Associated Complementary Dual and Quantum Codes of Skew Constacyclic Codes

1. Introduction

Rich algebraic structures and ease of practical application are two of the most well-known attributes of cyclic codes. On identifying a vector by a polynomial, one can consider any cyclic code over ${\mathbb{F}}_q$ as a submodule of the ${\mathbb{F}}_q\left[x\right]$-module ${\mathbb{F}}_q\left[x\right]/\langle x^n-1\rangle$ for the length n. Further, these codes can also be considered as ideals generated by divisors of $x^n-1$. It is worth noting that ${\mathbb{F}}_q\left[x\right]$ is a unique factorization domain, a fact which restricts the divisors of $x^n-1$ in ${\mathbb{F}}_q\left[x\right]$. As an extension of these codes, constacyclic codes may be regarded as ideals of ${\mathbb{F}}_q\left[x\right]/\langle x^n-\gamma \rangle$ for some non-zero element $\gamma \in {\mathbb{F}}_q$. Further, these codes were examined over some finite commutative rings [1,2]. Later, by relaxing the linearity condition, additive codes were considered over mixed alphabets [3,4].

On the other hand, in the desire to obtain more factorizations of $x^n-1$ than over a factorial ring, skew cyclic codes [5,6] were introduced with the notion of skew polynomial rings [7]. These codes over a finite field ${\mathbb{F}}_q$ are basically left submodules of the module ${\mathbb{F}}_q[x,\mathsf{\Theta }]/\langle x^n-1\rangle$ for an automorphism $\mathsf{\Theta }$ of ${\mathbb{F}}_q$. Later, these codes were investigated by Abualrub and Seneviratne [8] over ${\mathbb{F}}_q+v{\mathbb{F}}_q$, where $v^2=v$. Additionally, Gao [9], and Gursoy et al. [10] presented skew cyclic codes by considering different automorphisms. Later, as a continuation of these efforts, skew constacyclic codes were examined [11,12].

In 1992, Massey [13] proposed LCD codes ($\mathcal{V}\cap {\mathcal{V}}^{\perp }=\left\{\mathbf{0}\right\}$). It was demonstrated that these codes were the best linear coding solution to 2-BAC. Sufficient and necessary condition for the complementary duality of cyclic codes over finite fields was derived in 1994 by Yang and Massey [14]. Additionally, they deduced a relationship among the reversible and LCD cyclic codes. Later, DNA applications made full use of the former idea [15,16]. On the other hand, the Gilbert-Varshamov bound is satisfied by LCD codes, as demonstrated by Sendrier [17] in 2004. Subsequently, these codes were examined over chain rings in [18]. In 2016, these codes were shown to have applications in cryptosystems [19]. Afterwards, LCD codes were studied over different commutative rings in [20,21,22,23,24,25,26] , and applications of these codes were presented in Multi-secret Sharing Schemes [27]. Recently, LCD codes were studied in [28,29] using the skew polynomial rings. Besides, additive codes were investigated for the complementary duality over the structure ${\mathbb{Z}}_2{\mathbb{Z}}_2\left[u^3\right]$ in [30], and they were called additive complementary dual (ACD) codes.

In the last decades of the 20th century, it was noticed that quantum mechanics might improve the complexity of certain classical algorithms, like the DFT transform, list searching, or integer factorization. This last algorithm is a key ingredient in the RSA cryptosystem. For this reason, quantum computation got the attention of many researchers. To securitize quantum computation, Shor [31] introduced Quantum Error-Correcting Codes (QECCs) in 1995. In 1998, classical error-correcting codes were employed to obtain QECCs via some constructions such as CSS construction [32]. Afterwards, linear codes over different commutative rings were used to obtain good QECCs in [33,34,35,36]. As an extension of these works, additive codes over the commutative structure were also utilized [37]. Recently, cyclic and constacyclic codes using non-commutative rings have been employed for obtaining QECCs due to more possibility of factorization of a polynomial. Many good QECCs were obtained from cyclic and constacyclic codes [38,39,40,41] . Motivated by these works, we consider skew constacyclic and additive codes for obtaining new and better LCD and quantum codes.

This paper has been arranged as follows: Firstly, the structure of linear codes over ${\mathcal{B}}_e$ is presented in Section 2. Section 3 presents skew constacyclic codes over ${\mathcal{B}}_e$, defines a Gray map and analyzes its properties. In Section 4, we derive some results for LCD codes. Further, we derive quantum codes by utilizing CSS construction in Section 5. Section 6 presents the structure and properties of additive skew constacyclic codes over ${\mathbb{F}}_q{\mathcal{B}}_e$. In Section 7, we present various LCD codes as well as new quantum codes. Our work is concluded in Section 8.

2. Preliminaries

Let us suppose that ${\mathbb{F}}_q$ denotes a finite field having cardinality q. Let us consider a ring ${\mathcal{B}}_e={\mathbb{F}}_q+u{\mathbb{F}}_q+\cdots +u^{e-1}{\mathbb{F}}_q$, $u^e=u$, with $q\equiv 1(mode-1)$ ($e\ge 2$). We represent the collection of units in ${\mathcal{B}}_e$ by ${\mathcal{B}}_e^{\ast }$.

Consider a primitive element $\alpha \in {\mathbb{F}}_q$ and take $\xi ={\alpha }^{{\displaystyle \frac{q-1}{e-1}}}$. Following [42], consider

$$\begin{array}{cc}\hfill {ϵ}_1& =1-u^{e-1},\hfill \\ \hfill {ϵ}_2& ={\displaystyle \frac{1}{e-1}}(u+u^2+\cdots +u^{e-2}+u^{e-1}),\hfill \\ \hfill {ϵ}_3& ={\displaystyle \frac{1}{e-1}}(\xi u+{\xi }^2{u}^2+\cdots +{\xi }^{e-2}{u}^{e-2}+u^{e-1}),\hfill \\ \hfill {ϵ}_4& ={\displaystyle \frac{1}{e-1}}({\xi }^{2}u+{\left({\xi }^2\right)}^2{u}^2+\cdots +{\left({\xi }^2\right)}^{e-2}{u}^{e-2}+u^{e-1}),\hfill \\ \hfill & ⋮\hfill \\ \hfill {ϵ}_e& ={\displaystyle \frac{1}{e-1}}({\xi }^{e-2}u+{\left({\xi }^{e-2}\right)}^2{u}^2+\cdots +{\left({\xi }^{e-2}\right)}^{e-2}{u}^{e-2}+u^{e-1}).\hfill \end{array}$$

Then, we have ${\mathcal{B}}_e\cong {⨁}_{i=1}^e{ϵ}_i{\mathbb{F}}_q$ and each $s\in {\mathcal{B}}_e$ has a unique representation $s={\sum }_{i=1}^e{ϵ}_i{s}_i$, $s_i\in {\mathbb{F}}_q$. Moreover, any linear code $\mathcal{V}$ over ${\mathcal{B}}_e$ with length n has a representation

$$\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e,$$

for some linear codes ${\mathcal{V}}_i=\{r_i\in {\mathbb{F}}_q^n:\exists r_j\in {\mathbb{F}}_q^n,1\le j\ne i\le e\mathrm{such}\mathrm{that}{\sum }_{k=1}^e{ϵ}_k{r}_k\in \mathcal{V}\}$ over ${\mathbb{F}}_q$ ($1\le i\le e$) and the dual code of $\mathcal{V}$ is ${\mathcal{V}}^{\perp }={ϵ}_1{\mathcal{V}}_1^{\perp }\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e^{\perp }$. Subsequently, the following result holds.

Theorem 1. 

Let us assume that $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e$ is any γ-constacyclic code over ${\mathcal{B}}_e$ ($\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i\in {\mathcal{B}}_e^{\ast }$). Then $\mathcal{V}$ has $g\left(x\right)={\sum }_{i=1}^e{ϵ}_i{g}_i\left(x\right)$ as its generator polynomial, where each ${\mathcal{V}}_i$ has $g_i\left(x\right)$ as their generator polynomial ($1\le i\le e$).

3. Skew Constacyclic Codes over ${\mathcal{B}}_e$

The current section presents the generator polynomials for skew constacyclic codes over ${\mathcal{B}}_e$ and the dual codes. In order to proceed with the investigation, we first require an automorphism of ${\mathcal{B}}_e$ as defined below.

Define an automorphism ${\mathsf{\Theta }}_l:{\mathcal{B}}_e\to {\mathcal{B}}_e$ by

$${\mathsf{\Theta }}_l({ϵ}_1{a}_1+{ϵ}_2{a}_2+\cdots +{ϵ}_e{a}_e)={ϵ}_1{\mathsf{\Theta }}_l\left(a_1\right)+{ϵ}_2{\mathsf{\Theta }}_l\left(a_2\right)+\cdots +{ϵ}_e{\mathsf{\Theta }}_l\left(a_e\right)={ϵ}_1{a}_1^{p^l}+{ϵ}_2{a}_2^{p^l}+\cdots +{ϵ}_e{a}_e^{p^l},$$

for $0\le l\le m-1$. Using this, the definition for skew constacyclic codes is as mentioned below.

Definition 2. 

Suppose ${\mathsf{\Theta }}_l$ is an automorphism of the ring ${\mathcal{B}}_e$, $\gamma \in {\mathcal{B}}_e^{\ast }$. Then a linear code $\mathcal{V}$ over ${\mathcal{B}}_e$ is called a skew constacyclic (or skew ${\mathsf{\Theta }}_l-\gamma$- constacyclic) code over ${\mathcal{B}}_e$, if ${\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(\mathcal{V}\right)=\mathcal{V}$, where ${\tau }_{{\mathsf{\Theta }}_l,\gamma }:{\mathcal{B}}_e^n⟶{\mathcal{B}}_e^n$ is the skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic shift given by

$${\tau }_{{\mathsf{\Theta }}_l,\gamma }(c_0,c_1,\cdots ,c_{n-1})=(\gamma {\mathsf{\Theta }}_l\left(c_{n-1}\right),{\mathsf{\Theta }}_l\left(c_0\right),{\mathsf{\Theta }}_l\left(c_1\right),\cdots ,{\mathsf{\Theta }}_l\left(c_{n-2}\right)).$$

For $\gamma =-1$ (resp. $\gamma =1$), these codes are said to be skew negacyclic (resp. skew cyclic) codes, respectively. Further, for identity automorphism ${\mathsf{\Theta }}_l$ and $\gamma =-1$ (resp. $\gamma =1$), these codes are said to be negacyclic (resp. cyclic) codes with the corresponding negacyclic and cyclic shifts represented by ${\tau }_{-1}$ and ${\tau }_1$, respectively.

