2,n-1}, \\ &\qquad \cdots ,a_{2k+1,0},a_{2k+1,1}\cdots ,a_{2k+1,n-1}) \\ &\quad = \bigl(\sigma (a_{1,0},a_{1,1}\cdots ,a_{1,n-1}), \sigma (a_{2,0},a_{2,1} \cdots ,a_{2,n-1}), \\ &\qquad \cdots ,\sigma (a_{2k+1,0},a_{2k+1,1}\cdots ,a_{2k+1,n-1}) \bigr). \end{aligned}$$

Proposition 1

Let σ be the cyclic shift operator on $S_{k}^{n}$, let $\psi _{2k+1}$ be the quasi-cyclic shift on $({\mathbb{F}}_{q}^{n})^{2k+1}$ defined as above. Then $\phi _{k}\sigma =\psi _{2k+1}\phi _{k}$.

Proof

Let $(a_{0},a_{1},\ldots ,a_{n-1})\in S_{k}^{n}$, where $a_{j}=a_{1,j}e_{1}+a_{2,j}e_{2}+\cdots +a_{2k+1,j}e_{2k+1}\in S_{k}$, $j=0,1,2,\ldots ,n-1$, $a^{(i)}=(a_{i,0},a_{i,1},\ldots ,a_{i,n-1})$, $i=1,2,\ldots ,2k+1$.

$$\begin{aligned}& \phi _{k}(a_{0},a_{1},\ldots ,a_{n-1})= \bigl(a^{(1)}A,a^{(2)}A,\ldots ,a^{(2k+1)}A \bigr), \\& \sigma (a_{0},a_{1},\ldots ,a_{n-1})=(a_{n-1},a_{0}, \ldots ,a_{n-2}). \end{aligned}$$

If we apply $\phi _{k}$, we can have

$$\begin{aligned} \phi _{k} \bigl(\sigma (a_{0},a_{1}, \ldots ,a_{n-1}) \bigr)&=\phi _{k}(a_{n-1},a_{0}, \ldots ,a_{n-2}) \\ &= \bigl((a_{1,n-1},a_{1,0},\ldots ,a_{1,n-2})A,(a_{2,n-1},a_{2,0}, \ldots ,a_{2,n-2})A, \\ & \quad \cdots ,(a_{2k+1,n-1},a_{2k+1,0},\ldots ,a_{2k+1,n-2})A \bigr). \end{aligned}$$

On the other hand,

$$\begin{aligned} \psi _{2k+1} \bigl(\phi _{k}(a_{0},a_{1}, \ldots ,a_{n-1}) \bigr)&=\psi _{2k+1} \bigl(a^{(1)}A,a^{(2)}A, \ldots ,a^{(2k+1)}A \bigr) \\ &= \bigl(\sigma \bigl(a^{(1)}A \bigr),\sigma \bigl(a^{(2)}A \bigr),\ldots ,\sigma \bigl(a^{(2k+1)}A \bigr) \bigr) \\ &= \bigl((a_{1,n-1},a_{1,0},\ldots ,a_{1,n-2})A, \\ &\quad (a_{2,n-1},a_{2,0},\ldots ,a_{2,n-2})A, \\ &\quad \cdots ,(a_{2k+1,n-1},a_{2k+1,0},\ldots ,a_{2k+1,n-2})A \bigr) \\ &=\phi _{k} \bigl(\sigma (a_{0},a_{1}, \ldots ,a_{n-1}) \bigr). \end{aligned}$$

Thus $\phi _{k}\sigma =\psi _{2k+1}\phi _{k}$. □

Proposition 2

Let σ and $\psi _{2k+1}$ be defined as above, then a linear code C of length n over $S_{k}$ is a cyclic code if and only if $\phi _{k}(C)$ is a quasi cyclic code of index $2k+1$ of length $(2k+1)n$ over ${\mathbb{F}}_{q}$.

Proof

If C is a cyclic code of length n over $S_{k}$. Then $\sigma (C)=C$. We can have $\phi _{k}(\sigma (C))=\phi _{k}(C)$.

By Proposition 1,

$$ \phi _{k} \bigl(\sigma (C) \bigr)=\psi _{2k+1} \bigl(\phi _{k}(C) \bigr)=\phi _{k}(C). $$

So, $\phi _{k}(C)$ is a quasi-cyclic code of index $2k+1$ of length $(2k+1)n$ over ${\mathbb{F}}_{q}$.

Conversely, suppose $\phi _{k}(C)$ is a quasi-cyclic code of index $2k+1$ of length $(2k+1)n$ over ${\mathbb{F}}_{q}$, then $\psi _{2k+1}(\phi _{k}(C))=\phi _{k}(C)$.

By Proposition 1, we have $\psi _{2k+1}(\phi _{k}(C))=\phi _{k}(\sigma (C))=\phi _{k}(C)$.

Since $\phi _{k}$ is a bijective linear map, so $\sigma (C)=C$. □

Theorem 1

Let $\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1}$ be a unit of $S_{k}$. Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a linear code of length n over $S_{k}$, then C is a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic code over $S_{k}$ if and only if $C_{i}$ is a $\lambda _{i}$-constacyclic code over ${\mathbb{F}}_{q}$, where $i=1,2,\ldots ,2k+1$.

