Quantum codes from $\sigma$-dual-containing constacyclic codes over $\mathfrak{R}_{l, k}$

1.   Introduction

Constructing quantum codes is an important subject in the quantum information. The CSS construction and the Hermitian construction were introduced to construct quantum codes from classical error-correcting codes in ^[1,2,3,4]. Constructing quantum codes needs Euclidean dual-containing or Hermitian dual-containing codes with respect to the Euclidean or Hermitian inner product using the CSS construction or Hermitian construction. Fan and Zhang ^[5] introduced the Galois inner products to generalize the Euclidean inner product and the Hermitian inner product. In ^[6], Hermitian LCD $\lambda$-constacyclic codes over ${\mathbb F}_{q}$ were addressed by studying the Galois inner product. Galois hulls of linear codes over finite fields were addressed by studying the Galois dual in ^[7]. In ^[8], $\sigma$-self-orthogonal constacyclic codes over ${\mathbb F}_{p^m}+u{\mathbb F}_{p^m}$ were studied by generalizing the notion of self-orthogonal codes to $\sigma$-self-orthogonal codes over an arbitrary finite ring. Fu and Liu ^[9] extended constacyclic codes to obtain Galois self-dual codes.

Now, many quantum codes have been constructed by studying the algebraic structure of cyclic and constacyclic codes over finite fields, finite chain rings and finite non-chain ring. Huang et al. ^[10] constructed quantum codes from Hermitian dual-containing codes by applying the Hermitian construction. In ^[11], some quantum codes were obtained from constacyclic over ${\mathbb F}_{q}[u, v]/\langle u^{2}-\gamma u, v^{2}-\delta v, uv = vu = 0\rangle$ by using the CSS construction. Gowdhaman et al. ^[12] studied the the structure of cyclic and $\lambda$-constacyclic codes over $\frac{{\mathbb F}_{p}[u, v]}{\langle v^{3}-v, u^{3}-u, uv-vu\rangle}$ and constructed quantum codes over ${\mathbb F}_{p}$ using the CSS construction. Islam and Prakash ^[13] obtained quantum codes from cyclic codes over a finite non-chain ring ${\mathbb F}_{q}[u, v]/\langle u^{2}-\alpha u, v^{2}-1, uv-vu\rangle$ using the CSS construction. Kong and Zheng obtained some quantum codes from constacyclic codes over ${\mathbb F}_{q}[u_1, u_2, \cdots, u_k]/\langle u^{3}_{i} = u_{i}, u_iu_j = u_ju_i\rangle$ ^[14] and ${\mathbb F}_{q}[u_1, u_2, \cdots, u_k]/\langle u^{3}_{i} = u_{i}, u_iu_j = u_ju_i = 0\rangle$ ^[15] using the CSS construction. As an application, Galois inner product can be applied in the constructions of quantum codes. Some entanglement-assisted quantum codes were obtained from Galois dual codes in ^[16,17,18].

Motivated by these works, we define a new non-chain ring $\mathfrak{R}_{l, k}$ by generalizing ^[15] and study the algebraic structure of $\sigma$-self-orthogonal and $\sigma$-dual-containing constacyclic codes over $\mathfrak{R}_{l, k}$ based on the $\sigma$-inner product. Then, we give the necessary and sufficient conditions for $\lambda$-constacyclic codes over $\mathfrak{R}_{l, k}$ to satisfy $\sigma$-self-orthogonal and $\sigma$-dual-containing. Finally, we obtain some new quantum codes from $\sigma$-dual-containing constacyclic codes over $\mathfrak{R}_{l, k}$ using the CSS construction or Hermitian construction and compare these codes better with the existing codes that appeared in some recent papers.

2.   Preliminaries

Let $R$ be a finite commutative ring, $I$ is an ideal of $R$ and $I$ is generated by one element, then $I$ is called a principal ideal. If all the ideas of $R$ are principal, $R$ is called a principal ideal ring. If $R$ has a unique maximal ideal, $R$ is called a local ring. If the ideals of $R$ are linearly ordered by inclusion, $R$ is called a chain ring.

Let $\mathfrak{R}_{l, k} = {\mathbb F}_{p^m}[u_1, u_2, \cdots, u_k]/ \langle u_{i}^{l} = u_{i}, u_iu_j = u_ju_i = 0 \rangle$, where $p$ is a prime, $l$ is a positive integer, $(l-1)\mid(p-1)$ and $1\leq i, j\leq k$. It is a commutative non-chain ring with $p^{m(l-1)k+m}$ elements. Since $(l-1)\mid(p-1)$, then

where $\alpha_i\in{\mathbb F}_{p^m}$ for $i = 1, 2, \cdots, l$.