Consider the skew-polynomial ring

$${\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]=\{b_0+b_{1}x+\cdots +b_{n^{\prime }}{x}^{n^{\prime }}|b_i\in {\mathcal{B}}_e,n^{\prime }\in \mathbb{N}\}$$

in which the multiplication is given by $x\ast b={\mathsf{\Theta }}_l\left(b\right)x$. Then the center $Z\left({\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]\right)$ of ${\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]$ (which will be used to study skew constacyclic codes) can be obtained by the below-mentioned result.

Lemma 3. 

Let us assume that $\gamma \in {\mathcal{B}}_e^{\ast }$ such that ${\mathsf{\Theta }}_l\left(\gamma \right)=\gamma$, where ${\mathsf{\Theta }}_l$ is an automorphism of ${\mathcal{B}}_e$. The order of ${\mathsf{\Theta }}_l$ divides n iff $x^n-\gamma$ lies in $Z\left({\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]\right)$.

Proof. 

Let ${\mathsf{\Theta }}_l\left(\gamma \right)=\gamma$ and order of ${\mathsf{\Theta }}_l$ divides n, i.e., ${\mathsf{\Theta }}_l^n\left(s\right)=s\forall s\in {\mathcal{B}}_e$. Then, for any $s_0+s_{1}x+\cdots +s_{n^{\prime }}{x}^{n^{\prime }}\in {\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]$, we have

$$\begin{array}{cc}\hfill (x^n-\gamma )(s_0+s_{1}x+\cdots +s_{n^{\prime }}{x}^{n^{\prime }})=& {\mathsf{\Theta }}_l^n\left(s_0\right)x^n+{\mathsf{\Theta }}_l^n\left(s_1\right)x^{n+1}+\cdots +{\mathsf{\Theta }}_l^n\left(s_{n^{\prime }}\right)x^{n+n^{\prime }}\hfill \\ & -\gamma (s_0+s_{1}x+\cdots +s_{n^{\prime }}{x}^{n^{\prime }})\hfill \\ \hfill =& s_0{x}^n+s_1{x}^{n+1}+\cdots +s_{n^{\prime }}{x}^{n+n^{\prime }}\hfill \\ & -(\gamma s_0+\gamma s_{1}x+\cdots +\gamma s_{n^{\prime }}{x}^{n^{\prime }})\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill (s_0+s_{1}x+\cdots +s_{n^{\prime }}{x}^{n^{\prime }})(x^n-\gamma )=& s_0{x}^n+s_1{x}^{n+1}+\cdots +s_{n^{\prime }}{x}^{n+n^{\prime }}\hfill \\ & -(s_0\gamma +s_1{\mathsf{\Theta }}_l\left(\gamma \right)x+\cdots +s_{n^{\prime }}{\mathsf{\Theta }}_l^{n^{\prime }}\left(\gamma \right)x^{n^{\prime }})\hfill \\ \hfill =& s_0{x}^n+s_1{x}^{n+1}+\cdots +s_{n^{\prime }}{x}^{n+n^{\prime }}\hfill \\ & -(s_0\gamma +s_1\gamma x+\cdots +s_{n^{\prime }}\gamma x^{n^{\prime }}).\hfill \end{array}$$

That is, $(x^n-\gamma )(s_0+s_{1}x+\cdots +s_{n^{\prime }}{x}^{n^{\prime }})=(s_0+s_{1}x+\cdots +s_{n^{\prime }}{x}^{n^{\prime }})(x^n-\gamma )$ and hence $x^n-\gamma \in Z\left({\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]\right)$.

Conversely, assume that $x^n-\gamma \in Z\left({\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]\right)$. Then $(x^n-\gamma )\left(s{x}^i\right)=\left(s{x}^i\right)(x^n-\gamma )$, for all $s\in {\mathcal{B}}_e$ and $i\in \mathbb{N}$. Now,

$$(x^n-\gamma )\left(s{x}^i\right)=({\mathsf{\Theta }}_l^n\left(s\right)x^{n+i}-\gamma s{x}^i)\mathrm{and}\left(s{x}^i\right)(x^n-\gamma )=(s{x}^{i+n}-s{\mathsf{\Theta }}_l^i\left(\gamma \right)x^i)=(s{x}^{i+n}-s\gamma x^i).$$

It implies that ${\mathsf{\Theta }}_l^n\left(s\right)=s$ for all $s\in {\mathcal{B}}_e$. Therefore, we conclude that order of ${\mathsf{\Theta }}_l$ divides n. □

As we have already seen that for an automorphism ${\mathsf{\Theta }}_l$ of ${\mathcal{B}}_e$ whose order divides n and $\gamma \in {\mathcal{B}}_e^{\ast }$ such that $\gamma$ is fixed by ${\mathsf{\Theta }}_l$, we have $(x^n-\gamma )\in Z\left({\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]\right)$. Therefore, ${\displaystyle \frac{{\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]}{\langle x^n-\gamma \rangle }}$ is a ring. On the contrary, if n is not divisible by order of ${\mathsf{\Theta }}_l$ then ${\displaystyle \frac{{\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]}{\langle x^n-\gamma \rangle }}$ doesn’t form a ring but a left ${\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]$-module and any skew constacyclic code is characterized by the result given below.

Lemma 4. 

Any linear code $\mathcal{V}$ over ${\mathcal{B}}_e$ having length n is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code iff $\mathcal{V}$ forms a left ${\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]$-submodule of ${\displaystyle \frac{{\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]}{\langle x^n-\gamma \rangle }}$.

Theorem 5. 

Suppose $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i\in {\mathcal{B}}_e^{\ast }$ and $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e$ is a linear code over ${\mathcal{B}}_e$. Then $\mathcal{V}$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code iff each ${\mathcal{V}}_i$ is skew ${\mathsf{\Theta }}_l-{\alpha }_i$-constacyclic code ($1\le i\le e$).

Proof. 

Let us suppose that $\mathcal{V}$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code over ${\mathcal{B}}_e$ and $a_j=(a_{0,j},a_{1,j},\cdots ,a_{(n-1),j})\in {\mathcal{V}}_j$ for every j lying in the set $\{1,2,\cdots ,e\}$. Then $v=(v_0,v_1,\cdots ,v_{n-1})={\sum }_{j=1}^e{ϵ}_j{a}_j\in \mathcal{V}$, where $v_i={\sum }_{j=1}^e{ϵ}_j{a}_{i,j}$ for each i lying in the set $\{0,1,\cdots ,n-1\}$. As $\mathcal{V}$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code, we have ${\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(v\right)\in \mathcal{V}$. Note that

$$\begin{array}{cc}\hfill {\tau }_{{\mathsf{\Theta }}_l,\gamma }(v)& =(\gamma {\mathsf{\Theta }}_l(v_{n-1}),{\mathsf{\Theta }}_l(v_0),{\mathsf{\Theta }}_l(v_1),\cdots ,{\mathsf{\Theta }}_l(v_{n-2}))\hfill \\ & =(\gamma {\mathsf{\Theta }}_l(\sum _{j=1}^e{ϵ}_j{a}_{(n-1),j}),{\mathsf{\Theta }}_l(\sum _{j=1}^e{ϵ}_j{a}_{0,j}),\cdots ,{\mathsf{\Theta }}_l(\sum _{j=1}^e{ϵ}_j{a}_{(n-2),j})\hfill \\ & =(\sum _{j=1}^e{ϵ}_j{\alpha }_j{\mathsf{\Theta }}_l(a_{(n-1),j}),\sum _{j=1}^e{ϵ}_j{\mathsf{\Theta }}_l(a_{0,j}),\cdots ,\sum _{j=1}^e{ϵ}_j{\mathsf{\Theta }}_l(a_{(n-2),j})\hfill \\ & =\sum _{j=1}^e{ϵ}_j({\alpha }_j{\mathsf{\Theta }}_l(a_{(n-1),j}),{\mathsf{\Theta }}_l(a_{0,j}),\cdots ,{\mathsf{\Theta }}_l(a_{(n-2),j}).\hfill \end{array}$$

It implies that $({\alpha }_j{\mathsf{\Theta }}_l\left(a_{(n-1),j}\right),{\mathsf{\Theta }}_l\left(a_{0,j}\right),\cdots ,{\mathsf{\Theta }}_l\left(a_{(n-2),j}\right)\in {\mathcal{V}}_j$ for each $1\le j\le e$. Consequently, ${\mathcal{V}}_j$ is a skew ${\mathsf{\Theta }}_l-{\alpha }_j$-constacyclic code for each j lying in the set $\{1,2,\cdots ,e\}$.