Proof

$\forall c_{i}=(c_{i,0},c_{i,1},\ldots ,c_{i,n-1})\in C_{i}$, where $i=1,2,\ldots ,2k+1$.

$$ c=e_{1}c_{1}+e_{2}c_{2}+\cdots +e_{2k+1}c_{2k+1}= \Biggl(\sum_{i=1}^{2k+1}e_{i}c_{i,0}, \sum_{i=1}^{2k+1}e_{i}c_{i,1}, \ldots ,\sum_{i=1}^{2k+1}e_{i}c_{i,n-1} \Biggr) \in C. $$

$\forall \lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1} \in S_{k}$, it’s easy to know that $\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1} \in S_{k}$ is a unit if and only if $\lambda _{i}\neq 0$, that is, $\lambda _{i}$ is a unit over ${\mathbb{F}}_{q}$, where $i=1,2,\ldots ,2k+1$.

If $C_{i}$ is a $\lambda _{i}$-constacyclic code over ${\mathbb{F}}_{q}$, $i=1,2,\ldots ,2k+1$, then

$$ \sigma _{\lambda _{i}}(c_{i})=\sigma _{\lambda _{i}}(c_{i,0},c_{i,1}, \ldots ,c_{i,n-1})=(\lambda _{i}c_{i,n-1},c_{i,0}, \ldots ,c_{i,n-2}) \in C_{i}, $$

and

$$ \begin{aligned} &\sigma _{\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots + \lambda _{2k+1}e_{2k+1}}(c) \\ &\quad = \Biggl((\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1}) \sum_{i=1}^{2k+1}e_{i}c_{i,n-1}, \sum_{i=1}^{2k+1}e_{i}c_{i,0}, \ldots ,\sum_{i=1}^{2k+1}e_{i}c_{i,n-2} \Biggr) \\ &\quad =e_{1}\sigma _{\lambda _{1}}(c_{1})+e_{2} \sigma _{\lambda _{2}}(c_{2})+ \cdots +e_{2k+1}\sigma _{\lambda _{2k+1}}(c_{2k+1})\in C. \end{aligned} $$

So C is a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic code over $S_{k}$.

Conversely, if C is a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic code over $S_{k}$, we have

$$ \sigma _{\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1}}(c) =e_{1}\sigma _{\lambda _{1}}(c_{1})+e_{2} \sigma _{\lambda _{2}}(c_{2})+ \cdots +e_{2k+1}\sigma _{\lambda _{2k+1}}(c_{2k+1})\in C. $$

So $\sigma _{\lambda _{i}}(c_{i})\in C_{i}$, $C_{i}$ is a $\lambda _{i}$-constacyclic code over ${\mathbb{F}}_{q}$, $i=1,2,\ldots ,2k+1$. □

Theorem 2

Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic code of length n over $S_{k}$, then $C=\langle e_{1}g_{1}(x)+e_{2}g_{2}(x)+\cdots +e_{2k+1}g_{2k+1}(x) \rangle$, where $g_{i}$ is the generator polynomial of $C_{i}$, $i=1,2, \ldots ,2k+1$.

Proof

Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic n over $S_{k}$, by Theorem 1, we get that $C_{i}$ is a $\lambda _{i}$-constacyclic code over ${\mathbb{F}}_{q}$, $i=1,2,\ldots ,2k+1$.

Because the generator polynomial of $C_{i}$ is $g_{i}(x)$, $i=1,2,\ldots ,2k+1$. Then

$$ C= \bigl\langle e_{1}g_{1}(x),e_{2}g_{2}(x), \ldots ,e_{2k+1}g_{2k+1}(x) \bigr\rangle . $$

Let $C'=\langle e_{1}g_{1}(x)+e_{2}g_{2}(x)+\cdots +e_{2k+1}g_{2k+1}(x) \rangle$. So $C'\subseteq C$.

Because $e_{i}[e_{1}g_{1}(x)+e_{2}g_{2}(x)+\cdots +e_{2k+1}g_{2k+1}(x)]=e_{i}g_{i}(x),i=1,2, \ldots ,2k+1$. So $C\subseteq C'$.

So, we have $C= C'$, and the generator polynomial of C is

$$ g(x)=e_{1}g_{1}(x)+e_{2}g_{2}(x)+ \cdots +e_{2k+1}g_{2k+1}(x). $$

Because $g_{i}(x)$ is the generator polynomial of $C_{i}$, $g_{i}$ divides $x^{n}-\lambda _{i}$, $i=1,2,\ldots ,2k+1$. Let $g_{i}(x)f_{i}(x)=x^{n}-\lambda _{i}$, $i=1,2,\ldots ,2k+1$.

Then

$$ \begin{aligned} & \bigl[e_{1}g_{1}(x)+e_{2}g_{2}(x)+ \cdots +e_{2^{k}}g_{2k+1}(x) \bigr] \bigl[e_{1}f_{1}(x)+e_{2}f_{2}(x)+ \cdots +e_{2k+1}f_{2k+1}(x) \bigr] \\ &\quad =\lambda _{1}e_{1}+\lambda _{2}e_{2}+ \cdots +\lambda _{2k+1}e_{2k+1}. \end{aligned} $$

So

$$ e_{1}g_{1}(x)+e_{2}g_{2}(x)+ \cdots +e_{2k+1}g_{2k+1}(x)\mid x^{n}-( \lambda _{1}e_{1}+\lambda _{2}e_{2}+ \cdots +\lambda _{2k+1}e_{2k+1}). $$

 □

Theorem 3

Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a linear code of length n over $S_{k}$, let $C_{j}^{\bot}$ be the dual code of $C_{j}$, then $C^{\bot}=\sum_{j=1}^{2k+1}e_{j}C_{j}^{\bot}$, where $j=1,2,\ldots ,2k+1$.

