Skew-constacyclic codes over $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$

TL;DR

This paper analyzes the algebraic structure of skew-constacyclic codes over a specific ring, characterizes their automorphisms, and determines conditions for self-duality, advancing coding theory over non-field rings.

Contribution

It introduces a detailed study of skew-constacyclic codes over the ring rac{}_q[v]/\u2206<v^q - v>, including automorphism groups, generator polynomials, and self-duality conditions.

Findings

Determined the automorphism group of the ring.

Established generator polynomials for skew-constacyclic codes.

Provided necessary and sufficient conditions for self-dual codes.

Abstract

In this paper, the investigation on the algebraic structure of the ring $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ and the description of its automorphism group, enable to study the algebraic structure of codes and their dual over this ring. We explore the algebraic structure of skew-constacyclic codes, by using a linear Gray map and we determine their generator polynomials. Necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ are given.

Equations

$$\displaystyle\begin{array}[]{cccc}\phi_{i}:&R_{q}&\rightarrow&\left(\mathbb{F}_{q};+,\cdot;1,0\right)\\ &\sum\limits_{i=0}^{q-1}a_{i}\eta_{i}&\mapsto&a_{i},\end{array}$$

$$\text{x}\ast \text{y}:=(x_0{y}_0,x_1{y}_1,\cdots ,x_{q-1}{y}_{q-1})\in ({\mathbb{F}}_q{)}^q,$$

$$\text{x}+\text{y}:=(x_0+y_0,x_1+y_1,\cdots ,x_{q-1}+y_{q-1})\in ({\mathbb{F}}_q{)}^q.$$

$$\displaystyle\begin{array}[]{cccc}\phi:&R_{q}&\rightarrow&\left((\mathbb{F}_{q})^{q};+,\ast;\textbf{1},\textbf{0}\right)\\ &a&\mapsto&\left(\phi_{0}(a),\phi_{1}(a),\cdots,\phi_{q-1}(a)\right),\end{array}$$

$$\mathcal{U}(R_q)=\left\{\sum _{i=0}^{q-1}{a}_i{\eta }_i:a_i\ne 0\text{ for all }0\le i\le q-1\right\},$$

$$\displaystyle\begin{array}[]{cccc}\Theta_{\theta,\sigma}:&R_{q}&\rightarrow&R_{q}\\ &\sum\limits_{i=0}^{q-1}a_{i}\eta_{i}&\mapsto&\sum\limits_{i=0}^{q-1}\theta(a_{i})\eta_{\sigma(i)}.\end{array}$$

$$\text{Aut}(R_q)=\left\{{\Theta }_{\theta ,\sigma }:\theta \in \text{Aut}({\mathbb{F}}_q),\sigma \in {\mathbb{S}}_q\right\},$$

$$|\text{Aut}(R_q)|=r\cdot q\cdot (q-1)\cdot (q-2)\cdot \dots \cdot 2\cdot 1,$$

$$\{\Theta ({\eta }_i):0\le i\le q-1\}$$

$$\begin{array}[]{c c c c}\Theta_{\texttt{Id},\sigma}:&R_{2}&\longrightarrow&R_{2}\\ &a\eta_{0}+b\eta_{1}&\longmapsto&a\eta_{1}+b\eta_{0}.\end{array}$$

$${\eta }_0={\overline{v}}^3+1;{\eta }_1={\overline{v}}^3+\overline{v}+1;{\eta }_2={\overline{v}}^3+\alpha {\overline{v}}^2+{\alpha }^2\overline{v};{\eta }_4={\overline{v}}^3+{\alpha }^2{\overline{v}}^2+\alpha \overline{v}.$$

$$\begin{array}[]{c c c c}\Theta_{\theta,\sigma_{i}}:&R_{4}&\longrightarrow&R_{4}\\ &\sum\limits_{i=0}^{3}a_{i}\eta_{i}&\longmapsto&\sum\limits_{i=0}^{3}\theta(a_{i})\eta_{\sigma(i)},\end{array}$$

$$\displaystyle\begin{array}[]{c c c c}\Phi_{i}:&(R_{q})^{n}&\rightarrow&(\mathbb{F}_{q})^{n}\\ &\left(a_{0},a_{1},\cdots,a_{n-1}\right)&\mapsto&\left(\phi_{i}(a_{0}),\phi_{i}(a_{1}),\cdots,\phi_{i}(a_{n-1})\right),\end{array}$$

$$\displaystyle\begin{array}[]{c c c c}\Phi:&(R_{q})^{n}&\rightarrow&\left((\mathbb{F}_{q})^{n}\right)^{q}\\ &\textbf{a}&\mapsto&\left(\Phi_{0}(\textbf{a}),\Phi_{1}(\textbf{a}),\cdots,\Phi_{q-1}(\textbf{a})\right).\end{array}$$

$$C^{\perp }=\left\{\text{a}\in (R_q{)}^n;\text{a}\cdot \text{b}=0,\text{ for all }\text{b}\in C\right\}.$$

$$\text{a}\cdot \text{b}=0\Rightarrow \Phi (\text{a})\cdot \Phi (\text{b})=0,$$

$$\Phi (C^{\perp })\subseteq {\left(\Phi (C)\right)}^{\perp }.$$

$$(R_q{)}^n={\eta }_0(R_q{)}^n\oplus {\eta }_1(R_q{)}^n\oplus \cdots \oplus {\eta }_{q-1}(R_q{)}^n.$$

$$C_i:={\Phi }_i\left(C\right),$$

$$C={\eta }_0{C}_0\oplus {\eta }_1{C}_1\oplus \cdots \oplus {\eta }_{q-1}{C}_{q-1}.$$

$$C^{\perp }={\eta }_0{C}_0^{\perp }\oplus {\eta }_1{C}_1^{\perp }\oplus \cdots \oplus {\eta }_{q-1}{C}_{q-1}^{\perp }.$$