For the other side, assume that ${\mathcal{V}}_i$ is skew ${\mathsf{\Theta }}_l-{\alpha }_i$-constacyclic code over ${\mathbb{F}}_q$ and $v=(v_0,v_1,\cdots ,v_{n-1})\in \mathcal{V}$, where $v_j={\sum }_{i=1}^e{ϵ}_i{a}_{j,i}$ for each j lying in the set $\{0,1,\cdots ,n-1\}$. Then $a_i=(a_{0,i},a_{1,i},\cdots ,a_{(n-1),i})\in {\mathcal{V}}_i$ for each i lying in the set $\{1,2,\cdots ,e\}$ as $v={\sum }_{i=1}^e{ϵ}_i{a}_i$. Since ${\mathcal{V}}_i$ are skew ${\mathsf{\Theta }}_l-{\alpha }_i$-constacyclic codes, we have ${\tau }_{{\mathsf{\Theta }}_l,{\alpha }_i}\left(a_i\right)\in {\mathcal{V}}_i$ for each $1\le i\le e$ which further implies that ${\sum }_{i=1}^e{ϵ}_i{\tau }_{{\mathsf{\Theta }}_l,{\alpha }_i}\left(a_i\right)\in \mathcal{V}$. Now,

$$\begin{array}{cc}\hfill \sum _{i=1}^e{ϵ}_i{\tau }_{{\mathsf{\Theta }}_l,{\alpha }_i}(a_i)& =\sum _{i=1}^e{ϵ}_i({\alpha }_i{\mathsf{\Theta }}_l(a_{(n-1),i}),{\mathsf{\Theta }}_l(a_{0,i}),\cdots ,{\mathsf{\Theta }}_l(a_{(n-2),i}))\hfill \\ & =(\sum _{i=1}^e{ϵ}_i{\alpha }_i{\mathsf{\Theta }}_l(a_{(n-1),i}),\sum _{i=1}^e{ϵ}_i{\mathsf{\Theta }}_l(a_{0,i}),\cdots ,\sum _{i=1}^e{ϵ}_i{\mathsf{\Theta }}_l(a_{(n-2),i}))\hfill \\ & =(\gamma {\mathsf{\Theta }}_l(\sum _{i=1}^e{ϵ}_i{a}_{(n-1),i}),{\mathsf{\Theta }}_l(\sum _{i=1}^e{ϵ}_i{a}_{0,i}),\cdots ,{\mathsf{\Theta }}_l(\sum _{i=1}^e{ϵ}_i{a}_{(n-2),i}))\hfill \\ & =(\gamma {\mathsf{\Theta }}_l(v_{n-1}),{\mathsf{\Theta }}_l(v_0),{\mathsf{\Theta }}_l(v_1),\cdots ,{\mathsf{\Theta }}_l(v_{n-2}))\hfill \\ & ={\tau }_{{\mathsf{\Theta }}_l,\gamma }(v).\hfill \end{array}$$

Consequently, $\mathcal{V}$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code as ${\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(v\right)\in \mathcal{V}$. □

From [12], any skew ${\mathsf{\Theta }}_l-\alpha$-constacyclic code $\mathcal{V}$ having length n over ${\mathbb{F}}_q$ for $\alpha \in {\mathbb{F}}_q^{\ast }$ is principally generated by a polynomial $g\left(x\right)$ as a left ${\mathbb{F}}_q[x;{\mathsf{\Theta }}_l]$-submodule of the module ${\displaystyle \frac{{\mathbb{F}}_q[x;{\mathsf{\Theta }}_l]}{\langle x^n-\alpha \rangle }}$ and $\mathcal{V}=\langle g\left(x\right)\rangle$, where $g\left(x\right)$ divides $x^n-\alpha$ on the right and regarded as the generator polynomial of the code $\mathcal{V}$. Using these arguments and Theorem 5, the below mentioned conclusion can be derived.

Theorem 6. 

Suppose $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i\in {\mathcal{B}}_e^{\ast }$ and $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code over ${\mathcal{B}}_e$ with ${\mathcal{V}}_i=\langle g_i\left(x\right)\rangle$, where each $g_i\left(x\right)$ divides $x^n-{\alpha }_i$ on the right (i lying in the set $\{1,2,\cdots ,e\}$). Then $\mathcal{V}=\langle g\left(x\right)\rangle$, where $g\left(x\right)={\sum }_{i=1}^e{ϵ}_i{g}_i\left(x\right)$ divides $x^n-\gamma$ on the right.

We now consider a Gray map $\mathsf{\Phi }:{\mathcal{B}}_e^n\to {\mathbb{F}}_q^{en}$ defined as

$$\mathsf{\Phi }(r_0,r_1,\cdots ,r_{n-1})=[(s_{10},s_{20},\cdots ,s_{e0})M,(s_{11},s_{21},\cdots ,s_{e1})M,\cdots ,(s_{1(n-1)},s_{2(n-1)},\cdots ,s_{e(n-1)})M],$$

where $r_i={\sum }_{k=1}^e{ϵ}_k{s}_{ki}$ for each i lying in the set $\{0,1,\cdots ,n-1\}$ and a square matrix M satisfies $M{M}^t=\nu I$ for the identity matrix I and $\nu \in {\mathbb{F}}_q^{\ast }$. Then $\mathsf{\Phi }$ is a bijective linear map, and the following result can be deduced.

Lemma 7. 

Let us suppose that $\mathcal{V}$ is a linear code over ${\mathcal{B}}_e$ having dual code ${\mathcal{V}}^{\perp }$. Then $\mathsf{\Phi }\left({\mathcal{V}}^{\perp }\right)=\mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }.$

Proof. 

Suppose $v=(v_0,v_1,\cdots ,v_{n-1})\in \mathcal{V}$ and $x=(x_0,x_1,\cdots ,x_{n-1})\in {\mathcal{V}}^{\perp }$, where $v_i={\sum }_{j=1}^e{ϵ}_j{r}_{ji}$ and $x_i={\sum }_{j=1}^e{ϵ}_j{s}_{ji}$ for each i lying in the set $\{0,1,\cdots ,n-1\}$. Then

$$\langle v,x\rangle ={ϵ}_1\sum _{i=0}^{n-1}{r}_{1i}{s}_{1i}+{ϵ}_2\sum _{i=0}^{n-1}{r}_{2i}{s}_{2i}+\cdots +{ϵ}_e\sum _{i=0}^{n-1}{r}_{ei}{s}_{ei}=0,$$

which implies that ${\sum }_{i=0}^{n-1}{r}_{ji}{s}_{ji}=0$ for all $j\in \{1,\cdots ,e\}$. Now,

$$\mathsf{\Phi }\left(v\right)=((r_{10},r_{20},\cdots ,r_{e0})M,(r_{11},r_{21},\cdots ,r_{e1})M,\cdots ,(r_{1(n-1)},r_{2(n-1)},\cdots ,r_{e(n-1)})M),$$

$$\mathsf{\Phi }\left(x\right)=((s_{10},s_{20},\cdots ,s_{e0})M,(s_{11},s_{21},\cdots ,s_{e1})M,\cdots ,(s_{1(n-1)},s_{2(n-1)},\cdots ,s_{e(n-1)})M)$$

and

$$\begin{array}{cc}\hfill \langle \mathsf{\Phi }\left(v\right),\mathsf{\Phi }\left(x\right)\rangle =\mathsf{\Phi }\left(v\right){\left[\mathsf{\Phi }\left(x\right)\right]}^t& =\sum _{i=0}^{n-1}(r_{1i},r_{2i},\cdots ,r_{ei})M{M}^t{(s_{1i},s_{2i},\cdots ,s_{ei})}^t\hfill \\ & =\nu \sum _{i=0}^{n-1}(r_{1i},r_{2i},\cdots ,r_{ei}){(s_{1i},s_{2i},\cdots ,s_{ei})}^t\hfill \\ & =\nu \sum _{i=0}^{n-1}{r}_{1i}{s}_{1i}+\nu \sum _{i=0}^{n-1}{r}_{2i}{s}_{2i}+\cdots +\nu \sum _{i=0}^{n-1}{r}_{ei}{s}_{ei}\hfill \\ & =0,\hfill \end{array}$$

i.e., $\mathsf{\Phi }\left(x\right)\in \mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }$. Therefore, $\mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }\supseteq \mathsf{\Phi }\left({\mathcal{V}}^{\perp }\right)$. Moreover, $|\mathsf{\Phi }\left({\mathcal{V}}^{\perp }\right)|=|\mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }|$ as $\mathsf{\Phi }$ is a bijective map. Hence $\mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }=\mathsf{\Phi }\left({\mathcal{V}}^{\perp }\right).$ □

4. LCD Codes

This section first presents the dual of a skew constacyclic code, and then obtains conditions under which these codes must be LCD. Further, we study their Gray images.

For any $p\left(x\right)={\sum }_{i=0}^r{p}_i{x}^i\in {\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]$ of degree r with $p_0\in {\mathcal{B}}_e^{\ast }$, its left monic skew reciprocal polynomial is defined as $p^{♮}\left(x\right)={\left({\mathsf{\Theta }}_l^r\left(p_0\right)\right)}^{-1}{\sum }_{i=0}^r{\mathsf{\Theta }}_l^i\left(p_{r-i}\right)x^i$. The polynomial $p\left(x\right)$ is called as self-reciprocal iff $p\left(x\right)=p^{♮}\left(x\right)$. This monic skew reciprocal polynomial is useful for the investigation of dual code ${\mathcal{V}}^{\perp }$. If ${\mathsf{\Theta }}_l\left(\alpha \right)=\alpha$ and n is divisible by order of ${\mathsf{\Theta }}_l$, then the dual ${\mathcal{V}}^{\perp }$ of a skew ${\mathsf{\Theta }}_l-\alpha$-constacyclic code $\mathcal{V}=\langle g\left(x\right)\rangle$ over ${\mathbb{F}}_q$ is a skew ${\mathsf{\Theta }}_l-{\alpha }^{-1}$-constacyclic code over ${\mathbb{F}}_q$, which is given by ${\mathcal{V}}^{\perp }=\langle h^{♮}\left(x\right)\rangle$, where $g\left(x\right)h\left(x\right)=h\left(x\right)g\left(x\right)=x^n-\alpha$ for the length n of $\mathcal{V}$. Now, under the same conditions on n and considering ${\alpha }_i$ ( $1\le i\le e$) fixed by ${\mathsf{\Theta }}_l$, we obtain the following result.

Theorem 8. 

Suppose $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i\in {\mathcal{B}}_e^{\ast }$ and $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code over ${\mathcal{B}}_e$ having the generator polynomial $g\left(x\right)={\sum }_{i=1}^e{ϵ}_i{g}_i\left(x\right)$, where $g_i\left(x\right)$ is the generator polynomial of ${\mathcal{V}}_i$ and $g_i\left(x\right)h_i\left(x\right)=h_i\left(x\right)g_i\left(x\right)=x^n-{\alpha }_i$ for each $1\le i\le e$. Then the generator polynomial of ${\mathcal{V}}^{\perp }$ is $h^{♮}\left(x\right)$, where $h^{♮}\left(x\right)={\sum }_{i=1}^e{ϵ}_i{h}_i^{♮}\left(x\right)$ for $h\left(x\right)={\sum }_{i=1}^e{ϵ}_i{h}_i\left(x\right)$.

Definition 9 

(LCD codes). If $\mathcal{V}\cap {\mathcal{V}}^{\perp }=\left\{\mathbf{0}\right\}$ for any linear code $\mathcal{V}$ over ${\mathcal{B}}_e$, then the code $\mathcal{V}$ is considered as LCD (or complementary dual).