Proof

Let $\tilde{C}=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}^{\bot}$, $\forall x=\sum_{j=1}^{2k+1}e_{j}x_{j}\in C$, $\forall \tilde{x}=\sum_{j=1}^{2k+1}e_{j}\tilde{x_{j}}\in \tilde{C}$, where $x_{j}\in C_{j}$, $\tilde{x_{j}}\in C_{j}^{\bot}$.

Since $x_{j}\tilde{x_{j}}=0$, it follows that $x\cdot \tilde{x}=\sum_{j=1}^{2k+1}(x_{j}\tilde{x_{j}})e_{j}=0$.

So, $\tilde{C}\subseteq C^{\bot}$.

Since $| C| | C^{\bot}| =| S_{k}| ^{n}$, we have

$$ \vert \tilde{C} \vert =\prod_{j=1}^{2k+1} \bigl\vert C_{j}^{\bot} \bigr\vert =\prod _{j=1}^{2k+1} \frac{q^{n}}{ \vert C_{j} \vert }= \frac{ \vert S_{k} \vert ^{n}}{ \vert C \vert }= \bigl\vert C^{\bot} \bigr\vert . $$

So

$$ C^{\bot}=\tilde{C}. $$

 □

Theorem 4

Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic code of length n over $S_{k}$, then

$$ C^{\perp}= \bigl\langle e_{1}f_{1}^{*}(x)+e_{2}f_{2}^{*}(x)+ \cdots +e_{2k+1}f_{2k+1}^{*}(x) \bigr\rangle , \bigl\vert C^{\perp } \bigr\vert =q^{(\sum _{i=1}^{2k+1}\operatorname{deg}(g_{i}))}, $$

$f_{i}^{*}(x)$ is the reciprocal polynomial of $f_{i}(x)=(x^{n}-\lambda _{i})/g_{i}(x)$ which is defined as $f_{i}^{*}(x)=x^{\operatorname{deg}(f_{i})}f_{i}(x^{-1})$, where $g_{i}$ is the generator polynomial of $C_{i}$, $i=1,2, \ldots ,2k+1$.

Proof

Let $C_{i}=\langle g_{i}(x) \rangle$ be a $\lambda _{i}$-constacyclic code of length n over ${\mathbb{F}}_{q}$, $i=1,2,\ldots ,2k+1$. $\forall x=(x_{0},x_{1},\ldots ,x_{n-1})\in C_{i}^{\perp}$, $\forall y=(y_{0},y_{1},\ldots ,y_{n-1})\in C_{i}$, then $\sigma ^{n-1}_{\lambda _{i}}(y)=(\lambda _{i} y_{1},\lambda _{i} y_{2}, \ldots ,\lambda _{i} y_{n-1},y_{0})\in C_{i}$, and

$$\begin{aligned} 0&=x\cdot \sigma ^{n-1}_{\lambda _{i}}(y)=\lambda _{i} x_{0}y_{1}+ \lambda _{i} x_{1}y_{2}+\cdots +\lambda _{i} x_{n-2}y_{n-1}+x_{n-1}y_{0} \\ &=\lambda _{i} \bigl(x_{0}y_{1}+x_{1}y_{2}+ \cdots + x_{n-2}y_{n-1}+\lambda _{i}^{-1}x_{n-1}y_{0} \bigr) \\ &=\lambda _{i} \sigma _{\lambda _{i}^{-1}}(x)\cdot y. \end{aligned}$$

So, $\sigma _{\lambda _{i}^{-1}}(x)\in C_{i}^{\perp}$, $C_{i}^{\perp}$ is a $\lambda ^{-1}_{i}$-constacyclic code over ${\mathbb{F}}_{q}$.

Let $\tilde{C_{i}}=\langle f_{i}^{*}(x) \rangle$,

$$\begin{aligned} f_{i}^{*}(x)g_{i}^{*}(x)&=x^{\operatorname{deg}(f_{i})}f_{i} \bigl(x^{-1} \bigr)x^{ \operatorname{deg}(g_{i})}g_{i} \bigl(x^{-1} \bigr) \\ &=x^{\operatorname{deg}(f_{i})} \bigl(x^{-n}-\lambda _{i} \bigr)/g_{i} \bigl(x^{-1} \bigr)x^{ \operatorname{deg}(g_{i})}g_{i} \bigl(x^{-1} \bigr) \\ &=1-x^{n}\lambda _{i}=-\lambda _{i} \bigl(x^{n}-\lambda _{i}^{-1} \bigr) \end{aligned}$$

we have $f_{i}^{*}(x) \mid (x^{n}-\lambda _{i}^{-1})$, so $\tilde{C_{i}}\subseteq C_{i}^{\perp}$.

