Quantum codes from cyclic codes over the ring ${\mathbb {F}}_q+v_1{\mathbb {F}}_q+\cdots +v_r{\mathbb {F}}_q$

Abstract

Let $R = {{\mathbb {F}}_q} + {v_1}{{\mathbb {F}}_q} + \cdots + {v_r}{{\mathbb {F}}_q},$ where q is a power of a prime, $v_i^2=v_i,\; v_iv_j=v_jv_i=0$ for $1\le i,j \le r$ and $r\ge 1$. In this paper, the structure of cyclic codes over the ring R is studied and a Gray map $\phi$ from ${R^n}$ to ${\mathbb {F}}_q^{(r + 1)n}$ is given. We give a construction of quantum codes from cyclic codes over the ring R. We derive Euclidean dual containing codes over ${\mathbb {F}}_q$ and Hermitian dual containing codes over ${\mathbb {F}}_{p^{2m}}$ as Gray images of cyclic codes over R. In particular, we use $r+1$ codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code. Furthermore, some new non-binary quantum codes are obtained.

1 Introduction

Quantum error-correcting codes were used in quantum communication and quantum computation to protect quantum information from errors due to the decoherence and other quantum noise. Quantum error-correcting codes provided an efficient way to overcome decoherence. After the great discovery in [7, 25], the construction of quantum error-correcting codes from classical cyclic codes and their generalizations over the finite field ${\mathbb {F}}_q$ has developed rapidly.

Cyclic codes form an important class of linear codes due to their good algebraic structures in coding theory and decoding theory. There are some papers on quantum codes construction from cyclic codes (see [1,2,3,4, 10, 15,16,17,18, 21,22,23]).

The study of coding theory over finite commutative rings was first started in 1970s. Hammons et al. [12] proved that important families of binary non-linear codes were in fact images under a Gray map of linear codes over ${\mathbb {Z}}_4$. Codes over finite commutative rings have been developed rapidly after the study of Hammons et al., such as [1,2,3, 8, 10, 11, 15, 21, 24].

As an application, codes over finite commutative rings can be used to construct quantum codes. Qian et al. [23] provided a construction for quantum error-correcting codes starting from cyclic codes over finite chain ring ${\mathbb {F}}_2+u{\mathbb {F}}_2$, where $u^2=0$. Later, a construction of quantum codes from cyclic codes of odd length over finite chain ring ${\mathbb {F}}_4+u{\mathbb {F}}_4$ was given by Kai and Zhu [15]. In [24], Qian presented a new method of constructing quantum codes from cyclic codes over finite non-chain ring ${\mathbb {F}}_2+v{\mathbb {F}}_2$, where $v^2=v$. Motivated by [24], Ashraf and Mohammad [1] gave a construction of quantum codes from cyclic codes over finite non-chain ring ${\mathbb {F}}_3+v{\mathbb {F}}_3$, where $v^2=1$. Furthermore, quantum codes from cyclic codes over finite non-chain rings ${\mathbb {F}}_p+v{\mathbb {F}}_p$ [2] and ${{\mathbb {F}}_q} + u{{\mathbb {F}}_q} + v{{\mathbb {F}}_q} + uv{{\mathbb {F}}_q}$ with $u^2=u, v^2=v,uv=vu$ [3] were also studied by Ashraf and Mohammad. Gao [10] investigated quantum codes from cyclic codes over the ring ${\mathbb {F}}_q+v {\mathbb {F}}_q+v^2{\mathbb {F}}_q+v^3{\mathbb {F}}_q$, where $q=p^r$, p is a prime, $3|(p-1)$ and $v^4=v$. Further, u-constacyclic codes over ${\mathbb {F}}_p+u {\mathbb {F}}_p$ with $u^2=1$ and their applications of constructing new non-binary quantum codes were studied by Gao and Wang. Recently, Özen et al. [21] established a construction for quantum codes from cyclic codes over finite non-chain ring ${{\mathbb {F}}_3} + u{{\mathbb {F}}_3} + v{{\mathbb {F}}_3} + uv{{\mathbb {F}}_3}$, where $u^2=1$, $v^2=1$ and $uv=vu$. At the same time, Dertli et al. [8] studied the structure of cyclic and quasi-cyclic codes over the finite ring ${\mathbb {F}}_2+v_1{\mathbb {F}}_2+\cdots +v_r{\mathbb {F}}_2$, where $v_i^2=v_i,\; v_iv_j=v_jv_i=0$ for $1\le i,j \le r$ and $r\ge 1$.

In this paper, motivated by the previous work [8, 21], we study quantum codes construction from cyclic codes over the finite ring ${\mathbb {F}}_q+v_1{\mathbb {F}}_q+\cdots +v_r{\mathbb {F}}_q$, where $v_i^2=v_i,\; v_iv_j=v_jv_i=0$ for $1\le i,j \le r$ and $r\ge 1$.

The paper is organized as follows. In Sect. 2, the structure of cyclic codes ${\mathbf {C}}$ over the ring R is studied and a Gray map $\phi$ from ${R^n}$ to ${\mathbb {F}}_q^{(r + 1)n}$ is given. Furthermore, the relationship between ${\mathbf {C}}$ and $\phi ({\mathbf {C}})$ is studied. Section 3 gives a method to derive Euclidean dual containing codes over ${\mathbb {F}}_q$ as Gray images of cyclic codes over R. A necessary and sufficient condition for cyclic codes over R to be Euclidean dual containing is presented. From these linear codes, we obtain some new non-binary quantum codes. A construction of Hermitian dual containing codes over ${\mathbb {F}}_{p^{2m}}$ as Gray images of cyclic codes over R is introduced in Sect. 4. Some new non-binary quantum codes are obtained.

2 Cyclic codes over the finite ring ${\mathbb {F}}_q+v_1{\mathbb {F}}_q+\cdots +v_r{\mathbb {F}}_q$

In this section, we study the structure of cyclic codes over the ring R and give a Gray map $\phi$ from ${R^n}$ to ${\mathbb {F}}_q^{(r + 1)n}$. Furthermore, the relationship between ${\mathbf {C}}$ and $\phi ({\mathbf {C}})$ is studied, where ${\mathbf {C}}$ is a linear code of length n over R. The notations ${\mathbf {C}}$ and C denote codes over rings and fields, respectively, in the whole paper.

Let ${\mathbb {F}}_q$ be a finite field of cardinality q, where q is a power of a prime. Let $R = {{{{\mathbb {F}}_q}[{v_1},{v_2}, \ldots ,{v_r}]}\big / {\left\langle {v_i^2 - {v_i},{v_i}{v_j} = {v_j}{v_i}}=0 \right\rangle }} = {{\mathbb {F}}_q} + {v_1}{{\mathbb {F}}_q} + \cdots + {v_r}{{\mathbb {F}}_q},$ where $1 \le i,j \le r$ and $r\ge 1$. It is clear that R is a finite commutative ring with $q^{r+1}$ elements. This ring is a semi-local ring. There are $r+1$ maximal ideals of R. For any element $\alpha$ of R, we have $\alpha =\alpha _0+\alpha _1 v_1+\alpha _2 v_2+\cdots +\alpha _r v_r$, where $\alpha _i \in {\mathbb {F}}_q$ for $0 \le i \le r$. By the Chinese Remainder Theorem, we have that any element $\alpha \in R$ can be expressed uniquely as

$$\begin{aligned} \alpha =(1 - {v_1} - {v_2} - \cdots - {v_r})\beta _0+\beta _1 v_1+\beta _2 v_2+\cdots +\beta _r v_r, \end{aligned}$$

where $\beta _0, \beta _1, \ldots , \beta _r\in {\mathbb {F}}_q$.