In order to check the complementary duality of skew constacyclic codes over ${\mathcal{B}}_e$, we first need a criterion for evaluating the complementary duality of a skew constacyclic code over finite field.

Lemma 10. 

[28, Theorem 4.1] Suppose that n is divisible by order of ${\mathsf{\Theta }}_l$, $\alpha \in {\mathbb{F}}_q^{\ast }$ such that ${\alpha }^2=1$ and $\mathcal{V}$ is a skew ${\mathsf{\Theta }}_l-\alpha$-constacyclic code over finite field with skew generator polynomial $g\left(x\right)$. Choose $h\left(x\right)$ in order to ensure $g\left(x\right)h\left(x\right)=h\left(x\right)g\left(x\right)=x^n-\alpha$. Then $\mathcal{V}$ is a Euclidean LCD code iff $gcrd(g\left(x\right),h^{♮}\left(x\right))=1$, where $gcrd(g\left(x\right),h^{♮}\left(x\right))$ denotes the greatest common right divisor of $h^{♮}\left(x\right)$ and $g\left(x\right)$.

Once we get a link among the complementary duality of a linear code over ${\mathcal{B}}_e$ with its constituent codes, we can get requirements for complementary duality of any skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code over ${\mathcal{B}}_e$. So, first we establish that link which is presented in the below mentioned result.

Proposition 11. 

Suppose $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e$ is a linear code over ${\mathcal{B}}_e$. Then $\mathcal{V}$ is an LCD code iff each ${\mathcal{V}}_i$ is an LCD code over ${\mathbb{F}}_q$.

Proof. 

Suppose $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e$ is a linear code over ${\mathcal{B}}_e$. The dual code is ${\mathcal{V}}^{\perp }={ϵ}_1{\mathcal{V}}_1^{\perp }\oplus {ϵ}_2{\mathcal{V}}_2^{\perp }\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e^{\perp }$ and

$$\mathcal{V}\cap {\mathcal{V}}^{\perp }={ϵ}_1({\mathcal{V}}_1\cap {\mathcal{V}}_1^{\perp })\oplus {ϵ}_2({\mathcal{V}}_2\cap {\mathcal{V}}_2^{\perp })\oplus \cdots \oplus {ϵ}_e({\mathcal{V}}_e\cap {\mathcal{V}}_e^{\perp }).$$

Consequently, it is evident that $\mathcal{V}\cap {\mathcal{V}}^{\perp }=\left\{\mathbf{0}\right\}$ iff ${\mathcal{V}}_i\cap {\mathcal{V}}_i^{\perp }=\left\{\mathbf{0}\right\}$ for every $1\le i\le e$. □

From now onwards, we work under the assumption that ${\alpha }_i$’s are fixed by ${\mathsf{\Theta }}_l$ for $1\le i\le e$, and the order of ${\mathsf{\Theta }}_l$ divides n. Using the Proposition 11 and Lemma 10, we now obtain requirements for skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code over ${\mathcal{B}}_e$ to be complementary dual, where $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i\in {\mathcal{B}}_e^{\ast }$ for ${\alpha }_i=\pm 1$.

Theorem 12. 

Suppose $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i\in {\mathcal{B}}_e^{\ast }$ for ${\alpha }_i\in \{1,-1\}$ and n is a multiple of order of ${\mathsf{\Theta }}_l$. Then a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code $\mathcal{V}=\langle {ϵ}_1{g}_1\left(x\right)+{ϵ}_2{g}_2\left(x\right)+\cdots +{ϵ}_e{g}_e\left(x\right)\rangle$ having length n over ${\mathcal{B}}_e$ is an LCD code iff $gcrd(g_i\left(x\right),h_i^{♮}\left(x\right))=1$, where $gcrd(g_i\left(x\right),h_i^{♮}\left(x\right))$ denotes the greatest common right divisor of $h_i^{♮}\left(x\right)$ and $g_i\left(x\right)$ for each $1\le i\le e$.

Proof. 

Suppose ${\mathcal{V}}_i=\langle g_i\left(x\right)\rangle$ is a skew ${\mathsf{\Theta }}_l-{\alpha }_i$ constacyclic code over ${\mathbb{F}}_q$ and $h_i\left(x\right)$ is such that $g_i\left(x\right)h_i\left(x\right)=h_i\left(x\right)g_i\left(x\right)=x^n-{\alpha }_i$ for $1\le i\le e$. Then, by Lemma 10, ${\mathcal{V}}_i$ is an LCD code iff $gcrd(g_i\left(x\right),h_i^{♮}\left(x\right))=1$ for each $1\le i\le e$. Therefore, $\mathcal{V}$ is an LCD code iff $gcrd(g_i\left(x\right),h_i^{♮}\left(x\right))=1$ for each $1\le i\le e$. □

Next, we present a relation between the Gray image of the intersection of a code and its dual with the intersection of their Gray images, which can be proved by using Lemma 7.

Lemma 13. 

Suppose $\mathcal{V}$ is a linear code having length n over ${\mathcal{B}}_e$. Then $\mathsf{\Phi }(\mathcal{V}\cap {\mathcal{V}}^{\perp })=\mathsf{\Phi }\left(\mathcal{V}\right)\cap \mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }$, where Φ is the Gray map defined in Section 3.

Proof. 

Suppose $c\in \mathsf{\Phi }\left(\mathcal{V}\right)\cap \mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }$. Then there exist $a\in \mathcal{V}$ and $b\in {\mathcal{V}}^{\perp }$ such that $c=\mathsf{\Phi }\left(a\right)=\mathsf{\Phi }\left(b\right)$ as $\mathsf{\Phi }\left({\mathcal{V}}^{\perp }\right)=\mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }$ by Lemma 7. But $\mathsf{\Phi }$ is injective, so we have $a=b$ and hence $c\in \mathsf{\Phi }(\mathcal{V}\cap {\mathcal{V}}^{\perp })$. Therefore, $\mathsf{\Phi }\left(\mathcal{V}\right)\cap \mathsf{\Phi }\left({\mathcal{V}}^{\perp }\right)\subseteq \mathsf{\Phi }(\mathcal{V}\cap {\mathcal{V}}^{\perp })$. Conversely, assume that $c\in \mathsf{\Phi }(\mathcal{V}\cap {\mathcal{V}}^{\perp })$. Then $c=\mathsf{\Phi }\left(a\right)$ for some $a\in \mathcal{V}\cap {\mathcal{V}}^{\perp }$. Further, $c=\mathsf{\Phi }\left(a\right)\in \mathsf{\Phi }\left(\mathcal{V}\right)\cap \mathsf{\Phi }\left({\mathcal{V}}^{\perp }\right)$. Using Lemma 7, we get $c=\mathsf{\Phi }\left(a\right)\in \mathsf{\Phi }\left(\mathcal{V}\right)\cap \mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }$ as $\mathsf{\Phi }\left({\mathcal{V}}^{\perp }\right)=\mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }$. Therefore, $\mathsf{\Phi }(\mathcal{V}\cap {\mathcal{V}}^{\perp })\subseteq \mathsf{\Phi }\left(\mathcal{V}\right)\cap \mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }$ and the result follows. □

By applying this lemma, we are able to derive a relationship between the complementary duality of a linear code over ${\mathcal{B}}_e$ and its Gray image under the map $\mathsf{\Phi }$.

Theorem 14. 

Suppose $\mathcal{V}$ is a linear code over ${\mathcal{B}}_e$. Then $\mathcal{V}$ is an LCD code iff $\mathsf{\Phi }\left(\mathcal{V}\right)$ is an LCD code over ${\mathbb{F}}_q$.

Proof. 

As $\mathsf{\Phi }$ is injective, the proof follows using Lemma 13. □

5. Quantum Codes

In this section, we consider $\gamma$ to be a unit in ${\mathcal{B}}_e$ which is fixed by ${\mathsf{\Theta }}_l$ and n be divisible by order of ${\mathsf{\Theta }}_l$. We obtain the dual containing conditions for a skew $\gamma$-constacyclic code over ${\mathcal{B}}_e$ and then establish the existence of a quantum code by applying the CSS construction on their Gray images. In this direction, we first recall some preliminary definitions and results.

Following [34], for a Hilbert space ${\mathbb{H}}^q$ of dimension q over the field of complex numbers $\mathbb{C}$, ${\left({\mathbb{H}}^q\right)}^{\otimes n}={\mathbb{H}}^q\otimes \cdots \otimes {\mathbb{H}}^q$ is also a Hilbert space of dimension $q^n$. Further, any $q^k$-dimensional subspace of the Hilbert space ${\left({\mathbb{H}}^q\right)}^{\otimes n}$ is called a quantum code denoted by ${\left[[n,k,d]\right]}_q$, where d is the minimum distance of the code. Further, for the comparison of two quantum codes with parameters ${\left[[n,k,d]\right]}_q$ and ${\left[[n^{\prime },k^{\prime },d^{\prime }]\right]}_q$, we have the following conditions:

• $d=d^{\prime }$ and ${\displaystyle \frac{k}{n}}>{\displaystyle \frac{k^{\prime }}{n^{\prime }}}$
• $d>d^{\prime }$ and ${\displaystyle \frac{k}{n}}={\displaystyle \frac{k^{\prime }}{n^{\prime }}}$
• $d>d^{\prime }$ and ${\displaystyle \frac{k}{n}}>{\displaystyle \frac{k^{\prime }}{n^{\prime }}}$.

If any one of the above conditions is satisfied, then we say that the quantum code with the parameters ${\left[[n,k,d]\right]}_q$ is better than the one with the parameters ${\left[[n^{\prime },k^{\prime },d^{\prime }]\right]}_q$. Now, we recall the condition for a skew constacyclic code over ${\mathbb{F}}_q$ to contain its dual.

Lemma 15. 

[41, Lemma 5.3] Suppose $\mathcal{V}=\langle g\left(x\right)\rangle$ is a skew ${\mathsf{\Theta }}_l-\alpha$-constacyclic code having length n over ${\mathbb{F}}_q$, where $x^n-\alpha =h\left(x\right)g\left(x\right)$ for $\alpha =\pm 1$ and n be a multiple of the order of ${\mathsf{\Theta }}_l$. Then ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ iff $x^n-\alpha$ is right divisible by $h^{♮}\left(x\right)h\left(x\right)$.

