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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 [210] and many good quantum codes constructed by using cyclic codes over finite rings [1114]. 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. 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. 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.

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All data generated or analysed during this study are included in this published article.

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Acknowledgements

The authors would like to thank the referees and the editor for their careful reading the paper and valuable comments and suggestions, which improved the presentation of this manuscript.

Funding

This work was supported by the Key Technologies Research and Development Program of Henan Province (No. 212102210573) and Zhengzhou Special Fund for Basic Research and applied basic research (No. ZZSZX202111).

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Kong, B., Zheng, X. Quantum codes from constacyclic codes over \(S_{k}\). EPJ Quantum Technol. 10, 3 (2023). https://doi.org/10.1140/epjqt/s40507-023-00160-7

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MSC

  • 94B05
  • 94B15
  • 94B60

Keywords

  • Constacyclic codes
  • Quantum codes
  • Gray map
  • Dual-containing codes
  • CSS construction