$$|D|$$

$$\mathrm{G}={\eta }_0{\tilde{\mathrm{G}}}_0+{\eta }_1{\tilde{\mathrm{G}}}_1+\cdots +{\eta }_{q-1}{\tilde{\mathrm{G}}}_{q-1}$$

$$\Phi(\mathrm{G}):=\left(\begin{array}[]{ccccc}\mathrm{G}_{0}&0&0&\cdots&0\\ 0&\mathrm{G}_{1}&0&\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 0&\cdots&0&\mathrm{G}_{q-2}&0\\ 0&\cdots&0&0&\mathrm{G}_{q-1}\end{array}\right).$$

$$C={\eta }_0{C}_0\oplus {\eta }_1{C}_1\oplus \cdots \oplus {\eta }_{q-1}{C}_{q-1},$$

$${\eta }_0={\overline{v}}^3+1;{\eta }_1={\overline{v}}^3+\overline{v}+1;{\eta }_2={\overline{v}}^3+\alpha {\overline{v}}^2+{\alpha }^2\overline{v};{\eta }_4={\overline{v}}^3+{\alpha }^2{\overline{v}}^2+\alpha \overline{v}.$$

$$G_{1}:=\left(\begin{array}[]{llllll}1&0&0&\alpha^{2}&\alpha^{2}&1\\ 0&1&0&\alpha^{2}&0&\alpha\\ 0&0&1&1&\alpha&\alpha\\ \end{array}\right);~{}~{}G_{2}:=\left(\begin{array}[]{llllll}1&0&0&\alpha&\alpha&1\\ 0&1&0&\alpha&0&\alpha^{2}\\ 0&0&1&1&\alpha^{2}&\alpha^{2}\\ \end{array}\right);$$

$${\text{T}}_{\Theta ,\lambda }((a_0,a_1,\cdots ,a_{n-1}))=(\lambda \Theta (a_{n-1}),\Theta (a_0),\cdots ,\Theta (a_{n-2})).$$

$$\phi :(a_0,a_1,\cdots ,a_{n-1})⟼a_0+a_{1}x+\cdots +a_{n-1}{x}^{n-1}.$$

$$f(x)\left(g(x)+⟨x^n-\lambda ⟩\right)=f(x)g(x)+⟨x^n-\lambda ⟩,$$

Full text

Keywords: Non-chain ring, Skew-constacyclic codes, Gray map, Self-dual skew codes, Skew-multi-twisted codes.

AMS Subject Classification: 16P10, 94B05, 94B60.

1. Introduction

Codes over finite commutative chain rings have been extensively studied. In the last years, some special rings which are non-chain rings are used as alphabet for codes. Recently, using a (non-commutative) skew polynomial ring, which is a particular case of well-known re polynomial rings, D. Boucher et al. developed some classes of linear codes called skew-constacyclic codes [8, 9, 10, 11]. In analogous to classical constacyclic codes, skew constacyclic codes over a finite commutative ring have rich algebraic structures and can be seen as left-submodules of a certain class of module. These codes have received much attention in recent years [13, 20, 23] and have been studied for various automorphisms over some non-chain rings [16, 17, 21, 25].

In [1, 2], Abualrub et al. studied cyclic, and skew cyclic codes over $\mathbb{F}_{2}[v]/\langle\,v^{2}-v\,\rangle.$ Bayram et al. investigated cyclic codes over $\mathbb{Z}_{3}[v]/\langle\,v^{3}-v\,\rangle$ [5] and constacyclic codes over $\mathbb{F}_{p}[v]/\langle\,v^{p}-v\,\rangle$ [4]. In [14], Dertli et al. explored skew-constacyclic and skew-quasi constacyclic codes over $\mathbb{Z}_{3}[v]/\langle\,v^{3}-v\,\rangle.$ Recently, Shi et al., in [22], studied skew cyclic codes over $\mathbb{F}_{p^{s}}[v]/\langle\,v^{m}-v\,\rangle$, where $p$ is a prime and $m-1$ divides $p-1.$ The motivation for studying this ring is lying under the facts that first this ring is as a natural generalization of codes over the ring $\mathbb{F}_{p}[v]/\langle\,v^{p}-v\,\rangle.$ Second important fact regarding this ring is that a linear Gray map is defined and hence codes over fields are obtained. To the best of our knowledge, the study of skew constacyclic codes over $\mathbb{F}_{q}[v]/\langle\,v^{q}-v\,\rangle$ has not been considered by any coding scientist. Our objection is to characterize self-dual skew-constacyclic codes over $\mathbb{F}_{q}[v]/\langle\,v^{q}-v\,\rangle.$

This paper is organized as follows. In Section 2, we first give some properties about the ring $\mathbb{F}_{q}[v]/\langle\,v^{q}-v\,\rangle$ and describe its ring-automorphism group. In Section 3, we explore linear codes over $\mathbb{F}_{q}[v]/\langle\,v^{q}-v\,\rangle$ using a linear Gray map and we show that the Gray-image of any skew constacyclic code of length $n$ over $\mathbb{F}_{q}[v]/\langle\,v^{q}-v\,\rangle$ under this linear Gray map is a skew multi-twisted code of length $qn$ over $\mathbb{F}_{q}.$ In Section 4, we determine the structure of skew-constacyclic codes and characterize self-dual skew-constacyclic codes over $\mathbb{F}_{q}[v]/\langle\,v^{q}-v\,\rangle.$ We also give necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over $\mathbb{F}_{q}[v]/\langle\,v^{q}-v\,\rangle.$