The following result presents a relation between the dual containing property of a skew constacyclic code over ${\mathcal{B}}_e$ in terms of its constituent codes in the decomposition.

Lemma 16. 

Suppose $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length n over ${\mathcal{B}}_e$. Then ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ iff ${\mathcal{V}}_i^{\perp }\subseteq {\mathcal{V}}_i$ for $1\le i\le e$.

Proof. 

Suppose $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length n over ${\mathcal{B}}_e$, ${\mathcal{V}}^{\perp }={ϵ}_1{\mathcal{V}}_1^{\perp }\oplus {ϵ}_2{\mathcal{V}}_2^{\perp }\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e^{\perp }$ be its dual code such that ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$. Then multiplying by ${ϵ}_i$ in the equation

$${ϵ}_1{\mathcal{V}}_1^{\perp }\oplus {ϵ}_2{\mathcal{V}}_2^{\perp }\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e^{\perp }\subseteq {ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e,$$

we get ${ϵ}_i{\mathcal{V}}_i^{\perp }\subseteq {ϵ}_i{\mathcal{V}}_i$ for each $1\le i\le e$. As ${\mathcal{V}}_i$’s are codes over ${\mathbb{F}}_q$, it further implies that ${\mathcal{V}}_i^{\perp }\subseteq {\mathcal{V}}_i$ for each $1\le i\le e$.

Conversely, assume that ${\mathcal{V}}_i^{\perp }\subseteq {\mathcal{V}}_i$ for each $1\le i\le e$. Then ${ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e\subseteq {ϵ}_1{\mathcal{V}}_1^{\perp }\oplus {ϵ}_2{\mathcal{V}}_2^{\perp }\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e^{\perp }$, i.e., ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ which completes the proof. □

Using this relation and Lemma 15, we now obtain conditions for skew constacyclic code over ${\mathcal{B}}_e$ to contain its dual code.

Theorem 17. 

Suppose $\mathcal{V}=\langle g\left(x\right)\rangle$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length n over ${\mathcal{B}}_e$, where n is a multiple of the order of ${\mathsf{\Theta }}_l$, $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i$ for ${\alpha }_i=\pm 1$, $g\left(x\right)={\sum }_{i=1}^e{ϵ}_i{g}_i\left(x\right)$ such that $x^n-{\alpha }_i=h_i\left(x\right)g_i\left(x\right)$ for $1\le i\le e$. Then ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ iff $x^n-{\alpha }_i$ is right divisible by $h_i^{♮}\left(x\right)h_i\left(x\right)$ for each $1\le i\le e$.

Proof. 

Suppose $x^n-{\alpha }_i$ is right divisible by $h_i^{♮}\left(x\right)h_i\left(x\right)$ for each $1\le i\le e$. Then ${\mathcal{V}}_i^{\perp }\subseteq {\mathcal{V}}_i$ by Lemma 15, where ${\mathcal{V}}_i=\langle g_i\left(x\right)\rangle$ for $1\le i\le e$. It implies that

$${\mathcal{V}}^{\perp }={ϵ}_1{\mathcal{V}}_1^{\perp }\oplus {ϵ}_2{\mathcal{V}}_2^{\perp }\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e^{\perp }\subseteq {ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e=\mathcal{V}.$$

Conversely, assume that ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$, i.e.,

$${ϵ}_1{\mathcal{V}}_1^{\perp }\oplus {ϵ}_2{\mathcal{V}}_2^{\perp }\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e^{\perp }\subseteq {ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2\oplus \cdots \oplus {ϵ}_e{\mathcal{V}}_e.$$

Then multiplying by ${ϵ}_i$ on both the sides, we get

$${ϵ}_i{\mathcal{V}}_i^{\perp }\subseteq {ϵ}_i{\mathcal{V}}_i\mathrm{for}\mathrm{each}1\le i\le e.$$

It further implies that ${\mathcal{V}}_i^{\perp }\subseteq {\mathcal{V}}_i$ as ${\mathcal{V}}_i$’s are linear codes over ${\mathbb{F}}_q$ and hence $x^n-{\alpha }_i$ is right divisible by $h_i^{♮}\left(x\right)h_i\left(x\right)$ for each $1\le i\le e$. □

To obtain quantum codes from dual containing codes, we use the CSS construction [35, Theorem 3], which is given below.

Lemma 18 

(CSS construction). Suppose ${\mathcal{V}}_1={[n,k_1,d_1]}_q$ and ${\mathcal{V}}_2={[n,k_2,d_2]}_q$ are linear codes over ${\mathbb{F}}_q$ with ${\mathcal{V}}_2^{\perp }\subseteq {\mathcal{V}}_1$. Assume that $d=min\{w_H\left(v\right):v\in ({\mathcal{V}}_1\setminus {\mathcal{V}}_2^{\perp })\cup ({\mathcal{V}}_2\setminus {\mathcal{V}}_1^{\perp })\}\ge min\{d_1,d_2\}$. Then there exists a QECC with parameters ${\left[[n,k_1+k_2-n,d]\right]}_q$. In particular, if ${\mathcal{V}}_1={\mathcal{V}}_2$ and let $d=min\{w_H\left(v\right):v\in ({\mathcal{V}}_1\setminus {\mathcal{V}}_1^{\perp })\}$, then there exists a QECC with parameters ${\left[[n,2k_1-n,d]\right]}_q$.

Now, we employ the above CSS construction to obtain quantum codes from the dual containing skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic codes over ${\mathcal{B}}_e$, where $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i$ for ${\alpha }_i=\pm 1$.

Theorem 19. 

Suppose $\mathcal{V}$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code over ${\mathcal{B}}_e$ such that ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ and $\mathsf{\Phi }\left(\mathcal{V}\right)$ have the parameters $[en,k,d]$. Then, the existence of a quantum code over ${\mathbb{F}}_q$ having the parameters ${\left[[en,2k-en,d]\right]}_q$ is guaranteed.

Proof. 

Suppose $\mathcal{V}$ is a skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code over ${\mathcal{B}}_e$ with $\mathcal{V}\supseteq {\mathcal{V}}^{\perp }$. Then $\mathsf{\Phi }\left(\mathcal{V}\right)$ is a linear code over ${\mathbb{F}}_q$ with $\mathsf{\Phi }\left(\mathcal{V}\right)\supseteq \mathsf{\Phi }\left({\mathcal{V}}^{\perp }\right)=\mathsf{\Phi }{\left(\mathcal{V}\right)}^{\perp }$. By using Lemma 18, the existence of a quantum code over ${\mathbb{F}}_q$ having the parameters ${\left[[en,2k-en,d]\right]}_q$ is guaranteed. □

6. Complementary Dual and Quantum Codes from ${\mathbb{F}}_q{\mathcal{B}}_e$-Additive Skew Constacyclic Codes

In this section, we examine ${\mathbb{F}}_q{\mathcal{B}}_e$-additive codes. We first obtain dual containing conditions for an additive skew constacyclic code over ${\mathbb{F}}_q{\mathcal{B}}_e$ and establish the existence of a quantum code with certain parameters. Further, we obtain conditions for an additive skew constacyclic code to be ACD (additive complementary dual) in some instances. Throughout this section, we consider $\gamma$ to be a unit in ${\mathcal{B}}_e$ which is fixed by ${\mathsf{\Theta }}_l$ and n is divisible by the order of ${\mathsf{\Theta }}_l$.

Consider the set ${\mathbb{F}}_q{\mathcal{B}}_e=\{(a,b):a\in {\mathbb{F}}_q,b\in {\mathcal{B}}_e\}$ which forms a group under componentwise addition. Define a projection map $\pi :{\mathcal{B}}_e\to {\mathbb{F}}_q$ as $\pi (s_0+u{s}_1+\cdots +u^{e-1}{s}_{e-1})=s_0$, where $s_i\in {\mathbb{F}}_q$ for $0\le i\le e-1$. Further, we define a multiplication $\ast :{\mathcal{B}}_e\times {\mathbb{F}}_q^m{\mathcal{B}}_e^n\to {\mathbb{F}}_q^m{\mathcal{B}}_e^n$ as $s\ast (a_0,a_1,\cdots ,a_{m-1},b_0,b_1,\cdots ,b_{n-1})=(\pi \left(s\right)a_0,\pi \left(s\right)a_1,\cdots ,\pi \left(s\right)a_{m-1},s{b}_0,s{b}_1,\cdots ,s{b}_{n-1})$, where $s,b_i\in {\mathcal{B}}_e$ and $a_j\in {\mathbb{F}}_q$ for $0\le i\le n-1,0\le j\le m-1$. Then, it can be checked that the set ${\mathbb{F}}_q^m{\mathcal{B}}_e^n$ defined as

$${\mathbb{F}}_q^m{\mathcal{B}}_e^n=\{(a,b):a\in {\mathbb{F}}_q^m,b\in {\mathcal{B}}_e^n\}$$

forms a ${\mathcal{B}}_e$-module under the componentwise addition and the multiplication defined by ‘*’. Further, we recall that any ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code $\mathcal{V}$ having length $(m,n)$ is a non-empty subset of ${\mathbb{F}}_q^m{\mathcal{B}}_e^n$ such that $\mathcal{V}$ forms a ${\mathcal{B}}_e$-submodule of the module ${\mathbb{F}}_q^m{\mathcal{B}}_e^n$. The dual code ${\mathcal{V}}^{\perp }$ of an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code $\mathcal{V}$, which is defined as

$${\mathcal{V}}^{\perp }=\{z\in {\mathbb{F}}_q^m{\mathcal{B}}_e^n:[z,c]=0\forall c\in \mathcal{V}\}$$

where

$$[z,c]=u^{e-1}\left(\sum _{i=0}^{m-1}{a}_i{a}_i^{\prime }\right)+\sum _{j=0}^{n-1}{b}_j{b}_j^{\prime }$$

for $z=(a_0^{\prime },a_1^{\prime },\cdots ,a_{m-1}^{\prime },b_0^{\prime },b_1^{\prime },\cdots ,b_{n-1}^{\prime })$ and $c=(a_0,a_1,\cdots ,a_{m-1},b_0,b_1,\cdots ,b_{n-1})$, is also a ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code having length $(m,n)$. We say that an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code $\mathcal{V}$ is an additive complementary dual (ACD) code if $\mathcal{V}\cap {\mathcal{V}}^{\perp }=\left\{\mathbf{0}\right\}$. Now, we define an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew constacyclic code.