Let

where $2\leq i\leq l$ and $1\leq j\leq k$.

We can get that $\sum_{ij}\varsigma_{ij} = 1$, $\varsigma_{ij}^2 = \varsigma_{ij}$ and $\varsigma_{ij}\varsigma_{i^{'}j^{'}} = 0$, where $(i, j)\neq (i^{'}, j^{'})$.

Let $e_1 = \varsigma_{21}, e_2 = \varsigma_{22}, \cdots, e_k = \varsigma_{2k}, \cdots, e_{(l-1)k} = \varsigma_{lk}, e_{(l-1)k+1} = \varsigma_{11}$. Let $s = (l-1)k+1$, thus, $1 = e_1+e_2+\cdots+e_{s}$, $e_i^2 = e_i$ and $e_ie_j = 0$, where $i\neq j$ and $i, j = 1, 2, \cdots, s$. By the Chinese Remainder Theorem, then

For any element $r\in \mathfrak{R}_{l, k}$, it can be expressed uniquely as

where $r_{i}\in {\mathbb F}_{p^m}$ for $i = 1, 2, \cdots, s$.

Let $\mathrm{Aut}(\mathfrak{R}_{l, k})$ be the ring automorphism group of $\mathfrak{R}_{l, k}$. If $\sigma\in \mathrm{Aut}(\mathfrak{R}_{l, k})$, then we can get a bijective map

Let $a = (a_0, a_1, \cdots, a_{n-1})$ and $b = (b_0, b_1, \cdots, b_{n-1})\in \mathfrak{R}_{l, k}^n$, the $\sigma$-inner product ^[8] of $a, b$ is defined as

When $\mathfrak{R}_{l, k} = {\mathbb F}_{p^m}$ and $\sigma$ is the identity map of ${\mathbb F}_{p^m}$, then $\sigma$-inner product is the Euclidean inner product. When $\mathfrak{R}_{l, k} = {\mathbb F}_{p^m}$ and $m$ is even, $\forall a\in {\mathbb F}_{p^m}$, if $\sigma(a) = a^{p^{\frac{m}{2}}}$, then $\sigma$-inner product is the Hermitian inner product. When $\mathfrak{R}_{l, k} = {\mathbb F}_{p^m}$ and $\forall a\in {\mathbb F}_{p^m}$, $\sigma(a) = a^{p^l}$, $0\leq l\leq m-1$, then $\sigma$-inner product is the Galois inner product ^[5].

When $\langle a, b\rangle_\sigma = 0$, $a$ and $b$ are called $\sigma$-orthogonal, for any code $C$ over $\mathfrak{R}_{l, k}$, the $\sigma$-dual code of $C$ is defined as

If $C\subseteq C^{\bot_\sigma}$, $C$ is $\sigma$-self-orthogonal, if $C^{\bot_\sigma}\subseteq C$, $C$ is $\sigma$-dual-containing, and if $C = C^{\bot_\sigma}$, $C$ is $\sigma$-self-dual.

Let $\theta_t$ be an automorphism of ${\mathbb F}_{p^m}$, $\theta_t$: ${\mathbb F}_{p^m} \rightarrow {\mathbb F}_{p^m}$ defined by $\theta_{t}(a) = a^{p^t}$, where $0\leq t\leq m-1$. We can define the automorphism of $\mathfrak{R}_{l, k}$ as follows:

Let $c = (c_0, c_1, \cdots, c_{n-1})\in \mathfrak{R}_{l, k}^{n}$ and $\lambda$ be a unit of $\mathfrak{R}_{l, k}$, then the constacyclic shift $\delta_{\lambda}$ of $c$ is defined as $\delta_{\lambda}(c) = (\lambda c_{n-1}, c_0, \cdots, c_{n-2})$. If $\delta_{\lambda}(C) = C$, $C$ is called a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$. In particular, when $\lambda = 1$, $C$ is called cyclic code and negative cyclic code when $\lambda = -1$.

Let $f(x) = a_0+a_1x+a_2x^2+\cdots+a_{r-1}x^{r-1}+x^r$ be a monic polynomial, the reciprocal polynomial of $f(x)$ is denoted by $f^*(x) = x^rf(x^{-1})$.