2. Ring-automorphism group of

$\mathbb{F}_{q}[v]/\langle v^{q}-v\rangle$

Throughout this paper, $\mathbb{F}_{q}$ is a finite field with $q$ elements and $\beta$ is a generator of multiplicative group $\mathbb{F}_{q}\backslash\{0\}.$ Write $\mathbb{F}_{q}:=\{\alpha_{0},\alpha_{1},\cdots,\alpha_{q-1}\}$ where $\alpha_{0}=0,\alpha_{i}=\beta^{i}$ for all $0\leq i\leq q-1.$ We denote the ring $\mathbb{F}_{q}[v]/\langle\,v^{q}-v\,\rangle$ by $R_{q}$ and $\mathcal{U}(R_{q})$ its unit group. Obviously, $\mathbb{F}_{q}$ is a subring of the ring $R_{q},$ and $R_{q}$ is a vector space over $\mathbb{F}_{q}$ with basis $\{1,\overline{v},\cdots,\overline{v}^{q-1}\}$ where $\overline{v}:=v+\langle\,v^{q}-v\,\rangle.$ Thus $R_{q}=\mathbb{F}_{q}\oplus\mathbb{F}_{q}\overline{v}\oplus\cdots\oplus\mathbb{F}_{q}\overline{v}^{q-1}$ and any element $r$ in $R_{q}$ can be uniquely written as: $r=r_{0}+r_{1}\overline{v}+\cdots+r_{q}\overline{v}^{q-1}$ with $r_{i}\in\mathbb{F}_{q}$, for all $0\leq i\leq q-1.$ Moreover, the ring $R_{q}$ is a non-chain principal ideal ring with maximal ideals $\langle\,v-\alpha_{i}\,\rangle/\langle\,v^{q}-v\,\rangle$ ($0\leq i<q$), since the factorization of $v^{q}-v$ in $\mathbb{F}_{q}[v]$ is $v^{q}-v=(v-\alpha_{0})(v-\alpha_{1})\cdots(v-\alpha_{q-1}).$

An element $\eta$ of $R_{q}$ is called idempotent if $\eta^{2}=\eta;$ two idempotents $\eta_{1},\eta_{2}$ are said to be orthogonal if $\eta_{1}\eta_{2}=0.$ An idempotent of $R_{q}$ is said to be primitive if it is non-zero and it can not be written as sum of non-zero orthogonal idempotents. A set $\{\eta_{0},\eta_{1},\cdots,\eta_{q-1}\}$ of idempotents of $R_{q}$ is complete if $\sum_{i=0}^{q-1}\eta_{i}=1.$ Let $f_{i}(v):=v-\alpha_{i},$ and $\widehat{f_{i}}(v):=\frac{v^{q}-v}{f_{i}(v)},$ where $i=0,1,\cdots,q-1.$ Then there exist $a_{i}(v)$ and $b_{i}(v)$ in $\mathbb{F}_{q}[v]$ such that $a_{i}(v)f_{i}(v)+b_{i}(v)\widehat{f_{i}}(v)=1.$ Let $\eta_{i}=b_{i}(\overline{v})\widehat{f_{i}}(\overline{v}),$ then $\eta_{i}^{2}=\eta_{i},$ $\eta_{i}\eta_{j}=0$ and $\sum\limits_{i=0}^{q-1}\eta_{i}=1,$ where $0\leq i\neq j\leq q-1.$ It is easy to see that any complete set of idempotents in $R_{q}$ is a basis of $\mathbb{F}_{q}$-vector space. Therefore, any element $r$ of $R_{q}$ can be represented as: $r=r_{0}\eta_{0}+r_{1}\eta_{1}+\cdots+r_{q-1}\eta_{q-1}$ [22].

For $i=0,1,\cdots,q-1,$ we consider the map

[TABLE]

which is a ring-epimorphism. We denote by $\ast$ the componentwise multiplication (or Schur product) and by $+$ the componentwise addition on $(\mathbb{F}_{q})^{q},$ i.e. for $\textbf{x}:=(x_{0},x_{1},\cdots,x_{q-1}),\textbf{y}:=(y_{0},y_{1},\cdots,y_{q-1})\in(\mathbb{F}_{q})^{q},$ we put

[TABLE]

[TABLE]

By the remainder Chinese theorem, the following map

[TABLE]

is a ring-isomorphism, where $\textbf{1}:=(1,1,\cdots,1)$ and $\textbf{0}:=(0,0,\cdots,0).$ It is easy to see that the ring $R_{q}$ is a principal ideal ring whose $2^{q}$ ideals are $\mathrm{I}_{A}:=\langle\sum\limits_{i\in A}\eta_{i}\rangle$ where $A$ is a subset of $\{0;1;\cdots;q-1\}.$ For any $a\in R_{q},$ we set $\texttt{Supp}(a):=\{i\in\{0;1;\cdots;q-1\}:\phi_{i}(a)\neq 0\};$ then $\langle\,a\,\rangle=I_{\texttt{Supp}(a)}$ and $|\langle a\rangle|=q^{|\texttt{Supp}(a)|}.$ Moreover, the group of units of $R_{q}$ is described as follows:

[TABLE]

and so $|\mathcal{U}(R_{q})|=(q-1)^{q}.$ For instance, the ring $R_{3}$ has $8$ units given by: $1,2,1+v^{2},1+v+2v^{2},1+2v+2v^{2},2+v+v^{2},2+2v+v^{2},2+2v^{2}$ [5].

Lemma 1**.**

The ring $R_{q}$ admits a unique complete set $\{\eta_{0},\eta_{1},\cdots,\eta_{q-1}\}$ of primitive pairwise orthogonal idempotents.

**Proof. **The only complete set of primitive pairwise orthogonal idempotents in the ring $\left((\mathbb{F}_{q})^{q};+,\ast;\textbf{1},\textbf{0}\right)$ is $\{e_{0},e_{1},\cdots,e_{q-1}\}$ where $e_{0}:=(1,0,\cdots,0);$ $e_{1}:=(0,1,0,\cdots,0);$ $e_{2}:=(0,0,1,0,\cdots,0);\cdots;e_{q-1}:=(0,0,\cdots,0,1).$ From the ring-isomorphism (2.6), the set $\{\eta_{0},\eta_{1},\cdots,\eta_{q-1}\}$ is also the complete set of primitive pairwise orthogonal idempotents in the ring $R_{q},$ where $\phi(\eta_{i})=e_{i},$ for all $0\leq i\leq q-1.$ ∎

Theorem 1**.**

Let $\theta$ be a ring-automorphism of $\mathbb{F}_{q}$ and $\sigma$ is a permutation of $\{0,1,\cdots,q-1\}.$ Then the map

[TABLE]

is a ring-automorphism group of $R_{q}.$ Moreover

[TABLE]

where $\mathbb{S}_{q}$ is the group of permutations of $\{0,1,\cdots,q-1\},$ and

[TABLE]

where $q=p^{r}$ with $p$ a prime number.