Definition 20. 

For a unit element $\gamma \in {\mathcal{B}}_e$ and an automorphism ${\mathsf{\Theta }}_l$, an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code $\mathcal{V}$ having length $(m,n)$ is said to be an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code if $({\tau }_1\left(a\right),{\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(b\right))\in \mathcal{V}$ for every $(a,b)\in \mathcal{V}$ where ${\tau }_1$, ${\tau }_{{\mathsf{\Theta }}_l,\gamma }$ are the cyclic and the skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic shifts, respectively.

For a unit element $\gamma \in {\mathcal{B}}_e$, consider the set $S_{(m,n)}={\displaystyle \frac{{\mathbb{F}}_q\left[x\right]}{\langle x^m-1\rangle }}\times {\displaystyle \frac{{\mathcal{B}}_e[x;{\mathsf{\Theta }}_l]}{\langle x^n-\gamma \rangle }}$. We identify an element $(a,b)\in {\mathbb{F}}_q^m{\mathcal{B}}_e^n$ by the polynomials $(a\left(x\right),b\left(x\right))\in S_{(m,n)}$ where $a=(a_0,a_1,\cdots ,a_{m-1}),b=(b_0,b_1,\cdots ,b_{n-1}),a\left(x\right)=a_0+a_{1}x+\cdots ,a_{m-1}{x}^{m-1}$ and $b\left(x\right)=b_0+b_{1}x+\cdots +b_{n-1}{x}^{n-1}$. Now, we define the corresponding multiplication $\ast :{\mathcal{B}}_e\left[x\right]\times S_{(m,n)}\to S_{(m,n)}$ as

$$z\left(x\right)\ast \left(a\right(x),b(x\left)\right)=\left(\pi \right(z\left(x\right)\left)a\right(x),z(x\left)b\right(x\left)\right)$$

where $\pi \left(z\left(x\right)\right)={\sum }_i\pi \left(z_i\right)x^i$ for any $z\left(x\right)={\sum }_i{z}_i{x}^i\in {\mathcal{B}}_e\left[x\right]$ and $(a\left(x\right),b\left(x\right))\in S_{(m,n)}$. The set $S_{(m,n)}$ forms a ${\mathcal{B}}_e\left[x\right]$-module with respect to the usual componentwise addition of polynomials and the multiplication defined by *. Now, under the above identification of vectors by polynomials and considering the above module structure of $S_{(m,n)}$, an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew constacyclic code can be seen as a ${\mathcal{B}}_e\left[x\right]$-submodule of $S_{(m,n)}$ as given below.

Theorem 21. 

Suppose $\mathcal{V}$ is an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code having length $(m,n)$. Then, it is an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length $(m,n)$ iff $\mathcal{V}$ is a ${\mathcal{B}}_e\left[x\right]$-submodule of the module $S_{(m,n)}$.

Proof. 

Suppose $\mathcal{V}$ is an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length $(m,n)$ and $\left(a\right(x),b(x\left)\right)\in \mathcal{V}$ with the vector representation $(a,b)$. Then $({\tau }_1\left(a\right),{\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(b\right))\in \mathcal{V}$. Note that

$$x\ast (a\left(x\right),b\left(x\right))=(a_{m-1}+a_{0}x+\cdots +a_{m-2}{x}^{m-1},\gamma {\mathsf{\Theta }}_l\left(b_{n-1}\right)+{\mathsf{\Theta }}_l\left(b_0\right)x+\cdots ,{\mathsf{\Theta }}_l\left(b_{n-2}\right)x^{n-1})$$

which corresponds to $({\tau }_1\left(a\right),{\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(b\right))\in \mathcal{V}$. It implies that $x\ast \left(a\right(x),b(x\left)\right)\in \mathcal{V}$ and hence $x^i\ast (a\left(x\right),b\left(x\right))\in \mathcal{V}$ for any non-negative integer i. Also, using the polynomial identification, we get $z\left(x\right)\ast \left(a\right(x),b(x\left)\right)\in \mathcal{V}$ for any $z\left(x\right)\in {\mathcal{B}}_e\left[x\right]$ as $\mathcal{V}$ is a ${\mathcal{B}}_e$-submodule of the module ${\mathbb{F}}_q^m{\mathcal{B}}_e^n$. Therefore, $\mathcal{V}$ is a ${\mathcal{B}}_e\left[x\right]$-submodule of the module $S_{(m,n)}$.

Conversely, assume that $\mathcal{V}$ is a ${\mathcal{B}}_e\left[x\right]$-submodule of the module $S_{(m,n)}$ and $(a,b)\in \mathcal{V}$ with the polynomial representation $\left(a\right(x),b(x\left)\right)$. Then $x\ast \left(a\right(x),b(x\left)\right)\in \mathcal{V}$. Note that the polynomial representation of $({\tau }_1\left(a\right),{\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(b\right))$ is $x\ast \left(a\right(x),b(x\left)\right)$. Therefore, $({\tau }_1\left(a\right),{\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(b\right))\in \mathcal{V}$ and hence $\mathcal{V}$ is an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code. □

Now, we define a Gray map $\mathsf{\Psi }:{\mathbb{F}}_q^m{\mathcal{B}}_e^n\to {\mathbb{F}}_q^{m+en}$ as

$$\mathsf{\Psi }(a,r)=(a,\mathsf{\Phi }(r\left)\right)$$

where $a\in {\mathbb{F}}_q^m,r\in {\mathcal{B}}_e^n$ and $\mathsf{\Phi }$ is the map defined in Section 3. It can be checked that $\mathsf{\Psi }$ is a linear map, which is also bijective. Further, $\mathsf{\Psi }$ preserves duality as stated in the next result which can be proved on similar lines to [37, Lemma 6].

Lemma 22. 

Suppose $\mathcal{V}$ is an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code having length $(m,n)$ with its dual code ${\mathcal{V}}^{\perp }$. Then $\mathsf{\Psi }\left({\mathcal{V}}^{\perp }\right)=\mathsf{\Psi }{\left(\mathcal{V}\right)}^{\perp }$.

Now, we define two projection maps ${\pi }_n:{\mathbb{F}}_q^m{\mathcal{B}}_e^n\to {\mathcal{B}}_e^n$, ${\pi }_m:{\mathbb{F}}_q^m{\mathcal{B}}_e^n\to {\mathbb{F}}_q^m$ as ${\pi }_n(a,b)=b,{\pi }_m(a,b)=a$ where $a\in {\mathbb{F}}_q^m$ and $b\in {\mathcal{B}}_e^n$. Then, for any ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code $\mathcal{V}$ having length $(m,n)$, ${\pi }_m\left(\mathcal{V}\right)$ and ${\pi }_n\left(\mathcal{V}\right)$ are linear codes having lengths $m,n$ over ${\mathbb{F}}_q$ and ${\mathcal{B}}_e$, respectively. Further, we call an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code $\mathcal{V}$ to be a separable code if it can be written in terms of the codes ${\pi }_m\left(\mathcal{V}\right)$ and ${\pi }_n\left(\mathcal{V}\right)$ as given below.

Definition 23. 

An ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code $\mathcal{V}$ having length $(m,n)$ is said to be a separable code, if $\mathcal{V}={\pi }_m\left(\mathcal{V}\right)\times {\pi }_n\left(\mathcal{V}\right)$.

If $\mathcal{V}$ is a separable code then its dual code is ${\mathcal{V}}^{\perp }={\pi }_m{\left(\mathcal{V}\right)}^{\perp }\times {\pi }_n{\left(\mathcal{V}\right)}^{\perp }$. Further, the below result can classify a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew constacyclic code.

Theorem 24. 

Suppose $\mathcal{V}$ is a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code having length $(m,n)$. Then, it is an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length $(m,n)$ iff ${\pi }_m\left(\mathcal{V}\right)$, ${\pi }_n\left(\mathcal{V}\right)$ are cyclic and skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic codes over ${\mathbb{F}}_q$ and ${\mathcal{B}}_e$, respectively.

Proof. 

Suppose $\mathcal{V}$ is a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length $(m,n)$, $a\in {\pi }_m\left(\mathcal{V}\right)$ and $b\in {\pi }_n\left(\mathcal{V}\right)$. Then $(a,b)\in \mathcal{V}$ and $({\tau }_1\left(a\right),{\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(b\right))\in \mathcal{V}$. That is, ${\tau }_1\left(a\right)\in {\pi }_m\left(\mathcal{V}\right)$ and ${\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(b\right)\in {\pi }_n\left(\mathcal{V}\right)$. Therefore, ${\pi }_m\left(\mathcal{V}\right)$, ${\pi }_n\left(\mathcal{V}\right)$ are cyclic and skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic codes, respectively.

Conversely, assume that ${\pi }_m\left(\mathcal{V}\right)$, ${\pi }_n\left(\mathcal{V}\right)$ be cyclic and skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic codes and $(a,b)\in \mathcal{V}$. Then $a\in {\pi }_m\left(\mathcal{V}\right)$ and $b\in {\pi }_n\left(\mathcal{V}\right)$. Therefore, ${\tau }_1\left(a\right)\in {\pi }_m\left(\mathcal{V}\right)$ and ${\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(b\right)\in {\pi }_n\left(\mathcal{V}\right)$ as ${\pi }_m\left(\mathcal{V}\right)$, ${\pi }_n\left(\mathcal{V}\right)$ are cyclic and skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic codes, respectively. Hence, $({\tau }_1\left(a\right),{\tau }_{{\mathsf{\Theta }}_l,\gamma }\left(b\right))\in \mathcal{V}$, i.e., $\mathcal{V}$ is an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code. □

The above classification is later used to obtain quantum codes from separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic codes. For achieving this, we first derive the necessary and sufficient conditions for a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code to contain its dual code.

Lemma 25. 

Suppose $\mathcal{V}={\pi }_m\left(\mathcal{V}\right)\times {\pi }_n\left(\mathcal{V}\right)$ is a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code having length $(m,n)$. Then ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ iff ${\pi }_m{\left(\mathcal{V}\right)}^{\perp }\subseteq {\pi }_m\left(\mathcal{V}\right)$ and ${\pi }_n{\left(\mathcal{V}\right)}^{\perp }\subseteq {\pi }_n\left(\mathcal{V}\right)$.