Let $R^n$ be the set of n-tuples over R, i.e., ${R^n} = \{ ({c_0},{c_1}, \ldots ,{c_{n - 1}})|{c_i} \in R,\;i = 0,1, \ldots ,n - 1\} ,$ where n is a positive integer. A linear code ${\mathbf {C}}$ of length n over R is defined to be an R-submodule of ${R^n}$. For a linear code ${\mathbf {C}}$ of length n over R, if it is invariant with respect to the cyclic shift operator $\sigma$, which maps the element $({c_0},{c_1}, \ldots ,{c_{n - 1}})$ of ${R^n}$ to the element $({c_{n - 1}},{c_0}, \ldots ,{c_{n - 2}})$, then we call ${\mathbf {C}}$ a cyclic code. Let ${\mathbf {C}}$ be a cyclic code of length n over R. It is not difficult to see that ${\mathbf {C}}$ is an ideal of ${\mathcal {R}}={{R[x]} \big / {\left\langle {{x^n} - 1} \right\rangle }}$.

Let ${\mathbf {C}}$ be a linear code of length n over R. Let $e_0=1 - {v_1} - {v_2} - \cdots - {v_r}$, $e_1=v_1$, $\ldots$, $e_r=v_r$. It is not difficult to verify that $e_ie_j=0$ for $i\ne j$, $e_ie_i=e_i$ and $e_0+e_1+\cdots +e_r=1$ in R, where $i=0,1,\ldots ,r$. This implies that $e_i$ is an idempotent in R and $R ={e_0}R \oplus {e_1}R \oplus \cdots \oplus {e_r}R= {e_0}{{\mathbb {F}}_q} \oplus {e_1}{{\mathbb {F}}_q} \oplus \cdots \oplus {e_r}{{\mathbb {F}}_q}.$ Then ${\mathbf {C}}$ can be uniquely expressed as

$$\begin{aligned} {\mathbf {C}} = e_0{C_0}+ e_1{C_1} + \cdots + e_r{C_{r }}, \end{aligned}$$

(1)

where $C_0, C_1, \ldots , C_r$ are linear codes of length n over ${\mathbb {F}}_q$.

For any $a=a_0+a_1v_1+\cdots +a_rv_r=e_0a(0)+e_1a(1)+\cdots +e_ra(r)\in R$, we identify the element a with the vector $\mathbf{a }$, i.e., $\mathbf{a } =(a(0),a(1),\ldots ,a(r))$. Let $GL_{r+1}({\mathbb {F}}_q)$ be the set of all $(r+1)\times (r+1)$ invertible matrices over ${\mathbb {F}}_q$. We define a Gray map

$$\begin{aligned} \begin{array}{l} \phi :\;R \rightarrow {\mathbb {F}}_q^{r + 1}\\ \mathbf{a }= (a(0),a(1), \ldots ,a(r)) \mapsto (a(0),a(1), \ldots ,a(r))M, \end{array} \end{aligned}$$

where $M\in GL_{r+1}({\mathbb {F}}_q)$. Clearly, $\phi$ is an ${\mathbb {F}}_q$-module isomorphism. For simplicity, we abbreviate the vector $(a(0),a(1), \ldots ,a(r))M$ as $\mathbf{a }M$ in the rest of this paper. Similarly, the Gray map $\phi$ can be extended to map from $R^n$ to ${\mathbb {F}}_q^{(r+1)n}$ as follows

$$\begin{aligned} \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\, \phi :\;{R^n} \rightarrow {\mathbb {F}}_q^{(r + 1)n}\\ (\mathbf{a } _0,\mathbf{a } _1, \ldots ,\mathbf{a } _{n-1}) \mapsto (\mathbf{a } _0M,\mathbf{a }_1 M, \ldots ,{\mathbf{a }} _{n-1}M). \end{array} \end{aligned}$$

For any element $\mathbf{a } =(a(0),a(1),\ldots ,a(r))\in R$, we denote the Hamming weight $wt_H(\mathbf{a } M)$ of $\mathbf{a } M$ as the Gray weight of $\mathbf{a }$, i.e., $wt_G(\mathbf{a })=wt_H(\mathbf{a } M)$. The Gray weight of any element $(\mathbf{a } _0,\mathbf{a } _1, \ldots ,\mathbf{a } _{n-1})\in R^n$ is defined to be the integral $\sum \limits _{i = 0}^{n-1} {{wt_G}(\mathbf{a } _i)}$. Let ${\mathbf {C}}$ be a linear code of length n over R, the Gray distance of $c_1, c_2 \in {\mathbf {C}}$ be defined to be ${d_G}({c_1},{c_2}) = w{t_G}({c_1} - {c_2}).$ The minimum Gray distance $d_G$ of ${\mathbf {C}}$ is defined as $d_G({\mathbf {C}}) = \min \{ wt_G(c)|0 \ne c \in {\mathbf {C}}\} .$

According to the above definitions, we have that $\phi ({\mathbf {C}})$ is a linear code of length $(r+1)n$ over ${\mathbb {F}}_q$. Furthermore, for any $c=(a_0,a_1,\ldots ,{a_{n-1}}), c_1,c_2 \in {\mathbf {C}}$, we have

$$\begin{aligned} w{t_G}(c)= & {} \sum \limits _{i = 0}^{n - 1} {w{t_G}({a_i})} = \sum \limits _{i = 0}^{n - 1} {w{t_H}({a_i}M)} = w{t_H}(cM),\\ {d_G}({c_1},{c_2})= & {} w{t_G}({c_1} - {c_2}) = w{t_H}(({c_1} - {c_2})M) = w{t_H}({c_1}M - {c_2}M)\\= & {} {d_H}({c_1}M,{c_2}M). \end{aligned}$$

This implies that the Gray map $\phi$ is a weight and distance preserving map from $R^n$ (Gray weight or Gray distance) to ${\mathbb {F}}_q^{(r + 1)n}$ (Hamming weight or Hamming distance).

Lemma 1

Let $\mathbf{a }=(a(0),a(1),\ldots ,a(r))\in R$. Then $\mathbf{a }$ is a unit if and only if $a(i) \ne 0\;(\bmod \;p)$ for $0\le i \le r$.