**Proof. **First, for all $\theta\in\texttt{Aut}(\mathbb{F}_{q})$ and $\sigma\in\mathbb{S}_{q},$ it is obvious to check that $\Theta_{\theta,\sigma}$ is a ring-automorphism of $R_{q}.$ Therefore $\left\{\Theta_{\theta,\sigma}\,:\,\theta\in\texttt{Aut}(\mathbb{F}_{q}),\sigma\in\mathbb{S}_{q}\right\}\subseteq\texttt{Aut}(R_{q}).$ Inversely, if $\Theta\in\texttt{Aut}(R_{q})$ then it is clear that the restriction of $\Theta$ over $\mathbb{F}_{q}$ is a ring-automorphism $\theta$ of $\mathbb{F}_{q}.$ Thus for any $a:=\sum\limits_{i=0}^{q-1}a_{i}\eta_{i}$ in $R_{q},$ we have $\Theta(a)=\sum\limits_{i=0}^{q-1}\theta(a_{i})\Theta(\eta_{i}).$ Now the set

[TABLE]

is another complete set of primitive pairwise orthogonal idempotents in $R_{q}.$ From Lemma 1, it follows that there is a permutation $\sigma$ of $\{0,1,\cdots,q-1\}$ such that $\Theta(\eta_{i})=\eta_{\sigma(i)}.$ Hence $\Theta(a)=\sum\limits_{i=0}^{q-1}\theta(a_{i})\eta_{\sigma(i)}$ and $\texttt{Aut}(R_{q})=\left\{\Theta_{\theta,\sigma}\,:\,\theta\in\texttt{Aut}(\mathbb{F}_{q}),\sigma\in\mathbb{S}_{q}\right\}.$ Finally, $\Theta_{\theta^{\prime},\sigma^{\prime}}\circ\Theta_{\theta,\sigma}=\Theta_{\theta^{\prime}\circ\theta,\sigma^{\prime}\circ\sigma},$ for all $\theta^{\prime},\theta\in\texttt{Aut}(\mathbb{F}_{q})$ and $\sigma^{\prime},\sigma\in\mathbb{S}_{q}.$ ∎

Example 1**.**

The ring-automorphism group of $R_{2}$ and $R_{4}:$

1. (1)

The idempotents in $R_{2}$ are $\eta_{0}:=\overline{v}$ and $\eta_{1}:=\overline{v}+1.$ Since $\mathbb{S}_{2}:=\{\texttt{id},\sigma:=(01)\}$ and $\texttt{Aut}(\mathbb{F}_{2})=\{\texttt{Id}\}$, out of the identity map, the only ring-automorphism over $R_{2}$ is given by:

[TABLE]

The automorphism $\Theta_{\texttt{Id},\sigma}$ is used in **[1]** to study skew $\Theta_{\texttt{Id},\sigma}$-cyclic codes over $R_{2}.$ 2. (2)

The complete set of primitive pairwise orthogonal idempotents of $R_{4}$ is given by:

[TABLE]

A ring-automorphism over $R_{4}$ is given by:

[TABLE]

for all $\theta\in\texttt{Aut}(\mathbb{F}_{4})$ and $\sigma\in\mathbb{S}_{4}.$ Thus there exist exactly $48$ automorphisms over $R_{4}.$

3. Linear Gray map and linear codes over $R_{q}$

The ring-morphism $\phi_{i}:R_{q}\rightarrow\mathbb{F}_{q}$ defined in (2.3) is naturally extended to $(R_{q})^{n}$ as follows:

[TABLE]

which is an epimorphism of vector spaces over $\mathbb{F}_{q}.$ We define a linear Gray map over $R_{q}$ to be

[TABLE]

This map $\Phi:(R_{q})^{n}\rightarrow\left((\mathbb{F}_{q})^{n}\right)^{q}$ is an isomorphism of vector spaces over $\mathbb{F}_{q}.$

Definition 1**.**

The Gray weight of any element a in $(R_{q})^{n}$ is defined as: $W_{G}(\textbf{a})=W_{H}(\Phi(\textbf{a})),$ where $W_{H}$ denotes the Hamming weight over $\mathbb{F}_{q}.$ The Gray distance between two elements $\textbf{a}_{1},\textbf{a}_{2}$ in $(R_{q})^{n}$ is given by: $d_{G}(\textbf{a}_{1},\textbf{a}_{2})=W_{G}(\textbf{a}_{1}-\textbf{a}_{2}).$

It is obvious that the linear Gray map $\Phi$ is a weight preserving map from $((R_{q})^{n},W_{G})$ to \biggl{(}\left((\mathbb{F}_{q})^{n}\right)^{q},W_{H}\biggr{)} and $W_{G}(\textbf{a})=\sum\limits_{i=0}^{q-1}W_{H}(\Phi_{i}(\textbf{a}))$ for $\textbf{a}\in(R_{q})^{n}.$

A code $C$ of length $n$ over $R_{q}$ is a nonempty subset of $(R_{q})^{n}.$ If in addition the code is a submodule of $R^{n}_{q}$, it is called linear code. The Euclidean inner product between two elements $\textbf{a}=(a_{0},a_{1},\cdots,a_{n-1}),$ and $\textbf{b}=(b_{0},b_{1},\cdots,b_{n-1})$ in $(R_{q})^{n}$ is defined by: $\textbf{a}\cdot\textbf{b}=\sum\limits_{i=0}^{n-1}a_{i}b_{i}.$ The Euclidean dual code (shortly dual code) of a code $C$ of length $n$ over $R_{q}$ is defined as:

[TABLE]