Because $| \tilde{C_{i}}| =q^{n-\operatorname{deg} f_{i}^{*}}=q^{ \operatorname{deg}g_{i}}=\frac{q^{n}}{| C_{i} |}=| C_{i}^{\bot } |$, we have $C_{i}^{\bot}=\tilde{C_{i}}=\langle f_{i}^{*}(x) \rangle$, $i=1,2,\ldots ,2k+1$.

By Theorem 3, $C^{\bot}=\sum_{j=1}^{2k+1}e_{j}C_{j}^{\bot}$, we have $| C^{\bot}| =\prod_{j=1}^{2k+1}| C_{j}^{\bot}| =q^{( \sum _{i=1}^{2k+1}\mathrm{deg}(g_{i}))}$, and we can get the form of $C^{\bot}$ is

$$ C^{\bot}= \bigl\langle e_{1}f_{1}^{*}(x),e_{2}f_{2}^{*}(x), \ldots ,e_{2k+1}f_{2k+1}^{*}(x) \bigr\rangle . $$

Let $\tilde{C'}=\langle e_{1}f_{1}^{*}(x)+e_{2}f_{2}^{*}(x)+\cdots +e_{2k+1}f_{2k+1}^{*}(x) \rangle$. Then $\tilde{C'}\subseteq C^{\bot}$.

Because

$$ e_{i} \bigl[ e_{1}f_{1}^{*}(x),e_{2}f_{2}^{*}(x), \ldots ,e_{2k+1}f_{2k+1}^{*}(x) \bigr]=e_{i}f_{i}^{*}(x),\quad i=1,2, \ldots ,2k+1. $$

So $C^{\bot}\subseteq \tilde{C'}$.

We have

$$ C^{\bot}=\tilde{C'}= \bigl\langle e_{1}f_{1}^{*}(x)+e_{2}f_{2}^{*}(x)+ \cdots +e_{2k+1}f_{2k+1}^{*}(x) \bigr\rangle . $$

 □

5 Quantum codes from constacyclic codes over $S_{k}$

Theorem 5

Let C be a linear code of length n over $S_{k}$, then

$$ {{\phi }_{k}} {{(C)}^{\bot }}={{\phi }_{k}} \bigl({{C}^{\bot }} \bigr),\qquad {{\varphi }_{k}} {{(C)}^{ \bot }}={{\varphi}_{k}} \bigl({{C}^{\bot }} \bigr). $$

Proof

Let $a=(a_{0},a_{1},\ldots ,a_{n-1})\in C$, $b=(b_{0},b_{1},\ldots ,b_{n-1}) \in C^{\bot}$, where $a_{j}=a_{1,j}e_{1}+a_{2,j}e_{2}+\cdots +a_{2k+1,j}e_{2k+1}$, $b_{j}=b_{1,j}e_{1}+b_{2,j}e_{2}+ \cdots +b_{2k+1,j}e_{2k+1}\in S_{k}$, $j=0,1,2,\ldots ,n-1$, $a^{(i)}=(a_{i,0},a_{i,1}, \ldots ,a_{i,n-1})$, $b^{(i)}=(b_{i,0},b_{i,1},\ldots ,b_{i,n-1})$, $i=1,2,\ldots ,2k+1$.

Then

$$ a\cdot b=\sum_{j=0}^{n-1}a_{j}b_{j}= \sum_{j=0}^{n-1} \sum _{i=1}^{2k+1}a_{i,j}b_{i,j}e_{i}= \sum_{i=1}^{2k+1}{a^{(i)}{b^{(i)}}^{T}}e_{i}=0. $$

So

$$ a^{(i)}{b^{(i)}}^{T}=0,\quad i=1,2,\ldots ,2k+1. $$

Since

$$ \phi _{k}(a)= \bigl(a^{(1)}A,a^{(2)}A,\ldots ,a^{(2k+1)}A \bigr),\qquad \phi _{k}(b)= \bigl(b^{(1)}A,b^{(2)}A, \ldots ,b^{(2k+1)}A \bigr). $$

It follows that

$$\begin{aligned} \phi _{k}(a)\cdot \phi _{k}(b)&=\phi _{k}(a)\phi _{k}(b)^{T} \\ &=\sum_{i=1}^{2k+1}{{a}^{(i)}A{A^{T}} {b^{(i)}}^{T}}=\sum_{i=1}^{2k+1}{a^{(i)} \lambda E_{n}{b^{(i)}}^{T}} \\ &=\lambda \sum_{i=1}^{2k+1}{a^{(i)}{b^{(i)}}^{T}}=0. \end{aligned}$$

So we have

$$ \phi _{k} \bigl(C^{\bot } \bigr)\subseteq \phi _{k}(C)^{\bot }. $$

As $\phi _{k}$ is a bijection, and

$$ \vert C \vert = \bigl\vert \phi _{k}(C) \bigr\vert . $$

Then

$$ \bigl\vert {{\phi }_{k}} \bigl({{C}^{\bot }} \bigr) \bigr\vert = \frac{{{q}^{(2k+1)n}}}{ \vert C \vert }= \frac{{{q}^{(2k+1)n}}}{ \vert {{\phi }_{k}}(C) \vert }= \bigl\vert {{\phi }_{k}} {{(C)}^{ \bot }} \bigr\vert . $$