Each codeword $(a_0, a_1, \cdots, a_{n-1})\in\mathfrak{R}_{l, k}^n$ can be represented by a polynomial $a_0+a_1x+\cdots+a_{n-1}x^{n-1}\in\mathfrak{R}_{l, k}[x]/ \langle x^n-\lambda \rangle$,

In polynomial representation, a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$ is defined as an ideal of $\mathfrak{R}_{l, k}[x]/ \langle x^n-\lambda \rangle$.

We define a Gray map $\phi_{l, k}: \mathfrak{R}_{l, k} \rightarrow {\mathbb F}_{p^m}^{s}$ by $a = \sum_{i = 1}^{s}a_{i}e_{i}\mapsto(a_1, a_2, \cdots, a_{s})$, and we extend $\phi_{l, k}$ as

where $a_i = a_{1, i}e_1+a_{2, i}e_2+\cdots+a_{s, i}e_{s}\in \mathfrak{R}_{l, k}$, $i = 0, 1, \cdots, n-1$.

$\forall r\in \mathfrak{R}_{l, k}$, the Gray weight of $r$ is defined as $w_{G}(r) = w_{H}(\phi_{l, k}(r))$, where $w_{H}(\phi_{l, k}(r))$ is the Hamming weight of the image of $r$ under $\phi_{l, k}$.

$\forall x = (x_1, x_2, \cdots, x_{n}), y = (y_1, y_2, \cdots, y_{n})\in \mathfrak{R}_{l, k}^n$, the Gray weight of $x-y$ is defined as $w_G(x-y) = \sum_{i = 1}^{n}w_G(x_i-y_i)$, the Gray distance of $x, y$ is defined as $d_G(x, y) = w_G(x-y)$, and the Gray distance of $C$ is defined as $d_G(C) = \mathrm {min}\{d_G(a, b), a, b\in C, a\neq b\}$.

For a linear code $C$ of length $n$ over $\mathfrak{R}_{l, k}$. Let

where $j = 1, 2, \cdots, s.$

Then, $C_1, C_2, \cdots, C_{s}$ are linear codes of length $n$ over ${\mathbb F}_{p^m}$, $C = \bigoplus_{j = 1}^{s}e_jC_j$ and $\mid C \mid = \prod_{j = 1}^{s} \mid C_j \mid$.

Lemma 2.1. An element $(\lambda_1e_1+\lambda_2e_2+\cdots+\lambda_{s}e_{s})\in \mathfrak{R}_{l, k}$ is a unit in $\mathfrak{R}_{l, k}$ if and only if $\lambda_i$ is a unit in ${\mathbb F}_{p^m}$ for $i = 1, 2, \cdots, s$.

Proof. This proof is the same as Lemma 3.1 in ^[14]. □

3.   $\lambda$-constacyclic codes over $\mathfrak{R}_{l, k}$

In this section, let $\lambda = \sum_{j = 1}^{s}\lambda_je_j$ be a unit in $\mathfrak{R}_{l, k}$, then $\lambda^{-1} = \sum_{j = 1}^{s}\lambda_j^{-1}e_j$, where $s = (l-1)k+1$, $\lambda_j\in{\mathbb F}_{p^m}^*$ for $j = 1, 2, \cdots, s$.

Lemma 3.1. (See ^[8]) Let $R$ be a finite commutative Frobenius ring with identity, $\sigma, \widetilde{\sigma}\in \mathrm{Aut}(R)$ and $C$ be a liner code of length $n$ over $R$, then

(1) $C^{\bot_\sigma}$ is a linear code over $R$.

(2) $C^{\bot_\sigma} = \sigma^{-1}(C^{\bot})$ and $\mid C\mid \mid C^{\bot_\sigma}\mid = \mid R\mid^n$.

(3) $(C^{\bot_\sigma})^{\bot_{\widetilde{\sigma}}} = \widetilde{\sigma}^{-1}\sigma^{-1}(C)$.

Theorem 3.1. $C = \bigoplus_{j = 1}^{s}e_jC_j$ is a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$ if and only if $C_j$ is a $\lambda_j$-constacyclic code of length $n$ over ${\mathbb F}_{p^m}$ for $j = 1, 2, \cdots, s$.

Proof. This proof is the same as Theorem 1 in ^[15]. □

Theorem 3.2. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a linear code of length $n$ over $\mathfrak{R}_{l, k}$. Then $C^{\bot_\sigma} = \sum_{j = 1}^{s}e_jC_j^{\bot_\sigma}$, where $C_j^{\bot_\sigma}$ is the $\sigma$ dual code of $C_j$ for $j = 1, 2, \cdots, s$.