Recall that Euclidean dual of linear code of length $n$ over $R_{q}$ is a linear code of length $n$ over $R_{q}.$

Example 2**.**

The linear codes of length $1$ over $R_{q}$ are ideals $I_{A}$ of $R_{q},$ where $A$ is a subset of $\{0;1;\cdots;q-1\}.$ Thus for all $a\in R_{q},$ $W_{G}(a)=|A|$ and $\langle a\rangle^{\perp}=\left\langle\sum\limits_{i\in\overline{A}}\eta_{i}\right\rangle$ where $A:=\texttt{Supp}(a)$ and $\overline{A}:=\left\{i\in\{0;1;\cdots;q-1\}\;:\;i\not\in A\right\}.$ Therefore $I_{A}$ is a linear code of length $1$ over $R_{q}$ with Gray weight $|A|$ and $\Phi(I_{A})$ is a $[q,1,|A|]$-code over $\mathbb{F}_{q}.$

Proposition 3.1**.**

Let $C$ be a linear code of length $n$ over $R_{q}.$ Then $\Phi(C^{\perp})=\left(\Phi(C)\right)^{\perp},$ where $\left(\Phi(C)\right)^{\perp}$ denotes the ordinary dual of $\Phi(C)$ as a linear code over $\mathbb{F}_{q}.$ Moreover, $C$ is a self-dual code of length $n$ over $R_{q}$ if and only if $\Phi(C)$ is a self-dual code of length $qn$ over $\mathbb{F}_{q}.$

Proof.

For all a and b in $(R_{q})^{n},$ we have

[TABLE]

where $\Phi(\textbf{a})\cdot\Phi(\textbf{b})$ denotes the usual standard inner product in $\left((\mathbb{F}_{q})^{n}\right)^{q}.$ An immediate consequence is the inclusion:

[TABLE]

Combining this with the fact that in Frobenius rings, $|C||C^{\perp}|=|R_{q}|^{n},$ we get $\Phi(C^{\perp})=\left(\Phi(C)\right)^{\perp}.$ Finally, since $\Phi$ is an isomorphism of vector spaces over $\mathbb{F}_{q},$ the equivalence is an immediate consequence of the equality $\Phi(C^{\perp})=\left(\Phi(C)\right)^{\perp}.$ ∎

Since $R_{q}=\eta_{0}R_{q}\oplus\eta_{1}R_{q}\oplus\cdots\oplus\eta_{q-1}R_{q}$, it follows that

[TABLE]

Let $C$ be a linear code of length $n$ over $R_{q}$ and $\textbf{a}=(a_{0},a_{1},\cdots,a_{n-1})\in C.$ Then $a_{i}=\sum\limits_{j=0}^{q-1}\eta_{j}\phi_{j}(a_{i})$ and $\textbf{a}=\sum\limits_{i=0}^{q-1}\eta_{i}\Phi_{i}(\textbf{a})=\sum\limits_{i=0}^{q-1}\eta_{i}\textbf{a}_{i},$ where $\textbf{a}_{i}:=\Phi_{i}(\textbf{a}).$ We let

[TABLE]

for $0\leq i\leq q-1.$ For this, it is straightforward to see that $C_{0},C_{1},\cdots,C_{q-1}$ are linear codes of length $n$ over $\mathbb{F}_{q}$ and

[TABLE]

Proposition 3.2**.**

Let $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\cdots\oplus\eta_{q-1}C_{q-1}$ be a linear code over $R_{q}$ of length $n.$ Then

[TABLE]

Moreover, $C$ is a self-dual code of length $n$ over $R_{q}$ if and only if $C_{i}$ is a self-dual code of length $n$ over $\mathbb{F}_{q}$ for all $0\leq i\leq q-1.$

**Proof. **Let $D=\eta_{0}C_{0}^{\perp}\oplus\eta_{1}C_{1}^{\perp}\oplus\cdots\oplus\eta_{q-1}C_{q-1}^{\perp}.$ It is clear that $D\subseteq C^{\perp}.$ Since $R_{q}$ is a principal ideal ring, consequently $R_{q}$ is a Frobenius ring, then from [24], we have $|C^{\perp}|=\frac{|R_{q}|^{n}}{|C|}=\frac{q^{qn}}{|C|}.$ Thus

[TABLE]

Hence $C^{\perp}=D.$

On the other hand, if $C$ is self-dual, then for all $0\leq i\leq q-1,$ the code $C_{i}$ is also self-dual, by Definition (3.7) of the code $C_{i}.$ Inversely, if each $C_{i}$ is also self-dual, then $C$ is also self-dual, since $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\cdots\oplus\eta_{q-1}C_{q-1}.$ ∎

Let $\textbf{v}_{1},\cdots,\textbf{v}_{k}$ be vectors in $(R_{q})^{n}.$ The vectors $\textbf{v}_{1},\cdots,\textbf{v}_{k}$ are said to modular independent, if $\Phi_{i}(\textbf{v}_{1}),\cdots,\Phi_{i}(\textbf{v}_{k})$ are linearly independent for some $i.$ The vectors $\textbf{v}_{1},\cdots,\textbf{v}_{k}$ are said to independent, if $\sum\alpha_{j}\textbf{v}_{j}=\textbf{0}$ implies that $\alpha_{j}\textbf{v}_{j}=\textbf{0}$ for all $j.$

Let $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\cdots\oplus\eta_{q-1}C_{q-1}$ be a linear code of length $n$ over $R_{q}.$ Then from obtained results in [15], it follows that $\texttt{rank}_{R_{q}}(C)=\texttt{max}\{\texttt{dim}_{\mathbb{F}_{q}}(C_{i})\;:\;0\leq i\leq q-1\},$ $\Phi(C)$ is a linear code of length $qn$ over $\mathbb{F}_{q}$ and $\Phi(C)=C_{0}\times C_{1}\times\cdots\times C_{q-1}$ with $C_{i}$ be a linear code of length $n$ over $\mathbb{F}_{q}.$ In [15], the codewords $\textbf{c}_{1},\cdots,\textbf{c}_{k}$ form a basis of $C,$ if they are independent, modular independent and generate $C.$ We well-know that the rows of a generator matrix for $C$ form a basis of $C.$ The following result is a direct consequence from the above discussion.