Proof

By the Chinese Remainder Theorem, we have that $\mathbf{a }$ is a unit over R$\Longleftrightarrow$$a(0), a(1), \ldots , a(r)$ are units over ${\mathbb {F}}_q$$\Longleftrightarrow$$a(i) \ne 0\;(\bmod \;p)$ for $0\le i \le r$. The proof is completed. $\square$

According to the decomposition (1) of ${\mathbf {C}}$ and the notations given above, we give an explicitly expression of $C_i$ as follows. Define

$$\begin{aligned} {C_i} = \{ {c_i} \in {\mathbb {F}}_q^n|\exists \;{c_0}, \ldots ,{c_{i - 1}},{c_{i + 1}}, \ldots ,{c_r} \in {\mathbb {F}}_q^n,\;{e_0}{c_0} + {e_1}{c_1} + \cdots + {e_r}{c_r} \in {\mathbf {C}}\} \end{aligned}$$

for any $0\le i \le r$. Then $C_i$ is a linear code of length n over ${\mathbb {F}}_q$ such that

$$\begin{aligned} {\mathbf {C}} ={e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}. \end{aligned}$$

Let G be the generator matrix of ${\mathbf {C}}$, then

$$\begin{aligned} G = \left( \begin{array}{l} {e_0}{G_0}\\ {e_1}{G_1}\\ \;\;\; \vdots \\ {e_r}{G_r} \end{array} \right) , \end{aligned}$$

(2)

where $G_0,G_1,\ldots ,G_r$ are generator matrices of linear codes $C_1, C_2, \ldots , C_r$, respectively.

According to the above discussion, we have the following results.

Lemma 2

Let ${\mathbf {C}}$ be a linear code over R with length n, dimension $\sum \nolimits _{i = 0}^r {{k_i}}$ and minimum Gray distance $d_G$, i.e., ${\mathbf {C}}$ is a $[n,\sum \nolimits _{i = 0}^r {{k_i}} ,d_G]_R$ linear code, where $k_i$ is the dimension of $C_i$ for any $0 \le i \le r$. Then $\phi ({\mathbf {C}})$ is a $[(r+1)n,\sum \nolimits _{i = 0}^r {{k_i}} ,d_G]_{{\mathbb {F}}_q}$ linear code.

Proof

By the definition of the map $\phi$, we have $M\in GL_{r+1}({\mathbb {F}}_q)$ is a $(r+1)\times (r+1)$ invertible matrix. Since all the rows of the generator matrix G (2) of ${\mathbf {C}}$ are linear independent, we have that the dimension of $\phi ({\mathbf {C}})$ is $\sum \nolimits _{i = 0}^r {{k_i}}$. Since $\phi$ is a weight and distance preserving map from $R^n$ to ${\mathbb {F}}_q^{(r + 1)n}$, we get $\phi ({\mathbf {C}})$ is a $[(r+1)n,\sum \nolimits _{i = 0}^r {{k_i}} ,d_G]_{{\mathbb {F}}_q}$ linear code. $\square$

Lemma 3

Let ${\mathbf {C}}={e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ be a linear code of length n over R. Then ${\mathbf {C}}$ is a cyclic code over R if and only if $C_0, C_1, \ldots , C_r$ are cyclic codes over ${\mathbb {F}}_q$.

Proof

Let $({c_{i,0}},{c_{i,1}}, \ldots ,{c_{i,n - 1}}) \in {C_i}$ and ${c_j} = {\sum \nolimits _{i = 0}^r {{e_i}c} _{i,j}}$ for $0 \le j \le n-1$. Then, we have

$$\begin{aligned} ({c_0},{c_1}, \ldots ,{c_{n - 1}}) \in {\mathbf {C}}. \end{aligned}$$

Since ${\mathbf {C}}$ is a cyclic code over R, we have $({c_{n - 1}},{c_0},{c_1}, \ldots ,{c_{n - 2}}) \in {\mathbf {C}}.$ It is easy to find that

$$\begin{aligned} ({c_{n - 1}},{c_0},{c_1}, \ldots ,{c_{n - 2}}) = \sum \limits _{i = 0}^r {{e_i}({c_{i,n - 1}},{c_{i,0}},{c_{i,1}}, \ldots ,{c_{i,n - 2}})} . \end{aligned}$$

By the uniqueness presentation of decomposition of linear codes over R, we get $({c_{i,n - 1}},{c_{i,0}},$${c_{i,1}}, \ldots ,{c_{i,n - 2}}) \in {C_i}$. This implies that $C_0, C_1, \ldots , C_r$ are cyclic codes over ${\mathbb {F}}_q$.

Conversely, suppose that $C_i$ is a cyclic code over ${\mathbb {F}}_q$ for any $0\le i\le r$. Let $({c_0},{c_1}, \ldots ,{c_{n - 1}}) \in {\mathbf {C}}$, where ${c_j} = {\sum \nolimits _{i = 0}^r {{e_i}c} _{i,j}}$ for $0 \le j \le n-1$. Then $({c_{i,0}},{c_{i,1}}, \ldots ,{c_{i,n - 1}}) \in {C_i}$. It is easy to show that $({c_{n - 1}},{c_0},{c_1}, \ldots ,{c_{n - 2}}) = \sum \nolimits _{i = 0}^r {{e_i}({c_{i,n - 1}},{c_{i,0}},{c_{i,1}}, \ldots ,{c_{i,n - 2}})} \in {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}={\mathbf {C}}$, which follows that ${\mathbf {C}}$ is a cyclic code over R. $\square$

Theorem 1

Let ${\mathbf {C}}= {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ be a cyclic code of length n over R. Then there exists a polynomial $g(x)\in R[x]$, $g(x)|({x^n} - 1)$ such that ${\mathbf {C}} = \left\langle {g(x)} \right\rangle$, where $g(x) = \sum \nolimits _{i = 0}^r {{e_i}} {g_i}(x)$ and $g_i(x)$ is the generator polynomial of $C_i$ for $0\le i \le r$.

Proof

Let ${\mathcal {C}}=\langle \sum \nolimits _{i = 0}^r {{e_i}} {g_i}(x) \rangle$ be a cyclic code of length n over R, where $g_i(x)$ is the generator polynomial of $C_i$ for $0\le i \le r$. According to the definition of ${\mathbf {C}}$, we have ${\mathcal {C}}\subseteq {\mathbf {C}}$. On the other hand, since $e_iC_i=e_i {\mathcal {C}}$, we have ${\mathbf {C}}\subseteq {\mathcal {C}}$. Thus we get ${\mathbf {C}}={\mathcal {C}}=\langle \sum \nolimits _{i = 0}^r {{e_i}} {g_i}(x) \rangle$.

By the fact that $g_i(x)$ is the generator polynomial of $C_i$ for $0\le i \le r$, we have ${g_i}(x)|({x^n} - 1)$ in ${\mathbb {F}}_q[x]$. Then, there exists a polynomial $h_i(x)\in {\mathbb {F}}_q[x]$ such that ${x^n} - 1={g_i}(x){h_i}(x)$ for $0\le i \le r$. Furthermore, it is not difficult to verify that $\left( {\sum \nolimits _{i = 0}^r {{e_i}} {g_i}(x)} \right) \left( {\sum \nolimits _{i = 0}^r {{e_i}} {h_i}(x)} \right) = {x^n} - 1$. This implies that $g(x)=\sum \nolimits _{i = 0}^r {{e_i}} {g_i}(x)|({x^n} - 1)$. $\square$

With the notations and results above, we have the following lemma.

Lemma 4

Let ${\mathbf {C}}=\langle g(x) \rangle$ be a cyclic code of length n over R, where $g(x) = \sum \nolimits _{i = 0}^r {{e_i}} {g_i}(x)$, $g_i(x)$ is the generator polynomial of $C_i$ and $\deg ({g_i}(x)) = {t_i}$ for $0\le i \le r$. Then, we have $|{\mathbf {C}}|=|\phi ({\mathbf {C}})|=q^{(r+1)n-(t_0+\cdots +t_r)}$.