Proof. Let $\tilde{C} = \sum_{j = 1}^{s}e_jC_j^{\bot_\sigma}$, $\forall x = \sum_{j = 1}^{s}e_jx_j\in C$ and $\forall \tilde{x} = \sum_{j = 1}^{s}e_j\tilde{x_j}\in \tilde{C}$, then

where $x_j\in C_j$, $\tilde{x_j}\in C_j^{\bot_\sigma}$.

So

For $\mathfrak{R}_{l, k}$ is a Frobenius ring, by Lemma 3.1,

Hence

So

□

Theorem 3.3. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$. Then $C^{\bot_\sigma} = \sum_{j = 1}^{s}e_jC_j^{\bot_\sigma}$ is a $\sigma(\lambda^{-1})$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$, and $C_j^{\bot_\sigma}$ is a $\sigma(\lambda_j^{-1})$-constacyclic code over ${\mathbb F}_{p^m}$ for $j = 1, 2, \cdots, s$.

Proof. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$. $\forall x = (x_0, x_1, \cdots, x_{n-1})\in C^{\bot_\sigma}$ and $\forall y = (y_0, y_1, \cdots, y_{n-1})\in C$, then

and

We can get that

So

We have $\delta_{\sigma(\lambda^{-1})}(x)\in C^{\bot_\sigma}$, so $C^{\bot_\sigma}$ is a $\sigma(\lambda^{-1})$-constacyclic code.

By Lemma 2.1, $\lambda^{-1} = \lambda^{-1}_1e_1+\lambda^{-1}_2e_2+\cdots+\lambda^{-1}_{s}e_{s}$, it implies that $C^{\bot_\sigma}$ is a $\sigma({\lambda^{-1}})$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$. By Theorem 3.1, we can have $C_j^{\perp}$ is a $\sigma(\lambda_j^{-1})$-constacyclic code over ${\mathbb F}_{p^m}$ for $j = 1, 2, \cdots, s$. □

Theorem 3.4. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$, then there exists a polynomial $g(x)\in\mathfrak{R}_{l, k}[x]$ such that $C = \langle g(x)\rangle$, where $g(x) = \sum_{j = 1}^{s}e_jg_j(x)$ is the divisor of $x^n-\lambda$, $g_j(x)\in {\mathbb F}_{p^m}[x]$ is the generator polynomial of $\lambda_j$-constacyclic over $C_j$, and $g_j(x)$ divides $x^n-\lambda_j$ for $j = 1, 2, \cdots, s$.

Proof. This proof is the same as Theorem 2 in ^[15]. □

Corollary 3.1. Let $C = \langle g(x)\rangle$ be a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$. Then $C^{\bot_\sigma} = \langle \sum_{j = 1}^{s}e_j\sigma^{-1}(f_j^{*}(x))\rangle$ and $\mid C^{\bot_\sigma} \mid = p^{m\sum_{j = 1}^{s}\mathrm {deg}(g_j)}$, where $f_j(x)g_j(x) = x^n-\lambda_j$, $j = 1, 2, \cdots, s$.

Proof. Let $C_j^{\bot} = \langle f_j^*(x)\rangle$. By Lemma 3.1, Theorem 3.3 and Theorem 3.4, we can have

and

where $j = 1, 2, \cdots, s$.

Then

and $C^{\bot_\sigma}$ has the form

Let

it is easy to see that, $\tilde{D}\subseteq C^{\perp}$.

On the other hand, for $j = 1, 2, \cdots, s$,

Thus $C^{\bot_\sigma}\subseteq \tilde{D}$, it implies that,

□

Theorem 3.5. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$, then $C$ is a $\sigma$-self-orthogonal code over $\mathfrak{R}_k$ if and only if $C_1, C_2, \cdots, C_{s}$ are $\sigma$-self-orthogonal codes over ${\mathbb F}_{p^m}$.