Theorem 2**.**

Let $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\cdots\oplus\eta_{q-1}C_{q-1}$ be a linear code over $R_{q}$ of rank $k,$ and $\mathrm{G}_{i}$ is a generator matrix of linear code $C_{i}$ over $\mathbb{F}_{q}$ of dimension $k_{i},$ for $0\leq i\leq q-1.$ Then a generator matrix $\mathrm{G}$ for $C$ is given by:

[TABLE]

where \widetilde{\mathrm{G}}:=\left(\begin{array}[]{c}\mathrm{G}_{i}\\ \mathrm{O}_{(k-k_{i})\times n}\\ \end{array}\right) with $\mathrm{O}_{(k-k_{i})\times n}$ is the $(k-k_{i})\times n$-matrix whose entries are zeros, and a generator matrix for $\Phi(C)$ is given by:

[TABLE]

Corollary 1**.**

Let $C$ be a linear code of length $n$ over $R_{q}$ such that

[TABLE]

where $C_{i}$ is an $[n,k_{i},d_{i}]$-linear code over $\mathbb{F}_{q}.$ Then $\Phi(C)$ is a $\left[qn,\sum\limits_{i=0}^{q-1}k_{i},\underset{0\leq i\leq q-1}{\min}\,d_{i}\right]$-linear code over $\mathbb{F}_{q}.$

Example 3**.**

Consider the ring $\mathbb{F}_{4}[v]/\langle\,v^{4}-v\,\rangle,$ where $\mathbb{F}_{4}=\{0,1,\alpha,\alpha^{2}\}$ is the finite field with four elements such that $\alpha^{2}+\alpha+1=0.$ The complete set of primitive pairwise orthogonal idempotents of $R_{4}$ is given by:

[TABLE]

*We let

[TABLE]

and G_{3}:=\left(\begin{array}[]{llllll}1&0&0&1&0&0\\ 0&1&0&0&1&0\\ 0&0&1&0&0&1\\ \end{array}\right). Let $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\eta_{2}C_{2}\oplus\eta_{3}C_{3}$ be a linear code of length 6 over $R_{4}$; where $C_{0},C_{1},C_{2}$ are self dual $[6,3,3]$-codes of length 6 over $\mathbb{F}_{4}$ with generator matrix $G_{1},G_{1},G_{2}$ respectively. The code $C_{3}$ generated by $G_{3}$ is a self-dual $[6,3,2]$-code over $\mathbb{F}_{4}$ [19]. From Proposition 3.2, $C$ is a self-dual code of length $6$ over $R_{4}.$ From Proposition 3.1, Theorem 2 and Corollary 1, the code $\Phi(C)$ is a self-dual $[24,12,2]$-code with generator matrix \Phi(G)=\left(\begin{array}[]{l l l l}G_{1}&0&0&0\\ 0&G_{1}&0&0\\ 0&0&G_{2}&0\\ 0&0&0&G_{3}\end{array}\right).

4. Skew-constacyclic codes over $R_{q}$

Let $\Theta$ be a ring-automorphism of $R_{q}$ such that $\Theta=\Theta_{\theta,\sigma}$ where $\theta\in\texttt{Aut}(\mathbb{F}_{q})$, $\sigma\in\mathbb{S}_{q}.$ The $\theta$-skew polynomial ring over $\mathbb{F}_{q},$ denoted $\mathbb{F}_{q}[x;\theta],$ is right Euclidean domain [18]. The least common left multiple (lclm) of nonzero $f_{1}$ and $f_{2},$ in $\mathbb{F}_{q}[x;\theta],$ denoted $\texttt{lclm}(f_{1},f_{2}),$ is the unique monic polynomial $h\in\mathbb{F}_{q}[x;\theta]$ of lowest degree such that there exist $u_{1},u_{2}\in\mathbb{F}_{q}[x;\theta]$ with $h=u_{1}f_{1}$ and $h=u_{2}f_{2}.$ The skew polynomial ring $R_{q}[x,\Theta]$ is the set $R_{q}[x]$ of formal polynomials over $R_{q}$, where the addition is defined as the usual addition of polynomials and the multiplication is defined using the rule $xa=\Theta(a)x$, which is extended to all elements of $R_{q}[x,\Theta]$ by associativity and distributivity. An element in $R_{q}[x,\Theta]$ is called a skew polynomial.

Let $\lambda$ be a unit of $R_{q}.$ For a given automorphism $\Theta$ of $R_{q}$, we define the $\Theta$-$\lambda$- constacyclic shift $\texttt{T}_{\Theta,\lambda}$ on $R^{n}_{q}$ by:

[TABLE]

A linear code $C$ of length $n$ over $R_{q}$ is said to be skew-$\Theta$-$\lambda$-constacyclic, if $\texttt{T}_{\Theta,\lambda}(C)=C.$ Similarly to classical constacyclic codes over finite rings, skew-$\Theta$-$\lambda$-constacyclic codes of length $n$ over $R_{q}$ are identified with the left $R_{q}[x,\Theta]$-submodule of $R_{q}[x,\Theta]/\langle\,x^{n}-\lambda\,\rangle$ by the identification:

[TABLE]

Indeed, it is straightforward to see that the set $R_{q}[x,\Theta]/\langle\,x^{n}-\lambda\,\rangle$ is a left $R_{q}[x,\Theta]$-module under the multiplication defined by

[TABLE]

with $f(x),g(x)\in R_{q}[x,\Theta]/\langle\,x^{n}-\lambda\,\rangle$ and by analogous methods that have been used for skew constacyclic codes over finite fields [11], we have the following fact.