3 Quantum codes from Euclidean dual containing codes

In this section, we provide a method to derive Euclidean dual containing codes over ${\mathbb {F}}_q$ as Gray images of cyclic codes over R. From these linear codes, some new non-binary quantum codes are obtained.

For any $x=(x_0, x_1, \ldots , x_{n-1}), y=(y_0, y_1, \ldots , y_{n-1})\in R^n$, the Euclidean inner product is defined as

$$\begin{aligned} x \cdot y = \sum \limits _{i = 0}^{n - 1} {{x_i}{y_i}} \end{aligned}$$

(3)

For a linear code ${\mathbf {C}}$ of length n over R, its Euclidean dual code ${\mathbf {C}}^{\bot _E}$ is defined as

$$\begin{aligned} {\mathbf {C}}^{\bot _E} = \{ x \in {R^n}|x \cdot c = 0\;\forall \;c \in {\mathbf {C}}\} . \end{aligned}$$

Furthermore, ${\mathbf {C}}$ is said to be self-orthogonal if ${\mathbf {C}}\subseteq {\mathbf {C}}^{\bot _E}$, dual containing if ${\mathbf {C}}^{\bot _E}\subseteq {\mathbf {C}}$ and self-dual if ${\mathbf {C}}={\mathbf {C}}^{\bot _E}$.

Theorem 2

Let ${\mathbf {C}}= {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ be a linear code of length n over R. Then

$$\begin{aligned} {\mathbf {C}}^{{ \bot _E}} = {e_0}C_0^{{ \bot _E}} \oplus {e_1}C_1^{{ \bot _E}} \oplus \cdots \oplus {e_r}C_r^{{ \bot _E}} \end{aligned}$$

(4)

Furthermore, ${\mathbf {C}}$ is Euclidean self-dual over R if and only if $C_0, C_1, \ldots , C_r$ are Euclidean self-dual over ${\mathbb {F}}_q$.

Proof

By the decomposition (4) of ${\mathbf {C}}^{{ \bot _E}}$ and the notations given above, we give a explicitly expression of $C_i^{{ \bot _E}}$ as follows. Define

$$\begin{aligned} {\mathfrak {C}}_i^{{ \bot _E}} = \{ {x_i} \in {\mathbb {F}}_q^n|\exists \;{x_0}, \ldots ,{x_{i - 1}},{x_{i + 1}}, \ldots ,{x_r} \in {\mathbb {F}}_q^n,\;{e_0}{x_0} + {e_1}{x_1} + \cdots + {e_r}{x_r} \in {C^{{ \bot _E}}}\} \end{aligned}$$

for any $0\le i \le r$. According to the definitions of $e_i$ and ${\mathfrak {C}}_i^{{ \bot _E}}$, we have ${\mathbf {C}}^{{ \bot _E}}=e_0{\mathfrak {C}}_0^{{ \bot _E}} \oplus e_1{\mathfrak {C}}_1^{{ \bot _E}}\oplus \cdots \oplus e_r{\mathfrak {C}}_r^{{ \bot _E}}$. It is obvious that ${\mathfrak {C}}_i^{{ \bot _E}} \subseteq C_i^{{ \bot _E}}$ for any $0\le i \le r$. Let $x_i\in C_i^{{ \bot _E}}$, for any $c_i\in C_i$, there exist $c_0, \ldots , c_{i-1}, c_{i+1}, \ldots , c_{r}\in {\mathbb {F}}_q^n$ such that $x_i\cdot (e_0c_0+\cdots +e_rc_r)=0$. With the uniqueness presentation of decomposition of linear codes over R, we have $x_i\in {\mathfrak {C}}_i^{{ \bot _E}}$, i.e., $C_i^{{ \bot _E}}\subseteq {\mathfrak {C}}_i^{{ \bot _E}}$. Then we get ${{\mathbf {C}}^{{ \bot _E}}} = {e_0}C_0^{{ \bot _E}} \oplus {e_1}C_1^{{ \bot _E}} \oplus \cdots \oplus {e_r}C_r^{{ \bot _E}}.$

If $C_0, C_1, \ldots , C_r$ are Euclidean self-dual over ${\mathbb {F}}_q$, then we have ${\mathbf {C}}$ is Euclidean self-dual over R. On the other hand, if ${\mathbf {C}}$ is Euclidean self-dual over R, then $C_i$ is self-orthogonal, i.e., $C_i\subseteq C_i^{{ \bot _E}}$. In fact, we have $C_i= C_i^{{ \bot _E}}$. Otherwise, there exists an element $x_i\in C_i^{{ \bot _E}}\backslash {C_i}$ and $x_j\in C_j$ for $i\ne j$ such that $({e_0}{x_0} + {e_1}{x_1} + \cdots + {e_r}{x_r} )^2\ne 0$. This contradicts the fact that ${\mathbf {C}}$ is Euclidean self-dual over R. Thus, we have $C_0, C_1, \ldots , C_r$ are Euclidean self-dual over ${\mathbb {F}}_q$. $\square$

According to Lemma 3 and Theorem 2, we get the following lemma directly.

Lemma 5

Let ${\mathbf {C}}= {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ be a cyclic code of length n over R. Then its Euclidean dual code ${\mathbf {C}}^{\bot _E}$ is also a cyclic code of length n over R.

For any polynomial $f(x)\in {\mathbb {F}}_q[x]$ of degree n, the reciprocal of f(x) is defined as ${f^ *(x) } = {x^n}f(x^{-1})$. Let C be a cyclic code of length n over ${\mathbb {F}}_q$ with generator polynomial $g(x)\in {\mathbb {F}}_q[x]$ and check polynomial $h(x) = {{({x^n} - 1)} \big / {g(x)}}$. Let $C^{\bot _E}$ be the Euclidean dual code of C. Then, according to [19, Theorem 4], we have the following result.

Lemma 6

The Euclidean dual code $C^{\bot _E}$ is cyclic and has generator polynomial

$$\begin{aligned} {g^{{ \bot _E}}}(x) = {x^{\deg (h(x))}}h({x^{ - 1}}) = {h^ * }(x). \end{aligned}$$

With the notations above, by Lemma 6, we give the ideal presentation of the Euclidean dual code ${\mathbf {C}}^{\bot _E}$ of ${\mathbf {C}}$ in the following theorem.

Theorem 3

Let ${\mathbf {C}}=\left\langle {{e_0}{g_0}(x) + {e_1}{g_1}(x) + \cdots + {e_r}{g_r}(x)} \right\rangle$ be a cyclic code of length n over R. Then

$$\begin{aligned} {{\mathbf {C}}^{{ \bot _E}}} = \left\langle {{e_0}h_0^ * (x) + {e_1}h_1^ * (x) + \cdots + {e_r}h_r^ * (x)} \right\rangle \end{aligned}$$

and $|{{\mathbf {C}}^{{ \bot _E}}}|=q^{\sum \nolimits _{i = 0}^r {\deg {g_i}(x)} }$, where $g_i(x)$ is the generator polynomial of $C_i$ and $h_i(x) = {{({x^n} - 1)} \big / {g_i(x)}}$ for $0\le i \le r$.