Proof. By Theorem 3.2, $C^{\bot_\sigma} = \sum_{j = 1}^{s}e_jC_j^{\bot_\sigma}$, so $C\subseteq C^{\bot_\sigma}$ if and only if $C_j\subseteq C_j^{\bot_\sigma}$, it implies that $C$ is a $\sigma$-self-orthogonal code over $\mathfrak{R}_{l, k}$ if and only if $C_1, C_2, \cdots, C_{s}$ are $\sigma$-self-orthogonal codes over ${\mathbb F}_{p^m}$. □

Lemma 3.2. Let $C$ be a $\lambda$-constacyclic code of length $n$ over ${\mathbb F}_{p^m}$, its generator polynomial is $g(x)$. Then $C$ is a $\sigma$-self-orthogonal code if and only if $f(x)\sigma^{-1}(f^*(x))$ is the divisor of $x^n-\lambda$, $\lambda = \pm1$, where $f^*(x)$ is the generator polynomial of $C^{\perp}$.

Proof. We can assume that $\sigma(\lambda^{-1}) = \lambda$.

Since $C$ is a $\lambda$-constantcyclic code of length $n$ over ${\mathbb F}_{p^m}$, and $C^{\bot_\sigma}$ is a $\sigma(\lambda^{-1})$-constantcyclic code of length $n$ over ${\mathbb F}_{p^m}$, so $C$ is a $\sigma$-self-orthogonal code must satisfy the condition $C\subseteq C^{\bot_\sigma}$.

Let $C^{\perp} = \langle f^*(x)\rangle$, where $f(x)g(x) = (x^n-\lambda)$ and $\lambda = \pm1$. $C^{\bot_\sigma} = \sigma^{-1}(C^{\bot}) = \langle \sigma^{-1}(f^*(x))\rangle$, $C$ is a $\sigma$-self-orthogonal code if and only if there exists a polynomial $h(x)\in {\mathbb F}_{p^m}[x]$, such that $\sigma^{-1}(f^*(x))h(x) = g(x)$, if and only if $f(x)\sigma^{-1}(f^*(x))h(x) = f(x)g(x) = (x^n-\lambda)$ if and only if $f(x)\sigma^{-1}(f^*(x))$ is the divisor of $x^n-\lambda$. □

By Theorem 3.5 and Lemma 3.2, it is easy to get the following theorem.

Theorem 3.6. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$, then $C$ is a $\sigma$-self-orthogonal code over $\mathfrak{R}_{l, k}$ if and only if $f_j(x)\sigma^{-1}(f_j^*(x))$ is the divisor of $x^n-\lambda_j$, where $f_j^*(x)$ is the generator polynomial of $C_j^{\perp}$ and $\sigma(\lambda_j^{-1}) = \lambda_j$ for $j = 1, 2, \cdots, s$.

4.   Quantum codes from $\sigma$-dual-containing constacyclic codes over $\mathfrak{R}_{l, k}$

Theorem 4.1. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a linear code of length $n$ over $\mathfrak{R}_{l, k}$, which order $\mid C \mid = p^{mk^{'}}$ and the minimum Gray distance of $C$ is $d_G$, where $s = (l-1)k+1$. Then $\phi_{l, k}(C)$ is a linear code $[sn, k^{'}, d_G]$ and ${{\phi }_{l, k}}{{(C)}^{\bot_\sigma }} = {{\phi }_{l, k}}({{C}^{\bot_\sigma }})$. If $C$ is a $\sigma$-self-orthogonal code over $\mathfrak{R}_{l, k}$, then $\phi_{l, k}(C)$ is a $\sigma$-self-orthogonal code over ${\mathbb F}_{p^m}$. Specifically, if $C$ is a $\sigma$-self-dual code over $\mathfrak{R}_{l, k}$, then $\phi_{l, k}(C)$ is a $\sigma$-self-dual code over ${\mathbb F}_{p^m}$.

Proof. By the definition of $\phi_{l, k}$, it is easy to know that $\phi_{l, k}$ is both a distance preserving map and a linear map from $\mathfrak{R}_{l, k}^n$ to ${\mathbb F}^{sn}_{q}$, it implies that $\phi_{l, k}(C)$ is a linear code $[sn, k^{'}, d_G]$.

If $C$ is a $\sigma$-self-orthogonal code, $\forall x = \sum_{j = 1}^{s}e_jx_j\in C$, $\forall y = \sum_{j = 1}^{s}e_jy_j\in C$, where $x_j, y_j\in {\mathbb F}^n_{p^m}$, because $C\subseteq C^{\bot_\sigma}$, so the $\sigma$-inner product of $x, y$ is $\langle x, y\rangle_\sigma = \sum_{j = 1}^{s}e_j\langle x_j, y_j\rangle_\sigma = 0$, which implies $\langle x_j, y_j\rangle_\sigma = 0$, so

So $\phi_{l, k}(C)$ is a $\sigma$-self-orthogonal code over ${\mathbb F}_{p^m}$.