Lemma 2**.**

Let $\lambda$ be a unit of $R_{q}$, $\Theta$ an automorphism of $R_{q}$ and $C$ be a linear code of length $n$ over $R_{q}.$ Then $C$ is a skew-$\Theta$-$\lambda$-constacyclic code if and only if $\varphi(C)$ is a left $R[x,\Theta]$-submodule of $R_{q}[x,\Theta]/\langle\,x^{n}-\lambda\,\rangle.$

In order to characterize skew-$\Theta$-$\lambda$-constacyclic codes over $R_{q}$, we recall some well-known results about skew-constacyclic codes over finite fields [11, 12, 16, 23]. The skew reciprocal polynomial of a polynomial $g=\sum\limits_{j=0}^{k}a_{j}x^{j}\in\mathbb{F}_{q}[x,\theta]$ of degree $k$ denoted by $g^{*}$ is defined as

[TABLE]

If $a_{0}\neq 0$, the left monic skew reciprocal polynomial of $g$ is $g^{\natural}:=\frac{1}{\theta^{k}(a_{0})}g^{*}$ [12, Definition 3].

Lemma 3**.**

Let $C$ be a skew-$\theta$-$\lambda$-constacyclic code of length $n$ over $\mathbb{F}_{q}.$ Then there exists a monic polynomial $g$ of minimal degree in $C$ such that $g(x)$ is a right divisor of $x^{n}-\lambda$ and $C=\langle\,g(x)\,\rangle.$

Let $g(x)=x^{m}+a_{m-1}x^{m-1}+\cdots+a_{0}$ be a generator of a skew-$\theta$-$\lambda$-constacyclic code of length $n$ over $\mathbb{F}_{q}$ . Since $x^{n}-\lambda=h(x)g(x)$ for some $h\in\mathbb{F}_{q}[x,\theta]$; then $a_{0}$ must be a non-zero element of $\mathbb{F}_{q}.$ From [11, Theorem 1], we have the following result.

Lemma 4**.**

Let $C$ be a skew-$\theta$-$\lambda$-constacyclic code of length $n$ over $\mathbb{F}_{q}$ generated by a monic polynomial $g$ of degree $n-k$ with $g(x)=x^{n-k}+\sum\limits_{j=0}^{n-k-1}a_{i}x^{i}.$ Let $\lambda^{*}=\frac{\theta^{n}(a_{0})}{a_{0}\theta^{n-k}(\lambda)}.$ Then $C^{\perp}$ is a skew-$\theta$-$\lambda^{*}$-constacyclic code of length $n$ over $\mathbb{F}_{q}$ such that $C^{\perp}=\langle\,h^{*}(x)\,\rangle$ where $h$ is a monic polynomial of degree $k$ such that $x^{n}-\theta^{-k}(\lambda)=g(x)h(x).$ Moreover $h^{*}(x)$ is a right divisor of $x^{n}-\lambda^{*}.$

We generalize the notion of multi-twisted codes which be defined in [3], by introducing skew-multi-twisted codes.

Definition 2**.**

Let $C$ be a linear code of length $nq$ over $\mathbb{F}_{q}$ and $(\theta,\sigma)\in\texttt{Aut}(\mathbb{F}_{q})\times\mathbb{S}_{q}.$ Let $\lambda_{0},\lambda_{1},\cdots,\lambda_{q-1}$ be units in $\mathbb{F}_{q}.$ The code $C$ is skew $(\lambda_{0},\lambda_{1},\cdots,\lambda_{q-1})$-multi-twisted w.r.t $(\theta,\sigma)$ if for any codeword $\textbf{c}:=\left(\textbf{c}_{0},\textbf{c}_{1},\cdots,\textbf{c}_{q-1}\right)\in C$ where $\textbf{c}_{i}:=\Phi_{i}(\textbf{c})$ for $0\leq i\leq q-1,$ the word

[TABLE]

is in $C.$

Theorem 3**.**

Let $\lambda=\lambda_{0}\eta_{0}+\lambda_{1}\eta_{1}+\cdots+\lambda_{q-1}\eta_{q-1}$ be a unit of $R_{q}$ and $C$ be a linear code of length $n$ over $R_{q}.$ Then $C$ is skew $\Theta_{\theta,\sigma}$-$\lambda$-constacyclic if and only if the linear code $\Phi(C)$ of length $qn$ over $\mathbb{F}_{q}$ is skew $(\lambda_{0},\lambda_{1},\cdots,\lambda_{q-1})$-multi-twisted w.r.t $(\theta,\sigma).$

**Proof. **Set $\Theta:=\Theta_{\theta,\sigma}.$ Let $\textbf{c}=(c_{0},c_{1},\cdots,c_{n-1})\in(R_{q})^{n}$ where $\textbf{c}_{j}:=c_{0,j}\eta_{0}+c_{1,j}\eta_{1}+\cdots+c_{q-1,j}\eta_{q-1};$ for all $0\leq j\leq n-1.$ Then for any $0\leq t\leq q-1,$

[TABLE]

Thus, it follows that

[TABLE]

Hence, $C$ is a skew $\Theta$-$\lambda$-constacyclic code of length $n$ over $R_{q}$ if and only if $\texttt{T}_{\Theta,\lambda}(C)=C,$ which happens if and only if $\Phi(\texttt{T}_{\Theta,\lambda}(C))=\Phi(C),$ which is equivalent to saying $\Phi(C)$ is skew $(\lambda_{0},\lambda_{1},\cdots,\lambda_{q-1})$-multi-twisted code w.r.t $(\theta,\sigma).$ ∎

The following results are direct consequences of the above theorem:

Corollary 2**.**

Let $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\cdots\oplus\eta_{q-1}C_{q-1}$ be a linear code of length $n$ over $R_{q}$ and $\lambda=\lambda_{0}\eta_{0}+\lambda_{1}\eta_{1}+\cdots+\lambda_{q-1}\eta_{q-1}$ be a unit of $R_{q}.$ Then $C$ is a skew $\Theta_{\theta,\sigma}$-$\lambda$-constacyclic code of length $n$ over $R_{q}$ if and only if $C_{i}$ is a skew $\theta$-$\lambda_{\sigma(i)}$-constacyclic code of length $n$ over $\mathbb{F}_{q},$ for all $0\leq i\leq q-1.$