Proof. 

As $\mathcal{V}$ is a separable code, we have ${\mathcal{V}}^{\perp }={\pi }_m{\left(\mathcal{V}\right)}^{\perp }\times {\pi }_n{\left(\mathcal{V}\right)}^{\perp }$. Therefore, ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ iff ${\pi }_m{\left(\mathcal{V}\right)}^{\perp }\subseteq {\pi }_m\left(\mathcal{V}\right)$ and ${\pi }_n{\left(\mathcal{V}\right)}^{\perp }\subseteq {\pi }_n\left(\mathcal{V}\right)$. □

The following result obtains the condition for constacyclic code over ${\mathbb{F}}_q$ to contain its dual code.

Lemma 26. 

[32] Suppose $\mathcal{V}=\langle f\left(x\right)\rangle$ is an α-constacyclic code having length m over ${\mathbb{F}}_q$ for $\alpha =\pm 1$. Then ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ iff $x^m-\alpha \equiv 0(modf\left(x\right)f^{\ast }\left(x\right))$.

Proposition 27. 

Suppose $\mathcal{V}={\pi }_m\left(\mathcal{V}\right)\times {\pi }_n\left(\mathcal{V}\right)$ is a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length $(m,n)$, where $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i\in {\mathcal{B}}_e^{\ast }$ for ${\alpha }_i=\pm 1$. Assume that ${\pi }_m\left(\mathcal{V}\right)=\langle f\left(x\right)\rangle$ and ${\pi }_n\left(\mathcal{V}\right)=\langle {\sum }_{i=1}^e{ϵ}_i{g}_i\left(x\right)\rangle$. Choose $h\left(x\right),h_i\left(x\right)$ for $1\le i\le e$ such that $x^m-1=f\left(x\right)h\left(x\right)$ and $x^n-{\alpha }_i=h_i\left(x\right)g_i\left(x\right)$ for $1\le i\le e$. Then ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ iff $x^n-{\alpha }_i$ is right divisible by $h_i^{♮}\left(x\right)h_i\left(x\right)$ for each $1\le i\le e$ and $x^m-1\equiv 0(modf\left(x\right)f^{\ast }\left(x\right))$.

Proof. 

From Lemma 25, ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$ iff ${\pi }_m{\left(\mathcal{V}\right)}^{\perp }\subseteq {\pi }_m\left(\mathcal{V}\right)$, ${\pi }_n{\left(\mathcal{V}\right)}^{\perp }\subseteq {\pi }_n\left(\mathcal{V}\right)$ where ${\pi }_m\left(\mathcal{V}\right)$, ${\pi }_n\left(\mathcal{V}\right)$ are cyclic and skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic codes having lengths $m,n$ over ${\mathbb{F}}_q$ and ${\mathcal{B}}_e$, respectively. Rest of the result follows using Lemma 26 and Theorem 17. □

Proposition 27 gives the necessary and sufficient condition for a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code to contain its dual. Using the CSS construction and Proposition 27, we now present the construction of quantum codes from separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic codes in the next theorem.

Theorem 28. 

Suppose $\mathcal{V}={\pi }_m\left(\mathcal{V}\right)\times {\pi }_n\left(\mathcal{V}\right)$ is a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length $(m,n)$ such that ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$, where $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i\in {\mathcal{B}}_e^{\ast }$ for ${\alpha }_i=\pm 1$. Assume that the parameters of the code $\mathsf{\Psi }\left(\mathcal{V}\right)$ be $[m+en,k,d_H]$. Then, there exists a quantum code with the parameters ${\left[[m+en,2k-(m+en),d_H]\right]}_q$ over ${\mathbb{F}}_q$.

Proof. 

Suppose $\mathcal{V}={\pi }_m\left(\mathcal{V}\right)\times {\pi }_n\left(\mathcal{V}\right)$ is a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code such that ${\mathcal{V}}^{\perp }\subseteq \mathcal{V}$. Then $\mathsf{\Psi }\left({\mathcal{V}}^{\perp }\right)\subseteq \mathsf{\Psi }\left(\mathcal{V}\right)$ which implies that $\mathsf{\Psi }{\left(\mathcal{V}\right)}^{\perp }\subseteq \mathsf{\Psi }\left(\mathcal{V}\right)$ as $\mathsf{\Psi }\left({\mathcal{V}}^{\perp }\right)=\mathsf{\Psi }{\left(\mathcal{V}\right)}^{\perp }$ by Lemma 22. That is, $\mathsf{\Psi }\left(\mathcal{V}\right)$ is a $[m+en,k,d_H]$ linear code over ${\mathbb{F}}_q$ which contains its dual. Therefore, by Lemma 18, there exists a quantum code with the parameters ${\left[[m+en,2k-(m+en),d_H]\right]}_q$ over ${\mathbb{F}}_q$. □

Besides getting quantum codes from additive codes over ${\mathbb{F}}_q{\mathcal{B}}_e$, these codes can be investigated to obtain conditions for complementary duality. In particular, we obtain conditions for an ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew constacyclic code to be ACD for which we need two basic results which are given below.

Lemma 29. 

[14] Suppose $\mathcal{V}$ is a cyclic code over ${\mathbb{F}}_q$ with generator polynomial $f\left(x\right)$. Then $\mathcal{V}$ is an LCD code iff $gcd(f\left(x\right),h^{\ast }\left(x\right))=1$, where $gcd(f\left(x\right),h^{\ast }\left(x\right))$ denotes the greatest common divisor of $f\left(x\right)$ and $h^{\ast }\left(x\right)$.

Proposition 30. 

Suppose $\mathcal{V}={\pi }_m\left(\mathcal{V}\right)\times {\pi }_n\left(\mathcal{V}\right)$ is a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive code having length $(m,n)$. Then $\mathcal{V}$ is an ACD code iff ${\pi }_m\left(\mathcal{V}\right)$ and ${\pi }_n\left(\mathcal{V}\right)$ are LCD codes over ${\mathbb{F}}_q$ and ${\mathcal{B}}_e$, respectively.

Proof. 

Suppose $\mathcal{V}={\pi }_m\left(\mathcal{V}\right)\times {\pi }_n\left(\mathcal{V}\right)$ is an ACD code and its dual code be ${\mathcal{V}}^{\perp }={\pi }_m{\left(\mathcal{V}\right)}^{\perp }\times {\pi }_n{\left(\mathcal{V}\right)}^{\perp }$. If $x\in {\pi }_m\left(\mathcal{V}\right)\cap {\pi }_m{\left(\mathcal{V}\right)}^{\perp }$ and $y\in {\pi }_n\left(\mathcal{V}\right)\cap {\pi }_n{\left(\mathcal{V}\right)}^{\perp }$ then $(x,y)\in \mathcal{V}\cap {\mathcal{V}}^{\perp }=\left\{0\right\}$. Therefore, $x=y=0$ which implies that ${\pi }_m\left(\mathcal{V}\right)$ and ${\pi }_n\left(\mathcal{V}\right)$ are LCD codes over ${\mathbb{F}}_q$ and ${\mathcal{B}}_e$, respectively.

Conversely, assume that ${\pi }_m\left(\mathcal{V}\right)$, ${\pi }_n\left(\mathcal{V}\right)$ be LCD codes over ${\mathbb{F}}_q$ and ${\mathcal{B}}_e$, respectively. For any $(x,y)\in \mathcal{V}\cap {\mathcal{V}}^{\perp }$, we have $x\in {\pi }_m\left(\mathcal{V}\right)\cap {\pi }_m{\left(\mathcal{V}\right)}^{\perp }=\left\{0\right\}$ and $y\in {\pi }_n\left(\mathcal{V}\right)\cap {\pi }_n{\left(\mathcal{V}\right)}^{\perp }=\left\{0\right\}$. That is, $(x,y)=(0,0)$ which implies that $\mathcal{V}$ is an ACD code. □

Using the above classification for complementary duality, we now present necessary and sufficient conditions for a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew constacyclic code to be ACD.

Theorem 31. 

Suppose $\mathcal{V}={\pi }_m\left(\mathcal{V}\right)\times {\pi }_n\left(\mathcal{V}\right)$ is a separable ${\mathbb{F}}_q{\mathcal{B}}_e$-additive skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic code having length $(m,n)$, where $\gamma ={\sum }_{i=1}^e{ϵ}_i{\alpha }_i\in {\mathcal{B}}_e^{\ast }$ for ${\alpha }_i=\pm 1$. Assume that ${\pi }_m\left(\mathcal{V}\right)=\langle f\left(x\right)\rangle$ and ${\pi }_n\left(\mathcal{V}\right)=\langle {\sum }_{i=1}^e{ϵ}_i{g}_i\left(x\right)\rangle$. Choose $h\left(x\right),h_i\left(x\right)$ such that $x^m-1=f\left(x\right)h\left(x\right)$ and $x^n-{\alpha }_i=h_i\left(x\right)g_i\left(x\right)$ for $1\le i\le e$. Then $\mathcal{V}$ is an ACD code iff the following conditions hold:

• $gcd(f\left(x\right),h^{\ast }\left(x\right))=1$, where $gcd(f\left(x\right),h^{\ast }\left(x\right))$ denotes the greatest common divisor of $f\left(x\right)$ and $h^{\ast }\left(x\right)$.
• $gcrd(g_i\left(x\right),h_i^{♮}\left(x\right))=1$, where $gcrd(g_i\left(x\right),h_i^{♮}\left(x\right))$ denotes the greatest common right divisor of $g_i\left(x\right)$ and $h_i^{♮}\left(x\right)$, for $1\le i\le e$.

Proof. 

It can be verified by using Lemma 29, Theorems 12 and 24. □

7. Examples

In the present section, we derive several LCD and quantum codes in support of our study. Most of the codes are either MDS or near MDS. Moreover, a comparison is made between the resulting quantum codes and the codes found in recent literature.