Proof

Let $C'=\left\langle {{e_0}h_0^ * (x) + {e_1}h_1^ * (x) + \cdots + {e_r}h_r^ * (x)} \right\rangle$. According to the definition of the Euclidean inner product (3) and Lemma 6, it is clear that

$$\begin{aligned} \begin{array}{l} ({e_0}{g_0}(x) + {e_1}{g_1}(x) + \cdots + {e_r}{g_r}(x)) \cdot {({e_0}h_0^ * (x) + {e_1}h_1^ * (x) + \cdots + {e_r}h_r^ * (x))^ * }\\ \;\; = \sum \limits _{i = 0}^r {{e_i}e_i^ * } {g_i}(x){h_i}(x) = \sum \limits _{i = 0}^r {{e_i}e_i^ * } ({x^n} - 1) = 0. \end{array} \end{aligned}$$

Then, we have $C' \subseteq {{\mathbf {C}}^{{ \bot _E}}}$. It is not difficult to show that $|C'|=q^{\sum \nolimits _{i = 0}^r {\deg {g_i}(x)} }$. As, by Lemma 4, $|{\mathbf {C}}|=q^{(r+1)n-\sum \nolimits _{i = 0}^r {\deg {g_i}(x)}}$, we have $|C'|=|{{\mathbf {C}}^{{ \bot _E}}}|$. This implies that ${{\mathbf {C}}^{{ \bot _E}}}=C'$. $\square$

Lemma 7

Let $C=\langle g'(x) \rangle$ be a cyclic code of length n over ${\mathbb {F}}_q$ and ${C^{{ \bot _E}}}=\langle h'^*(x) \rangle$, where $h'(x)={{({x^n} - 1)} \big / {g'(x)}}$ in ${\mathbb {F}}_q[x]$. Then ${C^{{ \bot _E}}} \subseteq C$ if and only if

$$\begin{aligned} {x^n} - 1 \equiv 0\;(\bmod \;h'(x){h'^ * }(x)). \end{aligned}$$

Proof

If ${C^{{ \bot _E}}} \subseteq C$, then we have $g'(x)|{h'^*}(x)$. Thus, there exists a polynomial $k(x)\in {\mathbb {F}}_q[x]$ such that ${h'^*}(x)=g'(x)k(x)$. As $h'(x){h'^*}(x)=h'(x)g'(x)k(x)=(x^n-1)k(x)$, we have ${x^n} - 1 \equiv 0\;(\bmod \;h'(x){h'^ * }(x))$.

If ${x^n} - 1 \equiv 0\;(\bmod \;h'(x){h'^ * }(x))$, then there exists a polynomial $k(x)\in {\mathbb {F}}_q[x]$ such that $h'(x){h'^ * }(x)=({x^n} - 1)k(x)=g'(x)h'(x)k(x)$. This implies that ${h'^ * }(x)\in \langle g'(x) \rangle =C$. Since ${h'^ * }(x)$ is the generator polynomial of ${C^{\bot _E}}$, we have ${C^{{ \bot _E}}}\subseteq C$. $\square$

The following theorem gives a necessary and sufficient condition cyclic codes over R to be Euclidean dual containing.

Theorem 4

Let ${\mathbf {C}}=\left\langle {{e_0}{g_0}(x),{e_1}{g_1}(x), \ldots ,{e_r}{g_r}(x)} \right\rangle$ be a cyclic code of length n over R and ${{\mathbf {C}}^{{ \bot _E}}} = \left\langle {{e_0}h_0^ * (x), {e_1}h_1^ * (x), \cdots , {e_r}h_r^ * (x)} \right\rangle$. Then ${\mathbf {C}}^{\perp _E} \subseteq {\mathbf {C}}$ if and only if ${x^n} - 1 \equiv 0\;(\bmod \;h_i(x){h_i^ * }(x))$ for $0\le i \le r$.

Proof

The proof process is similar to that of [8, Theorem 4.8]. $\square$

It is not difficult to show that Euclidean dual containing cyclic codes given in Theorem 4 exist. From the Ref. [7], we have

$$\begin{aligned} {x^n} - 1 = \prod \limits _i {{p_i}(x)} \prod \limits _j {{q_j}} (x)q_j^ * (x), \end{aligned}$$

where the $p_i(x), q_j(x)$ and $q_j^ * (x)$ are all distinct and $p_i^ * (x)=\beta p_i(x)$, $\beta$ is a unit in R. Then a divisor g(x) of $x^n-1$ generates a Euclidean dual containing cyclic code if and only if g(x) is divisible by each of $p_i(x)$’s and by at least one from each $q_j(x)$, $q_j^ * (x)$ pair.

From the results given above, we have the following corollary directly.

Corollary 1

Let ${\mathbf {C}}= {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ be a cyclic code of length n over R. Then ${\mathbf {C}}$ is Euclidean dual containing, i.e., ${\mathbf {C}}^{\perp _E}\subseteq {\mathbf {C}}$ if and only if $C_i^{\perp _E} \subseteq C_i$ for $0\le i \le r$.

Let ${\mathbf {C}}$ be a Euclidean dual containing code of length n over R. With the discussion above, the relationship between ${\mathbf {C}}$ and $\phi ({\mathbf {C}})$ is given as follows.

Theorem 5

Let ${\mathbf {C}}$ be a Euclidean dual containing code of length n over R and $M\in GL_{r+1}({\mathbb {F}}_q)$ such that $MM^T=\lambda I_{r+1}$, where $M^T$ is the transpose of M, $\lambda \in {\mathbb {F}}_q^ *$, $I_{r+1}$ is a $(r+1)\times (r+1)$ identity matrix. Then $\phi ({\mathbf {C}})$ is a dual containing code of length $(r+1)n$ over ${\mathbb {F}}_q$. Furthermore, if ${\mathbf {C}}$ is Euclidean self-dual over R, then $\phi ({\mathbf {C}})$ is also Euclidean self-dual over ${\mathbb {F}}_q$.

Proof

For any $c=(c_0, c_1, \ldots , c_{n-1})$, $d=(d_0, d_1, \ldots , d_{n-1})\in \phi ({\mathbf {C}})$, there exist $x=(x_0, x_1, \ldots , x_{n-1})$, $y=(y_0, y_1, \ldots , y_{n-1})\in {\mathbf {C}}$ and $M\in GL_{r+1}({\mathbb {F}}_q)$ such that $c=(x_0M, x_1M, \ldots , x_{n-1}M)$ and $d=(y_0M, y_1M, \ldots , y_{n-1}M)$. Then, we have

$$\begin{aligned} c \cdot d = c{d^T} = \sum \limits _{j = 0}^{n - 1} {{x_j}} M{M^T}y_j^T = \sum \limits _{j = 0}^{n - 1} {{x_j}} \lambda {I_{r + 1}}y_j^T = \lambda \sum \limits _{j = 0}^{n - 1} {{x_j}} y_j^T. \end{aligned}$$

Due to ${\mathbf {C}}$ is self-orthogonal over R, it follows that $x\cdot y=\sum \nolimits _{j = 0}^{n - 1} {{x_j}} y_j^T=0$. This implies that $c\cdot d=0$, i.e., $\phi ({\mathbf {C}})$ is a self-orthogonal code of length $(r+1)n$ over ${\mathbb {F}}_q$.