Let $a = (a_0, a_1, \cdots, a_{n-1})\in C$ and $b = (b_0, b_1, \cdots, b_{n-1})\in C^{\bot_\sigma}$, where $a_j = \sum_{i = 1}^{s}a_{i, j}e_i$ and $b_j = \sum_{i = 1}^{s}b_{i, j}e_i\in \mathfrak{R}_{l, k}$ for $j = 0, 1, 2, \cdots, n-1$, $a^{(i)} = (a_{i, 0}, a_{i, 1}, \cdots, a_{i, n-1})$ and $b^{(i)} = (b_{i, 0}, b_{i, 1}, \cdots, b_{i, n-1})$ for $i = 1, 2, \cdots, s$.

Then

So $\langle a^{(i)}, {b^{(i)}}\rangle_\sigma = 0$ for $i = 1, 2, \cdots, s.$

Since $\phi_{l, k}(a) = (a^{(1)}, a^{(2)}, \cdots, a^{(s)})$ and $\phi_{l, k}(b) = (b^{(1)}, b^{(2)}, \cdots, b^{(s)})$. It follows that

So we have

As $\phi_{l, k}$ is a bijection, and $\mid C \mid = \mid \phi_{l, k}(C)\mid$.

Then

We have

Suppose $C$ is a $\sigma$-self-dual code and $C = C^{\bot_\sigma }$, then

Therefore, $\phi_{l, k}(C)$ is a $\sigma$-self-dual code over ${\mathbb F}_{p^m}$. □

Lemma 4.1. Let $C$ be a $\lambda$-constacyclic code of length $n$ over ${\mathbb F}_{p^m}$, whose generator polynomial is $g(x)$. Then, $C$ is a $\sigma$-dual-containing code if and only if $x^n-\lambda$ is the divisor of $\sigma^{-1}(f^*(x))f(x)$, where $f^*(x)$ is the generator polynomial of $C^{\perp}$ and $\sigma(\lambda^{-1}) = \lambda$.

Proof. Let $C^{\perp} = \langle f^*(x)\rangle$, where $f(x)g(x) = (x^n-\lambda)$ and $\sigma(\lambda^{-1}) = \lambda$. Then $C^{\bot_\sigma} = \sigma^{-1}(C^{\bot}) = \langle \sigma^{-1}(f^*(x))\rangle$, so $C$ is a $\sigma$-dual-containing code if and only if there exists a polynomial $h(x)\in {\mathbb F}_{p^m}[x]$, such that $\sigma^{-1}(f^*(x)) = h(x)g(x)$, if and only if $\sigma^{-1}(f^*(x))f(x) = h(x)g(x)f(x) = h(x)f(x)g(x) = h(x)(x^n-\lambda)$ if and only if $x^n-\lambda$ is the divisor of $\sigma^{-1}(f^*(x))f(x)$. □

Theorem 4.2. (CSS construction ^[4]) Let $C = [n, k, d]$ be a linear codes over ${\mathbb F}_q$ with $C^\perp \subseteq C$, then there exists a quantum code $[[n, 2k-n, d]]_q$.

Theorem 4.3. (Hermitian construction ^[4]) Let $C = [n, k, d]$ be a linear code over ${\mathbb F}_{q^2}$ with $C^{\perp_H} \subseteq C$, then there exists a quantum code $[[n, 2k-n, d]]_q$.

By Lemma 4.1 and Theorem 3.3, we can have the following theorem.

Theorem 4.4. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$, where $s = (l-1)k+1$. Then $C^{\bot_\sigma} \subseteq C$ if and only if $x^n-\lambda_j$ is the divisor of $\sigma^{-1}(f_j^*(x))f_j(x)$, where $\sigma(\lambda_j^{-1}) = \lambda_j$ and $f_j^*(x)$ is the generator polynomial of $C_j^{\perp}$ for $j = 1, 2, \cdots, s$.

By Lemma 4.1 and Theorem 4.4, we can have the following corollary and theorem.

Corollary 4.1. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$. Then $C^{\perp_\sigma} \subseteq C$ if and only if $C_j^{\perp_\sigma}\subseteq C_j$ for $j = 1, 2, \cdots, s$.