So

$$ {{\phi }_{k}} {{(C)}^{\bot }}={{\phi }_{k}} \bigl({{C}^{\bot }} \bigr). $$

Let

$$ c=({{c}_{1}},{{c}_{2}},\ldots ,{{c}_{n}}) \in C, \qquad d=({{d}_{1}},{{d}_{2}}, \ldots ,{{d}_{n}})\in {{C}^{\bot }}, $$

then

$$ {{\varphi }_{k}}(c)=({{c}_{1}}B,{{c}_{2}}B, \ldots ,{{c}_{n}}B),\qquad {{ \varphi }_{k}}(d)=({{d}_{1}}B,{{d}_{2}}B, \ldots ,{{d}_{n}}B). $$

The vector forms of ${{c}_{i}}$ and ${{d}_{i}}$ are respectively

$$ {{c}_{i}}=({{c}_{i1}},{{c}_{i2}},\ldots ,{{c}_{i(2k+1)}}),\qquad {{d}_{i}}=({{d}_{i1}},{{d}_{i2}}, \ldots ,{{d}_{i(2k+1)}}),\quad i=1,2,\ldots ,n. $$

Then

$$\begin{aligned} {{\varphi }_{k}}(c)\cdot {{\varphi }_{k}}(d)&={{ \varphi }_{k}}(c){{ \varphi }_{k}} {{(d)}^{T}} \\ &=\sum_{i=1}^{n}{{{c}_{i}}B{{B}^{T}}d_{i}^{T}=} \sum_{i=1}^{n}{{{c}_{i}} \lambda E_{2k+1}d_{i}^{T}}=\lambda \sum _{i=1}^{n}{{{c}_{i}}d_{i}^{T}}=0. \end{aligned}$$

So we have

$$ {{\varphi }_{k}} \bigl({{C}^{\bot }} \bigr)\subseteq {{ \varphi }_{k}} {{(C)}^{ \bot }}. $$

As $\varphi _{k}$ is a bijection, and

$$ \vert C \vert = \bigl\vert \varphi _{k}(C) \bigr\vert . $$

Then

$$ \bigl\vert {{\varphi }_{k}} \bigl({{C}^{\bot }} \bigr) \bigr\vert = \frac{{{q}^{(2k+1)n}}}{ \vert C \vert }= \frac{{{q}^{(2k+1)n}}}{ \vert {{\varphi }_{k}}(C) \vert }= \bigl\vert {{ \varphi}_{k}} {{(C)}^{\bot }} \bigr\vert . $$

Therefore,

$$ {{\varphi }_{k}} {{(C)}^{\bot }}={{\varphi }_{k}} \bigl({{C}^{\bot }} \bigr). $$

 □

Theorem 6

Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a linear code of length n over $S_{k}$, then C is a self-orthogonal code over $S_{k}$ if and only if $C_{j}$ is a self-orthogonal code over ${\mathbb{F}}_{q}$, if C is a self-orthogonal code over $S_{k}$, then $\phi _{k}(C)$ and $\varphi _{k}(C)$ are self-orthogonal codes over ${\mathbb{F}}_{q}$, where $j=1,2,\ldots ,2k+1$.

Proof

By using Theorem 1, we have $C\subseteq C^{\bot}$ if and only if $C_{j}\subseteq C_{j}^{\bot}$, so C is a self-orthogonal code over $S_{k}$ if and only if $C_{j}$ is a self-orthogonal code over ${\mathbb{F}}_{q}$, where $j=1,2,\ldots ,2k+1$.

Let C be a self-orthogonal code, $\forall a=(a_{0},a_{1},\ldots ,a_{n-1})$, $b=(b_{0},b_{1},\ldots ,b_{n-1}) \in C$, $a_{j}=a_{1,j}e_{1}+a_{2,j}e_{2}+\cdots +a_{2k+1,j}e_{2k+1}$, $b_{j}=b_{1,j}e_{1}+b_{2,j}e_{2}+ \cdots +b_{2k+1,j}e_{2k+1}\in S_{k}$, $j=0,1,2,\ldots ,n-1$, $a^{(i)}=(a_{i,0},a_{i,1},\ldots ,a_{i,n-1})$, $b^{(i)}=(b_{i,0},b_{i,1},\ldots ,b_{i,n-1})$, $i=1,2,\ldots ,2k+1$.

Then

$$ a\cdot b=\sum_{j=0}^{n-1}a_{j}b_{j}= \sum_{j=0}^{n-1} \sum _{i=1}^{2k+1}a_{i,j}b_{i,j}e_{i}= \sum_{i=1}^{2k+1}{a^{(i)}{b^{(i)}}^{T}}e_{i}=0. $$

So,

$$ a^{(i)}{b^{(i)}}^{T}=0,\quad i=1,2,\ldots ,2k+1. $$

It follows that

$$\begin{aligned} \phi _{k}(a)\cdot \phi _{k}(b)&=\phi _{k}(a)\phi _{k}(b)^{T} \\ &=\sum_{i=1}^{2k+1}{{a}^{(i)}A{A^{T}} {b^{(i)}}^{T}}=\sum_{i=1}^{2k+1}{a^{(i)} \lambda E_{n}{b^{(i)}}^{T}}=\lambda \sum _{i=1}^{2k+1}{a^{(i)}{b^{(i)}}^{T}}=0. \end{aligned}$$

So $\phi _{k}(C)$ is a self-orthogonal code over ${\mathbb{F}}_{q}$.