Let ${\mathbf {C}}$ be a Euclidean self-dual code of length n over R satisfying the above properties. Since $\phi$ is an ${\mathbb {F}}_q$-module isomorphism, we have $|{\mathbf {C}}|=|\phi ({\mathbf {C}})|=(q^{r+1})^{{n \big / 2}}=q^{(r + 1){n \big / 2}}$. Then, we have $\phi ({\mathbf {C}})$ is Euclidean self-dual over ${\mathbb {F}}_q$. $\square$

A q-ary quantum code Q of length n and size K is a K-dimensional subspace of the $q^n$-dimensional Hilbert space $({\mathbb {C}}^q)^{\otimes n}$. Let $k={\log _q}(K)$. We use $[[n,k,d]]_q$ to denote a quantum code over ${\mathbb {F}}_q$ with length n, dimension k and minimum Hamming distance d.

According to [13, 14], we have the quantum singleton bound as follows.

Lemma 8

(Quantum Singleton Bound) Let C be an $[[n,k,d]]_q$ quantum code. Then $k\le n-2d+2$.

A quantum code $[[n,k,d]]_q$ achieving this quantum singleton bound, i.e., $n=k+2d-2$, is called a quantum MDS code. The following lemmas are useful for our results.

Lemma 9

[13, Lemma 20] (CSS Construction) Let $C_1$ and $C_2$ denote two classical linear codes with parameters $[n,k_1,d_1]_q$ and $[n,k_2,d_2]_q$ such that $C_2^ \bot \le {C_1}$. Then there exists an $[[n,k_1+k_2-n,d]]_q$ stabilizer code with minimum distance $d = \min \{ wt(c)|c \in ({C_1}\backslash C_2^ \bot ) \cup ({C_2}\backslash C_1^ \bot )\}$ that is pure to $min\{d_1, d_2\}$.

Lemma 10

[13, Corollary 21] If C is a classical linear $[n,k,d]_q$ code containing its dual, then there exists an $[[n,2k-n,\ge d]]_q$ stabilizer code that is pure to d.

According to Corollary 1, Theorem 5 and Lemmas 8 and 9, with the notations above, we have the existence of non-binary quantum error-correcting codes as follows.

Theorem 6

Let ${\mathbf {C}}={e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ be an $[n,k,d_G]_R$ cyclic code. If $C_i^ {\bot _E} \subseteq {C_i}$ for $0\le i \le r$, then ${\mathbf {C}}^ {\bot _E} \subseteq {\mathbf {C}}$ and there exists a quantum error-correcting code with parameters $[[(r+1)n,2k-(r+1)n, \ge d_G]]_q$, where $k=\sum \nolimits _{i = 0}^r {{k_i}}$ and $k_i$ is the dimension of $C_i$.

Next, we give two examples to construct quantum MDS codes over finite fields ${\mathbb {F}}_7$ and ${\mathbb {F}}_{11}$.

Example 1

Let $R={\mathbb {F}}_7+v_1{\mathbb {F}}_7$ and $n=3$. Then $x^3-1=(x+3)(x+5)(x+6)$ over ${\mathbb {F}}_7$. Let $g(x)=(1-v_1)g_0(x)+v_1g_1(x)$ with $g_0=x+5$ and $g_1=x+3$. By [6] and Lemma 6, we have that $C_0=\langle x+5 \rangle$ and $C_0^{\perp _E}=\langle 4x^2 + 2x + 1 \rangle =\langle (x+5)(4x+3) \rangle$. Then we have $C_0^{\perp _E}\subseteq C_0$. With a similar method, we have $C_1^{\perp _E}=\langle 2x^2 + 4x + 1 \rangle =\langle (x+3)(2x+5) \rangle \subseteq C_1=\langle x+3 \rangle$. Let ${\mathbf {C}}=\langle g(x) \rangle$, according to Theorem 3 and Corollary 1, we have ${\mathbf {C}}^{\perp _E} \subseteq {\mathbf {C}}$.

Let $M = \left( {\begin{array}{*{20}{c}} 6&{}2\\ 2&{}1 \end{array}} \right) \in G{L_2}({{\mathbb {F}}_7})$. Then we have $M{M^T} = \left( {\begin{array}{*{20}{c}} 5&{}0\\ 0&{}5 \end{array}} \right) .$ With the computational algebra systems Magma [6], we have that $\phi ({\mathbf {C}})$ is an [6, 4, 3] linear code over ${\mathbb {F}}_7$. By Lemma 7 and Theorem 6, we obtain a quantum MDS code with parameters $[[6,2,3]]_7$.

Example 2

Let $R={\mathbb {F}}_{11}+v_1{\mathbb {F}}_{11}$ and $n=5$. Then $x^5-1=(x+2)(x+6)(x+7)(x+8)(x+10)$ over ${\mathbb {F}}_{11}$. Let $g(x)=(1-v_1)g_0(x)+v_1g_1(x)$ with $g_0=x+6$ and $g_1=x+2$. Let $C=\langle g(x) \rangle$, $C_0=\langle x+6 \rangle$ and $C_1=\langle x+2 \rangle$. It is not difficult to verify that $C_0^{\perp _E}\subseteq C_0$ and $C_1^{\perp _E}\subseteq C_1$. By Theorem 3 and Corollary 1, we have ${\mathbf {C}}^{\perp _E} \subseteq {\mathbf {C}}$.

Let $M = \left( {\begin{array}{*{20}{c}} 10&{}2\\ 2&{}1 \end{array}} \right) \in G{L_2}({{\mathbb {F}}_{11}})$. Then we have $M{M^T} = \left( {\begin{array}{*{20}{c}} 5&{}0\\ 0&{}5 \end{array}} \right) .$ Using Magma [6], we have that $\phi ({\mathbf {C}})$ is an [10, 8, 3] linear code over ${\mathbb {F}}_{11}$. By Lemma 7 and Theorem 6, we obtain a quantum MDS code with parameters $[[10,6,3]]_{11}$.

At the last of this section, with the methods given in Examples 1 and 2, we list some new non-binary quantum codes for $r\ge 1$ and $q\le 11$ in Table 1. For simplicity, we list the coefficients of the polynomials in descending order in Table 1. For example, the polynomial $x^6+x^5+x^4+2x+3$ is represented by 1110023.

Table 1 New quantum codes $[[n,k,d]]_q$

4 Quantum codes from Hermitian dual containing codes

In this section, we introduce a construction of Hermitian dual containing codes over ${\mathbb {F}}_{p^{2m}}$ as Gray images of cyclic codes over R. Moreover, we construct some new non-binary quantum codes according to these linear codes.

Let $q=p^{2m}$, where p is a prime and m is a positive integer. In this section, we consider the ring $R = {{{{\mathbb {F}}_{p^{2m}}}[{v_1},{v_2}, \ldots ,{v_r}]} \big / {\left\langle {v_i^2 - {v_i},{v_i}{v_j} = {v_j}{v_i}} \right\rangle }} = {{\mathbb {F}}_{p^{2m}}} + {v_1}{{\mathbb {F}}_{p^{2m}}} + \cdots + {v_r}{{\mathbb {F}}_{p^{2m}}}.$ For any vectors $x=(x_0, x_1, \ldots , x_{n-1})$ and $y=(y_0, y_1, \ldots , y_{n-1})\in {\mathbb {F}}_{p^{2m}}^n$, their Hermitian inner product is defined as

$$\begin{aligned} {\left\langle {x,y} \right\rangle _H} = \sum \limits _{i = 0}^{n - 1} {{x_i}} y_i^{p^m}. \end{aligned}$$

Furthermore, x and y are called orthogonal with respect to the Hermitian inner product if ${\left\langle {x,y} \right\rangle _H}=0$.