To obtain the Gray image of any skew constacyclic code over ${\mathcal{B}}_2$ under $\mathsf{\Phi }$, we consider the matrix $M=\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$. Notably, all linear codes over a finite field having parameters $[n,k,d]$ satisfy the Singleton bound, which is determined by $d\le n-k+1$. If $d=n-k+1$, it is considered as an MDS code; if $d=n-k$, it is considered as near MDS. In case of an ${\left[[n,k,d]\right]}_q$ quantum code, we have the Singleton bound$k+2d\le n+2$. The quantum code satisfying the equality $k+2d=n+2$ is called as an MDS code, and the code satisfying $k+2d=n$ is called a near MDS code. Now, we go through a few examples of codes in detail.

Example 32. 

Take $n=12$, $e=2$. Consider the factorization of $x^{12}-1$ over ${\mathbb{F}}_{5^2}$ given by

$$\begin{array}{cc}\hfill x^{12}-1=& (x^9+2x^8+t^{20}{x}^7+3x^6+t^{22}{x}^5+2x^4+t{x}^3+t^{11}x+4)(x^3+3x^2+t^{11}x+1)\hfill \end{array}$$

and take ${\mathcal{V}}_1=\langle x^3+3x^2+t^{11}x+1\rangle$. Further, consider the factorization

$$\begin{array}{cc}\hfill x^{12}-1=& (x^{11}+t^{22}{x}^{10}+4x^9+t^{10}{x}^8+x^7+t^{22}{x}^6+4x^5+t^{10}{x}^4+x^3+t^{22}{x}^2+4x+t^{10})(x+t^2)\hfill \end{array}$$

and take ${\mathcal{V}}_2=\langle x+t^2\rangle$. Then $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2$ is a cyclic code over ${\mathbb{F}}_{5^2}+u{\mathbb{F}}_{5^2}$ and $\mathsf{\Phi }\left(\mathcal{V}\right)$ is an LCD code with the parameters ${[24,20,4]}_{5^2}$ which is a near MDS code.

Example 33. 

Take $n=30$, $e=2$. Consider the factorization of $x^{30}+1$ over ${\mathbb{F}}_{{11}^2}$ given by

$$\begin{array}{cc}\hfill x^{30}+1=& (x^{28}+t^{78}{x}^{27}+2x^{26}+t^{78}{x}^{25}+x^{24}+10x^{22}+t^{18}{x}^{21}+9x^{20}+t^{18}{x}^{19}+10x^{18}+x^{16}\hfill \\ & +t^{78}{x}^{15}+2x^{14}+t^{78}{x}^{13}+x^{12}+10x^{10}+t^{18}{x}^9+9x^8+t^{18}{x}^7+10x^6+x^4+t^{78}{x}^3\hfill \\ & +2x^2+t^{78}x+1)(x^2+t^{18}x+1)\hfill \end{array}$$

and take ${\mathcal{V}}_1=\langle x^2+t^{18}x+1\rangle$. Further, consider the factorization

$$\begin{array}{cc}\hfill x^{30}+1=& (x^{29}+t^{71}{x}^{28}+2x^{27}+t^{83}{x}^{26}+4x^{25}+t^{95}{x}^{24}+8x^{23}+t^{107}{x}^{22}+5x^{21}+t^{119}{x}^{20}+\hfill \\ & 10x^{19}+t^{11}{x}^{18}+9x^{17}+t^{23}{x}^{16}+7x^{15}+t^{35}{x}^{14}+3x^{13}+t^{47}{x}^{12}+6x^{11}+t^{59}{x}^{10}+\hfill \\ & x^9+t^{71}{x}^8+2x^7+t^{83}{x}^6+4x^5+t^{95}{x}^4+8x^3+t^{107}{x}^2+5x+t^{119})(x+t)\hfill \end{array}$$

and take ${\mathcal{V}}_2=\langle x+t\rangle$. Then $\mathcal{V}={ϵ}_1{\mathcal{V}}_1\oplus {ϵ}_2{\mathcal{V}}_2$ is a skew negacyclic code over ${\mathbb{F}}_{{11}^2}+u{\mathbb{F}}_{{11}^2}$ and the parameters of $\mathsf{\Phi }\left(\mathcal{V}\right)$ are ${[60,57,3]}_{{11}^2}$. By using Theorem 19, we obtain a quantum code having the parameters ${\left[[60,54,3]\right]}_{{11}^2}$ which is better than the code ${\left[[60,50,3]\right]}_{{11}^2}$ given in [40].

Example 34. 

Take $m=48,n=8$, $e=2$. Consider the factorization of $x^{48}-1$ over ${\mathbb{F}}_{7^2}$ given by

$$\begin{array}{cc}\hfill x^{48}-1=& (x^{46}+t^6{x}^{45}+t^{31}{x}^{44}+t^{25}{x}^{43}+t^{10}{x}^{42}+t^{19}{x}^{41}+6x^{40}+t^{28}{x}^{39}+t^{17}{x}^{38}+t^{35}{x}^{37}+\hfill \\ & t^{35}{x}^{36}+t^{30}{x}^{35}+t^{31}{x}^{34}+t^{18}{x}^{33}+t^{47}{x}^{32}+t^{20}{x}^{31}+t^{19}{x}^{30}+t^{33}{x}^{29}+t^{44}{x}^{28}+\hfill \\ & t^{19}{x}^{27}+t{x}^{26}+t^{29}{x}^{25}+t^{13}{x}^{24}+t^{20}{x}^{23}+t^{10}{x}^{22}+t^{23}{x}^{21}+5x^{20}+t^7{x}^{19}+t^{29}{x}^{18}+\hfill \\ & t^{15}{x}^{17}+t^{46}{x}^{16}+t^{44}{x}^{15}+t^{20}{x}^{14}+t^{36}{x}^{13}+t^{46}{x}^{12}+t^{42}{x}^{11}+t^{44}{x}^{10}+t^{41}{x}^9+t^{20}{x}^8\hfill \\ & +t^{28}{x}^7+t^{21}{x}^6+t^{13}{x}^5+t{x}^4+t^{13}{x}^3+2x^2+t^{36}x+t^{27})(x^2+t^{30}x+t^{45})\hfill \end{array}$$

and take $f\left(x\right)=x^2+t^{30}x+t^{45}$. Further, consider the factorizations

$$\begin{array}{cc}\hfill x^8+1=& (x^6+t^{30}{x}^4+t^{12}{x}^2+t^{42})(x^2+t^6)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill x^8-1=& (x^7+t^{12}{x}^6+x^5+t^{12}{x}^4+x^3+t^{12}{x}^2+x+t^{12})(x+t^{12})\hfill \end{array}$$

and take $g_1\left(x\right)=x^2+t^6$ and $g_2\left(x\right)=x+t^{12}$. Then, the code $\mathcal{V}$ given by Proposition 27 is a ${\mathbb{F}}_{7^2}({\mathbb{F}}_{7^2}+u{\mathbb{F}}_{7^2})-$ additive skew constacyclic code containing its dual. The Gray image $\mathsf{\Phi }\left(\mathcal{V}\right)$ of the code $\mathcal{V}$ is a ${[64,59,3]}_{7^2}$ code which contains its dual. Therefore, a quantum code having the parameters ${\left[[64,54,3]\right]}_{7^2}$ is obtained by using Theorem 28.

In Table 1 and Table 2, we present LCD and quantum codes obtained from skew ${\mathsf{\Theta }}_1-\gamma$-constacyclic code $\mathcal{V}=\langle {ϵ}_1{g}_1\left(x\right)+{ϵ}_2{g}_2\left(x\right)\rangle$ over ${\mathcal{B}}_2$, respectively. In both the tables, the first column represents the length of the code $\mathcal{V}$, and the second column represents $({\alpha }_1,{\alpha }_2)$ such that $\gamma ={ϵ}_1{\alpha }_1+{ϵ}_2{\alpha }_2$. The third column represents the polynomials $g_1\left(x\right),g_2\left(x\right)$ such that $\mathcal{V}$ is generated by ${ϵ}_1{g}_1\left(x\right)+{ϵ}_2{g}_2\left(x\right)$ while the parameters of $\mathsf{\Phi }\left(\mathcal{V}\right)$ are presented in the fourth column. In Table 2, the parameters of the quantum codes obtained from the code $\mathsf{\Phi }\left(\mathcal{V}\right)$ (using Theorem 19) are presented in the fifth column. In the sixth column, the comparison of these obtained codes is made with the known quantum codes that have been reported recently in the literature.

Further, we obtain quantum codes from ${\mathbb{F}}_q{\mathcal{B}}_2$-additive skew ${\mathsf{\Theta }}_1-\gamma$-constacyclic codes having length $(m,n)$ where $\gamma ={ϵ}_1{\alpha }_1+{ϵ}_2{\alpha }_2$. In Table 3, we tabulate ${\alpha }_1,{\alpha }_2$, the polynomials $f\left(x\right),g_1\left(x\right),g_2\left(x\right)$ such that ${\pi }_m\left(\mathcal{V}\right)=\langle f\left(x\right)\rangle$ and ${\pi }_n\left(\mathcal{V}\right)=\langle {\sum }_{i=1}^2{ϵ}_i{g}_i\left(x\right)\rangle$ where $f\left(x\right)|x^m-1$ and $g_i\left(x\right)$ is a right divisor of $x^n-{\alpha }_i$ for $i=1,2$. The parameters of the code $\mathsf{\Psi }\left(\mathcal{V}\right)$ are presented in the fifth column, whereas the parameters of the derived quantum codes are presented in the sixth column by using Theorem 28.

Remark: 

In the tables, the polynomials are represented by their coefficients in increasing powers of x. For instance, $t^{23}{t}^{11}1$ represents the polynomial $t^{23}+t^{11}x+x^2$, where t is the primitive element of the corresponding field ${\mathbb{F}}_q$.

8. Conclusions

In this article, we have investigated skew ${\mathsf{\Theta }}_l-\gamma$-constacyclic codes over the ring ${\mathcal{B}}_e,$ and obtained requirements for these codes to be LCD for some specific values of $\gamma$. Further, we have obtained requirements for a skew constacyclic code over ${\mathcal{B}}_e$ to satisfy the dual containing property, motivated by the CSS construction for quantum codes. We also defined a duality-preserving Gray map. This yielded several (MDS/near MDS) LCD codes and new quantum codes over finite fields. Moreover, we have also considered additive skew constacyclic codes, and obtained conditions for complementary duality and dual containing property.