Let $c=({{c}_{1}},{{c}_{2}},\ldots ,{{c}_{n}})\in C$, $d=({{d}_{1}},{{d}_{2}},\ldots ,{{d}_{n}})\in C$, then

$$\begin{aligned} &{{\varphi }_{k}}(c)=({{c}_{1}}B,{{c}_{2}}B, \ldots ,{{c}_{n}}B),\qquad {{ \varphi }_{k}}(d)=({{d}_{1}}B,{{d}_{2}}B, \ldots ,{{d}_{n}}B). \\ &c_{i}=c_{i,1}e_{1}+c_{i,2}e_{2}+ \cdots +c_{i,2k+1}e_{2k+1}\in S_{k}, \\ &d_{i}=d_{i,1}e_{1}+d_{i,2}e_{2}+ \cdots +d_{i,2k+1}e_{2k+1}\in S_{k}, \end{aligned}$$

where $i=1,2,\ldots ,n$.

The vector forms of ${{c}_{i}}$ and ${{d}_{i}}$ are respectively

$$ {{c}_{i}}=({{c}_{i,1}},{{c}_{i,2}},\ldots ,{{c}_{i,2k+1}}),\qquad {{d}_{i}}=({{d}_{i,1}},{{d}_{i,2}}, \ldots ,{{d}_{i,2k+1}}),\quad i=1,2,\ldots ,n. $$

Since C is a self-orthogonal code,

$$ c\cdot d=\sum_{j=1}^{n}c_{j}d_{j}= \sum_{i=1}^{n} \sum _{j=1}^{2k+1}c_{i,j}d_{i,j}e_{i}= \sum_{i=1}^{2k+1}{c_{i}{d_{i}}^{T}}e_{i}=0. $$

So,

$$ c_{i}{d_{i}}^{T}=0, \quad i=1,2,\ldots ,2k+1. $$

Then,

$$\begin{aligned} {{\varphi }_{k}}(c)\cdot {{\varphi }_{k}}(d)&={{ \varphi }_{k}}(c){{ \varphi }_{k}} {{(d)}^{T}} \\ &=\sum_{i=1}^{n}{{{c}_{i}}B{{B}^{T}}d_{i}^{T}=} \sum_{i=1}^{n}{{{c}_{i}} \lambda E_{2k+1}d_{i}^{T}}=\lambda \sum _{i=1}^{n}{{{c}_{i}}d_{i}^{T}}=0. \end{aligned}$$

So $\varphi _{k}(C)$ is a self-orthogonal code over ${\mathbb{F}}_{q}$. □

Lemma 5

Let C be a constacyclic code over ${\mathbb{F}}_{q}$, the generator polynomial is $g(x)$. Then, C contains its dual code if and only if $x^{n}-\lambda \equiv 0(\operatorname{mod} g(x)g^{*}(x))$, where $g^{*}(x)$ is the reciprocal polynomial of $g(x)$, $\lambda =\pm 1$.

Proof

Let $C^{\perp}=\langle f^{*}(x)\rangle$ be the dual code of C, where $f(x)=(x^{n}-\lambda )/g(x)$, $\lambda =\pm 1$. C contains its dual code if and only if there exists $h(x)\in {\mathbb{F}}_{q}[x]$, such that $f^{*}(x)=g(x)h(x)$ if and only if $g^{*}(x)g(x)=\frac{\lambda (x^{n}-\lambda ^{-1})}{f^{*}(x)}g(x)= \frac{\lambda (x^{n}-\lambda ^{-1})}{g(x)h(x)}g(x)= \frac{\lambda (x^{n}-\lambda )}{h(x)}$ if and only if $(x^{n}-\lambda )=\lambda ^{-1}g^{*}(x)g(x)h(x)\equiv 0(\mathrm{mod} g(x)g^{*}(x))$. □

Theorem 7

(CSS construction, [20])

Let $C_{1}=[n, k_{1}, d_{1}]q$ and $C_{2}=[n, k_{2}, d_{2}]q$ be linear codes over ${\mathbb{F}}_{q}$, with $C_{2}^{\perp }\subseteq C_{1}^{\perp}$. Let $d = \min{(d_{1}, d_{2})}$, then there exists a quantum error-correcting code C with parameters $C=[[n,k_{1}+k_{2}-n,\geq d]]_{q}$. In particular, if $C_{1}^{\perp }\subseteq C_{1}$, then there exists a quantum error-correcting code $C=[[n,2k_{1}-n,\geq d_{1}]]_{q}$.

Theorem 8

Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic code of length n over $S_{k}$, where $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$ is a unit in $S_{k}$. Then $C^{\perp}\subseteq C$ if and only if $x^{n}-\lambda _{i}\equiv 0(\mathrm{mod} g_{i}(x)\tilde{g_{i}}(x))$, where $g_{i}$ is the generator polynomial of $C_{i}$, $\tilde{g_{i}}(x)=\frac{1}{g_{i}(0)}g_{i}^{*}(x)=\frac{1}{g_{i}(0)}x^{ \mathrm{deg} g_{i}}g_{i}(x^{-1})$, $i=1,2,\ldots , 2k+1$.