Let ${\mathbf {C}}$ be a linear code over R. the Hermitian dual code of ${\mathbf {C}}$ is defined as

$$\begin{aligned} {{\mathbf {C}}^{{ \bot _H}}} = \{ x \in {R^n}|{\left\langle {x,c} \right\rangle _H} = 0\;\forall \;c \in {\mathbf {C}}\} . \end{aligned}$$

${\mathbf {C}}$ is called dual containing if ${\mathbf {C}}^{{ \bot _H}}\subseteq {\mathbf {C}}$ and self-dual if ${\mathbf {C}}={\mathbf {C}}^{{ \bot _H}}$. For any $c=(c_0,c_1,\ldots ,c_{n-1})\in {\mathbf {C}}$, we denote $c^{p^m}=(c_0^{p^m},c_1^{p^m},\ldots ,c_{n-1}^{p^m})$. Similarly, for any invertible matrix $M=({m_{ij}})_{0 \le i,j \le r} \in G{L_{r+1}}({{\mathbb {F}}_{p^{2m}}}),$ denote $M^{p^m}={({m_{ij}^{p^m}})}_{0 \le i,j \le r}$.

A $q^2$-cyclotomic coset modulo n containing a is defined by ${C_a} = \{ a{q^{2k}}(\bmod \;n):k \ge 0\} ,$ where a is not necessarily the least number in $C_a$. The set of $q^2$-cyclotomic cosets modulo n is denoted by ${C^{{q^2},n}}$. Let C be a cyclic code over ${\mathbb {F}}_{q^2}$. Then C is an ideal of ${{{{\mathbb {F}}_{{q^2}}}[x]} \big / {\left\langle {{x^n} - 1} \right\rangle }}$ and $C=\langle g(x) \rangle$, where $g(x)|(x^n-1)$. The defining set of C is defined as

$$\begin{aligned} Z = \{ i:g({\alpha ^i}) = 0,0 \le i < n\} , \end{aligned}$$

where $\alpha$ is a primitive nth root of unity in some extension of ${\mathbb {F}}_{q^2}$.

According to [4, 20], we have a necessary and sufficient condition for the existence of dual containing cyclic codes over ${\mathbb {F}}_{q^2}$ in the following lemma.

Lemma 11

[20, Lemma 4.1] Assume that $\gcd (q,n) = 1$. A cyclic code of length n over ${\mathbb {F}}_{q^2}$ with defining set Z contains its Hermitian dual code if and only if $Z \cap {Z^{ - q}} = \phi$, where ${Z^{ - q}} = \{ - qz (\bmod \;n): z \in Z\} .$

With the notations and properties above, we consider Hermitian dual containing cyclic codes over the ring R.

Theorem 7

Let ${\mathbf {C}}= {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ be a cyclic code of length n over R. Then

• (i) the Hermitian dual code of ${\mathbf {C}}$ is

  $$\begin{aligned} {{\mathbf {C}}}^{{ \bot _H}} = {e_0}C_0^{{ \bot _H}} \oplus {e_1}C_1^{{ \bot _H}} \oplus \cdots \oplus {e_r}C_r^{{ \bot _H}}. \end{aligned}$$

  Furthermore, $C_0^{{ \bot _H}}, C_1^{{ \bot _H}}, \ldots , C_r^{{ \bot _H}}$ are cyclic codes of length n over ${\mathbb {F}}_{p^{2m}}$ and ${\mathbf {C}}^{{ \bot _H}}$ is a cyclic code of length n over R.

• (ii) ${\mathbf {C}}$ is Hermitian self-dual over R if and only if $C_0, C_1, \ldots , C_r$ are Hermitian self-dual over ${\mathbb {F}}_{p^{2m}}$.

Proof

• (i) With a similar proof process of Theorem 2, we define $\mathcal {C}_i^{{ \bot _H}} = \{ {x_i} \in {\mathbb {F}}_{p^{2m}}^n|\exists \;{x_0},$$\ldots ,{x_{i - 1}},{x_{i + 1}}, \ldots ,{x_r} \in {\mathbb {F}}_{p^{2m}}^n,\;{e_0}{x_0} + {e_1}{x_1} + \cdots + {e_r}{x_r} \in {{\mathbf {C}}^{{ \bot _H}}}\}$ for any $0\le i \le r$. Then we have ${\mathbf {C}}^{{ \bot _H}}=e_0{\mathcal {C}}_0^{{ \bot _H}} \oplus e_1{\mathcal {C}}_1^{{ \bot _H}}\oplus \cdots \oplus e_r{\mathcal {C}}_r^{{ \bot _H}}$. It is not difficult to show that $C_i^{{ \bot _H}}={\mathcal {C}}_i^{{ \bot _H}}.$ Then we have

  $$\begin{aligned} {{\mathbf {C}}^{{ \bot _H}}} = {e_0}C_0^{{ \bot _H}} \oplus {e_1}C_1^{{ \bot _H}} \oplus \cdots \oplus {e_r}C_r^{{ \bot _H}}. \end{aligned}$$

  Since ${\mathbf {C}}= {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ is a cyclic code of length n over R, by Lemma 3, we have that $C_i$ is cyclic code of length n over ${\mathbb {F}}_{p^{2m}}$ for $0\le i \le r$. Then $C_i^{{ \bot _H}}$ is a cyclic code of length n over ${\mathbb {F}}_{p^{2m}}$. This implies that ${\mathbf {C}}^{{ \bot _H}}$ is a cyclic code of length n over R.

• (ii) If $C_0, C_1, \ldots , C_r$ are Hermitian self-dual over ${\mathbb {F}}_{p^{2m}}$, then we have ${\mathbf {C}}$ is Hermitian self-dual over R. Conversely, if ${\mathbf {C}}$ is Hermitian self-dual over R, then $C_i$ is self-orthogonal, i.e., $C_i\subseteq C_i^{{ \bot _H}}$. In fact, we have $C_i= C_i^{{ \bot _H}}$. Otherwise, there exists an element $x_i\in C_i^{{ \bot _H}}\backslash {C_i}$ and $x_j\in C_j$ for $i\ne j$ such that

  $$\begin{aligned} \langle ({e_0}{x_0} + {e_1}{x_1} + \cdots + {e_r}{x_r} ),({e_0}{x_0} + {e_1}{x_1} + \cdots + {e_r}{x_r} )\rangle _H \ne 0. \end{aligned}$$

  This leads to a contradiction. Then we have $C_0, C_1, \ldots , C_r$ are Hermitian self-dual over ${\mathbb {F}}_{p^{2m}}$.$\square$

Theorem 8

Let ${\mathbf {C}}= {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ be a cyclic code of length n over R. Let $Z_i$ be the defining set of $C_i$ and ${Z_i^{ - {p^m}} = \{ - {p^m}}z_i (\bmod \;n): z_i \in Z_i\}$ be the defining set of $C_i^{{ \bot _H}}$ for $0\le i \le r$. Then ${\mathbf {C}}^{{ \bot _H}}\subseteq {\mathbf {C}}$ if and only if $Z_i \cap {Z_i^{ - {p^m}}} = \phi$.