Theorem 4.5. Let $C = \bigoplus_{j = 1}^{s}e_jC_j$ be a a $\lambda$-constacyclic code of length $n$ over $\mathfrak{R}_{l, k}$, $C_j$ is $\lambda_j$-constacyclic code over ${\mathbb F}_{p^m}$ and $C_j^{\perp_\sigma}\subseteq C_j$, where $s = (l-1)k+1$, $\sigma(\lambda_j^{-1}) = \lambda_j$ for $j = 1, 2, \cdots, s$. Then ${{\phi }_{l, k}}{{(C)}^{\bot_\sigma }}\subseteq{{\phi }_{l, k}}(C)$. If $\theta_t$ is the identity map of ${\mathbb F}_{p^m}$, then there exists a quantum code $[[sn, 2k^{'}-sn, d_G]]_{p^m}$. If $m$ is even, $\forall a\in {\mathbb F}_{p^m}$, then $\sigma(a) = a^{p^{\frac{m}{2}}}$, then there exists a quantum code $[[sn, 2k^{'}-sn, d_G]]_{p^{m/2}}$, where $d_G$ is the minimum Gray weight of code $C$, and $k^{'}$ is the dimension of the linear code $\phi_{l, k}(C)$.

Proof. Let $C_j^{\perp_\sigma}\subseteq C_j$ and $\sigma(\lambda_j^{-1}) = \lambda_j$. By Corollary 4.1, we have $C^{\perp_\sigma} \subseteq C$, so $\phi_{l, k}(C^{\perp_\sigma}) \subseteq \phi_{l, k}(C)$. By Theorem 4.1, ${{\phi }_{l, k}}{{(C)}^{\bot_\sigma }} = {{\phi }_{l, k}}({{C}^{\bot_\sigma }})$, therefore $\phi_{l, k}(C)^{\perp_\sigma} \subseteq \phi_{l, k}(C)$, and by Theorem 4.1, $\phi_{l, k}(C)$ is a linear code $[sn, k^{'}, d]$.

If $\theta_t$ is the identity map of ${\mathbb F}_{p^m}$, so $\sigma$-inner product is the Euclidean inner product, by Theorem 4.2, there exists a quantum code $[[sn, 2k^{'}-sn, d_G]]_{p^m}$.

If $m$ is even, $\forall a\in {\mathbb F}_{p^m}$, then $\sigma(a) = a^{p^{\frac{m}{2}}}$, so $\sigma$-inner product is the Hermitian inner product. By Theorem 4.3, there exists a quantum code $[[sn, 2k^{'}-sn, d_G]]_{p^{m/2}}$. □

Example 4.1. Let $n = 8$ and $\mathfrak{R}_{2, 2} = {\mathbb F}_{17}[u_1, u_2]/ \langle u^{2}_{1} = u_{1}, u^{2}_{2} = u_{2}, u_1u_2 = u_2u_1 = 0\rangle$. In ${\mathbb F}_{17}[x]$,

If $\theta_t$ is the identity map of ${\mathbb F}_{p^m}$, then $\sigma$-inner product is the Euclidean inner product. Let $C$ be an $(e_1+e_2+(-1)e_3)$-constacyclic code of length $8$ over $\mathfrak{R}_{2, 2}$. Let $g(x) = e_1g_1+e_2g_2+e_3g_3$ be the generator polynomial of $C$, where $g_1(x) = (x+2)(x+4)(x+8), g_2(x) = g_3(x) = (x+3)$, then $C_1 = \langle g_1(x)\rangle$ is a cyclic code of length 8 over ${\mathbb F}_{17}$, $C_2 = \langle g_2(x)\rangle$ and $C_3 = \langle g_3(x)\rangle$ are negacyclic codes of length 8 over ${\mathbb F}_{17}$.

By Theorem 4.1, $\phi_{2, 2}(C)$ is a linear code over ${\mathbb F}_{17}$ with parameters $[24, 19, 4]$. By Theorem 4.5, we have $C^\perp\subseteq C$, we get a quantum code $[[24, 14, 4]]_{17}$.

Example 4.2. Let $n = 45$ and $\mathfrak{R}_{2, 2} = {\mathbb F}_{5}[u_1, u_2]/ \langle u^{2}_{1} = u_{1}, u^{2}_{2} = u_{2}, u_1u_2 = u_2u_1 = 0\rangle$. In ${\mathbb F}_{5}[x]$,

Let $C$ be an $(e_1+(-1)e_2+(-1)e_3)$-constacyclic code of length $45$ over $\mathfrak{R}_{2, 2}$. Let $g_1(x) = x^2+x+1$, $g_2(x) = g_3(x) = x+1$, then $C_1 = \langle g_1(x)\rangle$ is a cyclic code of length $45$, $C_2 = \langle g_2(x)\rangle$ and $C_3 = \langle g_3(x)\rangle$ are negacyclic codes of length $45$ over ${\mathbb F}_5$.