Proof

If $x^{n}-\lambda _{i}\equiv 0(\operatorname{mod} g_{i}(x)\tilde{g_{i}}(x))$, by Lemma 5, we have $C_{i}^{\perp}\subseteq C_{i}$, $i=1,2,\ldots , 2k+1$, then $e_{i}C_{i}^{\perp}\subseteq e_{i}C_{i}$, so $C^{\perp}=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}^{\perp}\subseteq \bigoplus_{j=1}^{2k+1}e_{j}C_{j}=C$.

Conversely, let $C^{\perp}\subseteq C$, then $C^{\perp}=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}^{\perp}\subseteq \bigoplus_{j=1}^{2k+1}e_{j}C_{j}=C$, we have $C_{i}^{\perp}\subseteq C_{i}$, by Lemma 5, we have $x^{n}-\lambda _{i}\equiv 0(\mathrm{mod} g_{i}(x)\tilde{g_{i}}(x))$ $i=1,2,\ldots , 2k+1$. □

By using Lemma 5 and Theorem 8, we can have the following corollary.

Corollary 1

Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic code of length n over $S_{k}$, where $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$ is a unit in $S_{k}$. Then $C^{\perp}\subseteq C$ if and only if $C_{i}^{\perp}\subseteq C_{i}$, where $C_{i}$ is a $\lambda _{i}$-constacyclic code of length n over ${\mathbb{F}}_{q}$, $\lambda _{i}=\pm 1$, $i=1,2,\ldots , 2k+1$.

By using Theorem 7 and Theorem 8 we can have the following theorems.

Theorem 9

Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic code of length n over $S_{k}$. Let $C_{i}$ be a $\lambda _{i}$-constacyclic code of length n over ${\mathbb{F}}_{q}$, $C_{i}^{\perp}\subseteq C_{i}$, where $\lambda _{i}=\pm 1$, $i=1,2,\ldots , 2k+1$, then $C^{\perp}\subseteq C$ and there exists a quantum error-correcting code with parameters $[[(2k+1)n,2l-(2k+1)n,\geq d]]_{q}$, where d is the minimum Gray weight of code C, and l is the dimension of the linear code $\phi _{k}(C)$.

Theorem 10

Let $C=\bigoplus_{j=1}^{2k+1}e_{j}C_{j}$ be a $(\lambda _{1}e_{1}+\lambda _{2}e_{2}+\cdots +\lambda _{2k+1}e_{2k+1})$-constacyclic code of length n over $S_{k}$. Let $C_{i}$ be a $\lambda _{i}$-constacyclic code of length n over ${\mathbb{F}}_{q}$, $C_{i}^{\perp}\subseteq C_{i}$, where $\lambda _{i}=\pm 1$, $i=1,2,\ldots , 2k+1$, then $C^{\perp}\subseteq C$ and there exists a quantum error-correcting code with parameters $[[(2k+1)n,2l-(2k+1)n,\geq d]]_{q}$, where d is the minimum Gray weight of code C, and l is the dimension of the linear code $\varphi _{k}(C)$.

Example 1

Let

$$ B= \begin{bmatrix} 1&-2& 2& 0& 0 \\ -2& 1& 2& 0& 0 \\ 2& 2& 1& 0& 0 \\ 0& 0& 0& 3& 0 \\ 0& 0& 0& 0& 3\end{bmatrix} , $$

$S_{2}={\mathbb{F}}_{5}[u_{1},u_{2}]/\langle u^{3}_{1}=u_{1},u^{3}_{2}=u_{2},u_{1}u_{2}=u_{2}u_{1}=0 \rangle$, $e_{1}=\frac{u_{1}^{2}+u_{1}}{2}$, $e_{2}=\frac{u_{1}^{2}-u_{1}}{2}$, $e_{3}=\frac{u_{2}^{2}+u_{2}}{2}$, $e_{4}=\frac{u_{2}^{2}-u_{2}}{2}$, $e_{5}=1-u_{1}^{2}-u_{2}^{2}$, when $n=30$,

$$\begin{aligned}& x^{30}+1=(x+2)^{5}(x+3)^{5}(x^{2} + 2x + 4)^{5}(x^{2} + 3x + 4)^{5}, \\& x^{30}-1=(x+1)^{5}(x+4)^{5}(x^{2} + x + 1)^{5}(x^{2} + 4x + 1)^{5} \quad \text{in } {\mathbb{F}}_{5}(x). \end{aligned}$$

Let C be a $(1-2u_{2}^{2})$-constacyclic code of length 30 over $S_{2}$ with generator polynomial $e_{1}g_{1}(x)+e_{2}g_{2}(x)+e_{3}g_{3}(x)+e_{4}g_{4}(x)+e_{5}g_{5}(x)$, where $g_{1}=x+1$, $g_{2}=x+4$, $g_{3}=x+2$, $g_{4}=x+3$, $g_{5}=x+1$, then $x^{n}-1\equiv 0(\mathrm{mod} g_{i}(x)\tilde{g_{i}}(x))$, when $i=1,2,5$, $x^{n}+1\equiv 0(\mathrm{mod} g_{i}(x)\tilde{g_{i}}(x))$, when $i=3,4$. By using Theorem 8, we have $C^{\perp}\subseteq C$ and $\phi _{2}(C)$ is a linear code over ${\mathbb{F}}_{5}$ with parameters $[150,145,2]$. By Theorem 9, we know that there is a quantum error correcting code with parameters $[[150,140,\geq 2]]_{5}$.