Proof

According to Lemma 10, if $Z_i \cap {Z_i^{ - {p^m}}} = \phi$ for $0\le i \le r$, then we have $C_i^{{ \bot _H}}\subseteq C_i$. It is obvious that $e_iC_i^{{ \bot _H}}\subseteq e_iC_i$. Then we have ${e_0}C_0^{{ \bot _H}} \oplus {e_1}C_1^{{ \bot _H}} \oplus \cdots \oplus {e_r}C_r^{{ \bot _H}}\subseteq {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$. This implies that ${\mathbf {C}}^{{ \bot _H}}\subseteq {\mathbf {C}}$.

On the other hand, if ${\mathbf {C}}^{{ \bot _H}}\subseteq {\mathbf {C}}$, then we have ${e_0}C_0^{{ \bot _H}} \oplus {e_1}C_1^{{ \bot _H}} \oplus \cdots \oplus {e_r}C_r^{{ \bot _H}}\subseteq {e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$. By taking $\bmod \;{e_0}$, $\bmod \;{e_1}, \ldots$, $\bmod \;{e_r}$, respectively, we have $C_i^{{ \bot _H}}\subseteq C_i$ for $0\le i \le r$. Then $Z_i \cap {Z_i^{ - {p^m}}} = \phi$. $\square$

Let ${\mathbf {C}}$ be a Hermitian dual containing code of length n over R. The following theorem gives the relationship between ${\mathbf {C}}$ and $\phi ({\mathbf {C}})$.

Theorem 9

Let ${\mathbf {C}}$ be a Hermitian dual containing code of length n over R and $M\in GL_{r+1}({\mathbb {F}}_{p^{2m}})$ such that $M(M^p)^T=\lambda I_{r+1}$, where $\lambda \in {\mathbb {F}}_{p^{2m}}^ *$, $I_{r+1}$ is a $(r+1)\times (r+1)$ identity matrix. Then $\phi ({\mathbf {C}})$ is a Hermitian dual containing code of length $(r+1)n$ over ${\mathbb {F}}_{p^{2m}}$. Furthermore, if ${\mathbf {C}}$ is Hermitian self-dual over R, then $\phi ({\mathbf {C}})$ is also Hermitian self-dual over ${\mathbb {F}}_{p^{2m}}$.

Proof

The proof process is similar to that of Theorem 5. $\square$

By [5, 13], we can construct quantum codes via the Hermitian construction given in the following lemma.

Lemma 12

(Hermitian Construction) If there exists a classical linear $[n,k,d]_{q^2}$ code C such that ${C^{{ \bot _H}}} \subseteq C$, then there exists an $[[n,2k-n,\ge d]]_q$ stabilizer code.

With the results given above, we obtain a method to construct quantum codes over ${\mathbb {F}}_{p}$ immediately.

Theorem 10

Let ${\mathbf {C}}={e_0}{C_0} \oplus {e_1}{C_1} \oplus \cdots \oplus {e_r}{C_r}$ be a $[n,k,d_G]_R$ cyclic code. If $C_i^ {\bot _H} \subseteq {C_i}$ for $0\le i \le r$, then ${\mathbf {C}}^ {\bot _H} \subseteq {\mathbf {C}}$ and there exists a quantum error-correcting code with parameters$[[(r+1)n,2k-(r+1)n,d_G]]_{p^m}$, where $k=\sum \nolimits _{i = 0}^r {{k_i}}$ and $k_i$ is the dimension of $C_i$.

With the notations and results above, we give some new quantum codes over ${\mathbb {F}}_{13}$ and ${\mathbb {F}}_{17}$.

Example 3

Let $R={\mathbb {F}}_{{13}^2}+v_1{\mathbb {F}}_{{13}^2}+v_2{\mathbb {F}}_{{13}^2}$ and $n=5$. Then $C_0^{169,5} = \{ 0\}$, $C_1^{169,5} = \{ 1,4\}$ and $C_2^{169,5} = \{ 2,3\}$. Let ${Z_0} = {Z_1}=\{ 1,4\}$ and ${Z_2} = \{ 2,3\}$ be the defining sets of $C_0$, $C_1$ and $C_2$, respectively. It is easy to show that $Z_0^{ - 13} = Z_1^{ - 13}=\{ 2,3\}$ and $Z_2^{ - 13} = \{ 1,4\}$. Then, we have $Z_i \cap {Z_i^{ - 13}} = \phi$ for $0\le i \le 2$, i.e., $C_i^{{ \bot _H}}\subseteq C_i$. According to Theorem 8, we have ${\mathbf {C}}={e_0}{C_0} \oplus {e_1}{C_1}\oplus {e_2}{C_2}$ is a Hermitian dual containing code over R, i.e., ${\mathbf {C}}^{{ \bot _H}}\subseteq {\mathbf {C}}$.

Let $g_0(x)=g_1(x)=(x-\rho ^1)(x-\rho ^4)=x^2 + w^{30}x + 1$ and $g_2(x)=(x-\rho ^2)(x-\rho ^3)=x^2 + w^{54}x + 1$, where w is a primitive element of ${\mathbb {F}}_{{13}^2} = {{{{\mathbb {F}}_{13}}[x]} \big / {\left\langle {{x^2} + 12x + 2} \right\rangle }}$ with $\mathrm{{ord}}(w) = {13^2} - 1 = 168$ and $\rho$ is a primitive 5th root of unity over the splitting field of $x^5 -1$ over ${\mathbb {F}}_{{13}^2}$. Then, $C_0=\langle g_0(x) \rangle$, $C_1=\langle g_1(x) \rangle$ and $C_2=\langle g_2(x) \rangle$.

Let $M = \left( {\begin{array}{*{20}{c}} 11&{}2&{}1\\ 12&{}11&{}2\\ 2&{}1&{}2 \end{array}} \right) \in G{L_2}({{\mathbb {F}}_{{13}^2}}).$ Then we have $M{(M^{13})^T} = \left( {\begin{array}{*{20}{c}} 9&{}0&{}0\\ 0&{}9&{}0\\ 0&{}0&{}9 \end{array}} \right) .$ With the computational algebra systems Magma [6], we have that $\phi ({\mathbf {C}})$ is a [15, 9, 5] linear code over ${\mathbb {F}}_{{13}^2}$. By Theorems 9 and 10, we obtain a quantum code with parameters $[[15,3,5]]_{13}$.

At the last of this example, we give some new quantum codes over ${\mathbb {F}}_{13}$ and ${\mathbb {F}}_{17}$ from cyclic codes over the ring R in Table 2, where $\eta$ is a primitive element of ${\mathbb {F}}_{{17}^2} = {{{{\mathbb {F}}_{17}}[x]} \big / {\left\langle {x^2 + 16x + 3} \right\rangle }}$ with $\mathrm{{ord}}(\eta ) = {17^2} - 1 = 288$.

Table 2 New quantum codes $[[n,k,d]]_{p^m}$