By Theorem 4.1, $\phi_{2, 2}(C)$ is a linear code over ${\mathbb F}_{5}$ with parameters $[135,131, 3]$. By Theorem 4.5, we have $C^\perp\subseteq C$, we can get a quantum code $[[135,127, 3]]_5$, which has larger dimension than $[[135, 63, 3]]_5$ in ^[12].

Example 4.3. Let $n = 30$ and $\mathfrak{R}_{2, 2} = {\mathbb F}_{5}[u_1, u_2]/ \langle u^{2}_{1} = u_{1}, u^{2}_{2} = u_{2}, u_1u_2 = u_2u_1 = 0\rangle$. In ${\mathbb F}_{5}[x]$,

Let $C$ be an $(e_1+(-1)e_2+(-1)e_3)$-constacyclic code of length $30$ over $\mathfrak{R}_{2, 2}$. Let $g_1(x) = x^2+x+1$, $g_2(x) = g_3(x) = x+3$, then $C_1 = \langle g_1(x)\rangle$ is a cyclic code of length $30$, $C_2 = \langle g_2(x)\rangle$ and $C_3 = \langle g_3(x)\rangle$ are negacyclic codes of length $30$ over ${\mathbb F}_5$.

By Theorem 4.1, $\phi_{2, 2}(C)$ is a linear code over ${\mathbb F}_{5}$ with parameters $[90, 86, 3]$. By Theorem 4.5, we have $C^{\perp} \subseteq C$, we get a quantum code $[[90, 82, 3]]_5$, which has larger dimension than $[[90, 68, 3]]_5$ in ^[11].

Example 4.4. Let $n = 8$ and $\mathfrak{R}_{1, 2} = {\mathbb F}_{9}[u_1]/ \langle u^{2}_{1} = u_{1}\rangle$, $\sigma(a) = a^{3}$. In ${\mathbb F}_{9}[x]$,

Let $C$ be an $(e_1+e_2)$-constacyclic code of length $8$ over $\mathfrak{R}_{1, 2}$. Let $g_1(x) = (x+w)(x+w^2)$ and $g_2(x) = x+w^3$, then $C_1 = \langle g_1(x)\rangle$ and $C_2 = \langle g_2(x)\rangle$ are cyclic codes of length 8 over ${\mathbb F}_9$.

By Theorem 4.1, $\phi_{1, 2}(C)$ is a linear code over ${\mathbb F}_{9}$ with parameters $[16, 13, 3]$. By Theorem 4.5, we have $C^{\perp_\sigma} \subseteq C$, we can get a quantum code $[[16, 10, 3]]_3$ satisfying $n-k+2-2d = 2.$

In Table 1, we provide some new quantum codes (in the eighth column) and compare the existing codes (in the ninth column) better (by means of larger code rate or larger distance) than ^[11,12,13]. Further, the fifth column gives the value of units $(\lambda_1, \cdots, \lambda_{s})$, the sixth column gives the generator polynomials $\langle g_1(x), \cdots, g_{s}(x)\rangle$, where $g_i(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ is denoted by $a_na_{n-1}\cdots a_1a_0$, e.g., "$11$" represents the polynomial "$x^2+x$", the seventh column gives parameters of $\phi_{l, k}(C)$.

5.   Conclusions

In this article, we construct quantum codes by studying the algebraic structure of $\sigma$-self-orthogonal constacyclic codes over a new finite non-chain ring $\mathfrak{R}_{l, k}$, and our results will enrich the code sources of constructing quantum codes. As an application, we obtain some new quantum codes from $\sigma$-dual-containing constacyclic codes over $\mathfrak{R}_{l, k}$ using the CSS construction or Hermitian construction and compare these codes better with the existing codes that appeared in some recent references.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported in part by the Zhengzhou Special Fund for Basic Research and Applied Basic Research under Grant ZZSZX202111, in part by the Postgraduate Education Reform and Quality Improvement Project of Henan Province under Grant YJS2023JD6, and in part by the Soft Science Research Project of Henan Province under Grant 232400411122.

We 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.

Conflict of interest

We declare no conflicts of interest.