Example 2

Let

$$ B= \begin{bmatrix} 1& 1& 1& 1& 0 \\ 1& 1&-1&-1& 0 \\ 1&-1& 1&-1& 0 \\ 1&-1&-1& 1& 0 \\ 0& 0& 0& 0& 2 \end{bmatrix} , $$

$S_{2}={\mathbb{F}}_{7}[u_{1},u_{2}]/\langle u^{3}_{1}=u_{1},u^{3}_{2}=u_{2},u_{1}u_{2}=u_{2}u_{1}=0 \rangle$, $e_{1}=\frac{u_{1}^{2}+u_{1}}{2}$, $e_{2}=\frac{u_{1}^{2}-u_{1}}{2}$, $e_{3}=\frac{u_{2}^{2}+u_{2}}{2}$, $e_{4}=\frac{u_{2}^{2}-u_{2}}{2}$, $e_{5}=1-u_{1}^{2}-u_{2}^{2}$, when $n=15$,

$$\begin{aligned}& \begin{aligned} x^{15}-1&=(x+3)(x+5)(x+6)(x^{4}+x^{3}+x^{2}+x+1) \\ &\quad {}\times (x^{4}+2x^{3}+4x^{2}+x+2)(x^{4}+4x^{3}+2x^{2}+x+4), \end{aligned} \\& \begin{aligned} x^{15}+1&=(x+1)(x+2)(x+4)(x^{4}+3x^{3}+2x^{2}+6x+4) \\ &\quad {}\times (x^{4}+5x^{3}+4x^{2}+6x+2)(x^{4}+6x^{3}+x^{2}+6x+1). \end{aligned} \end{aligned}$$

Let C be a $(1-2u^{2}_{1}-u^{2}_{2})$-constacyclic code of length 15 over $S_{2}$ with generator polynomial $e_{1}g_{1}(x)+e_{2}g_{2}(x)+e_{3}g_{3}(x)+e_{4}g_{4}(x)+e_{5}g_{5}(x)$, where $g_{1}=x^{4}+3x^{3}+2x^{2}+6x+4$, $g_{2}=x^{4}+5x^{3}+4x^{2}+6x+2$, $g_{3}=g_{4}=x^{4}+6x^{3}+x^{2}+6x+1$, $g_{5}=x^{4}+x^{3}+x^{2}+x+1$. By using Theorem 8, we have $C^{\perp}\subseteq C$ and $\varphi _{2}(C)$ is a linear code over ${\mathbb{F}}_{7}$ with parameters $[85,65,4]$. By Theorem 10, we know that there is a quantum error correcting code with parameters $[[85,45,\geq 4]]_{7}$.

Example 3

Let

$$ A= \begin{bmatrix} 1& 0& 1 \\ 1& 0&-1 \\ 0& 1& 0\end{bmatrix} , $$

$n=3$ and $S_{1}={\mathbb{F}}_{7}[u_{1}]/\langle u^{3}_{1}=u_{1}\rangle$, $e_{1}=\frac{u_{1}^{2}+u_{1}}{2}$, $e_{2}=\frac{u_{1}^{2}-u_{1}}{2}$, $e_{3}=1-u_{1}^{2}$, $x^{3}+1=(x+1)(x+2)(x+4)$, $x^{3}-1=(x+3)(x+5)(x+6)$.

Let C be a $(2u^{2}_{1}-1)$-constacyclic code of length 3 over $S_{1}$ with generator polynomial $e_{1}g_{1}(x)+e_{2}g_{2}(x)+e_{3}g_{3}(x)$, where $g_{1}=x+3$, $g_{2}=x+5$, $g_{3}=x+4$. By Theorem 8, we have $C^{\perp}\subseteq C$, and $\phi _{1}(C)$ is a linear code over ${\mathbb{F}}_{7}$ with parameters $[9,6,2]$. By Theorem 9, we know that there is a quantum error correcting code with parameters $[[9,3,\geq 2]]_{7}$.

In Table 1, we provide some new quantum codes $[[n,l,d]]_{q}$ (in the sixth column) and compare the constructed codes $[[n',l',d']]_{q}$ (in the seventh column) better (by means of larger code rate or larger distance) than the existing references [13, 16, 17]. Further, the first column represents the length n, the second column is parameter k for $S_{k}$, the third column gives the value of units $(\lambda _{1},\ldots , \lambda _{2k+1})$, the fourth column gives the generator polynomials $\langle g_{1}(x),\ldots ,g_{2k+1}(x)\rangle$, where $g_{i}(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}$ is denoted by $a_{n}a_{n-1}\cdots a_{1}a_{0}$, e.g., 112 represents the polynomial $x^{2}+x+2$, the fifth column gives parameters of $\varphi _{k}(C)$.

Table 1 New Quantum codes over $S_{k}$

6 Conclusion

In this paper, we study the structure of constacyclic codes over the non-chain rings $S_{k}={\mathbb{F}}_{q}[u_{1},u_{2},\ldots ,u_{k}]/\langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 \rangle$, and apply the CSS construction on Gray images of dual containing constacyclic codes to obtain some new quantum codes improving the existing codes that appeared in some recent references.