In the sequel, we only consider the automorphism $\Theta_{\theta}(=\Theta_{\theta,\texttt{id}})$ defined by

[TABLE]

where $\theta\in\texttt{Aut}(\mathbb{F}_{q}).$ Now, we give a generator of a skew-constacyclic code over $R_{q}.$

Theorem 4**.**

Let $\lambda=\lambda_{0}\eta_{0}+\lambda_{1}\eta_{1}+\cdots+\lambda_{q-1}\eta_{q-1}\in\mathcal{U}(R_{q})$ and $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\cdots\oplus\eta_{q-1}C_{q-1}$ be a skew $\Theta_{\theta}$-$\lambda$-constacyclic code over $R_{q}.$ Then there exist polynomials $g_{0},g_{1},\cdots,g_{q-1}\in\mathbb{F}_{q}[x,\theta]$ such that

[TABLE]

with $C_{i}=\langle\,g_{i}\,\rangle$ in $\frac{\mathbb{F}_{q}[x,\theta]}{\langle\,x^{n}-\lambda_{i}\,\rangle}.$

**Proof. **Let $D=\langle\,\eta_{0}g_{0},\eta_{1}g_{1},\cdots,\eta_{q-1}g_{q-1}\,\rangle.$ Let $\textbf{c}(x)\in C$ such that $\textbf{c}(x)=\sum\limits_{i=0}^{q-1}\eta_{i}\textbf{c}_{i}(x)$ with $\textbf{c}_{i}\in C_{i},~{}0\leq i\leq q-1.$ Since $C_{i}=\langle\,g_{i}\,\rangle$ as a left submodule of $\frac{\mathbb{F}_{q}[x,\theta]}{\langle\,x^{n}-\lambda_{i}\,\rangle}$; there exist $k_{0}(x),k_{1}(x),\cdots,k_{q-1}(x)\in\mathbb{F}_{q}[x,\theta]$ such that $\textbf{c}(x)=\sum\limits_{i=0}^{q-1}\eta_{i}k_{i}(x)g_{i}(x)$. Therefore $\textbf{c}\in D.$ Reciprocally, let $\textbf{d}\in D$, there exist $l_{0}(x),l_{1}(x),\cdots,l_{q-1}(x)\in\frac{R_{q}[x,\Theta_{\theta}]}{\langle\,x^{n}-\lambda\,\rangle}$ such that $\textbf{d}(x)=\sum\limits_{i=0}^{q-1}\eta_{i}l_{i}(x)g_{i}(x).$ Then for all $0\leq i\leq q-1,$ there exists $a_{i}(x)\in\mathbb{F}_{q}[x,\theta]$ such that $\eta_{i}l_{i}(x)=\eta_{i}a_{i}(x)$, hence $\textbf{d}(x)=\sum\limits_{i=0}^{q-1}\eta_{i}a_{i}(x)g_{i}(x)$ and $\textbf{d}\in C.$ ∎

Corollary 3**.**

Under the above assumptions; let $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\cdots\oplus\eta_{q-1}C_{q-1}$ be a skew $\Theta_{\theta}$-$\lambda$-constacyclic code over $R_{q}$ such that

[TABLE]

Then $C$ is principally generated with $C=\langle\,g(x)\,\rangle$, where $g(x)=\eta_{0}g_{0}(x)+\eta_{1}g_{1}(x)+\cdots+\eta_{q-1}g_{q-1}(x).$ Moreover $g(x)$ is a right divisor of $x^{n}-\lambda$ in $R_{q}[x,\Theta_{\theta}].$

**Proof. **It is clear that $\langle\,g(x)\,\rangle\subseteq C.$ Reciprocally, we have $\eta_{i}g(x)=\eta_{i}g_{i}(x)$ for all $0\leq i\leq q-1,$ which implies that $C\subseteq\langle\,g(x)\,\rangle,$ whence $C=\langle\,g(x)\,\rangle.$ Since $g_{i}$ is a right divisor of $x^{n}-\lambda_{i}$, for all $0\leq i\leq q-1,$ there exists $h_{i}(x)\in\mathbb{F}_{q}[x,\theta]$ such that $x^{n}-\lambda_{i}=h_{i}g_{i}.$ Since $\eta_{i}(x^{n}-\lambda)=\eta_{i}(x^{n}-\lambda_{i})$ therefore

[TABLE]

which implies that $\sum\limits_{i=0}^{q-1}\eta_{i}h_{i}$ is a right divisor of $x^{n}-\lambda$ over $R_{q}.$ ∎

By a similar work, and by Lemma 4 we have the corresponding result for skew-dual codes.

Lemma 5**.**

Under the above assumptions; let $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\cdots\oplus\eta_{q-1}C_{q-1}$ be a skew $\Theta_{\theta}$-$\lambda$- constacyclic code over $R_{q}$ such that $C=\langle\,\eta_{0}g_{0},\eta_{1}g_{1},\cdots,\eta_{q-1}g_{q-1}\,\rangle$ with $g_{i}=x^{n-k_{i}}+\sum_{j=0}^{k_{i}}a_{ij}x^{j}.$ Let $\lambda_{i}^{*}=\frac{\theta^{n}(a_{i0})}{a_{i0}\theta^{n-k_{i}}(\lambda_{i})}$, for all $0\leq i\leq q-1$ and $\lambda^{*}=\sum\limits_{i=0}^{q-1}\eta_{i}\lambda_{i}^{*}.$ Then $C^{\perp}$ is a skew-$\Theta_{\theta}$-$\lambda^{*}$-constacyclic codes over $R_{q}$ such that $C^{\perp}=\langle\,h^{*}(x)\,\rangle,$ where $h^{*}(x)=\sum\limits_{i=0}^{q-1}\eta_{i}h_{i}^{*}$ and $h_{0},h_{1},\cdots,h_{q-1}$ are skew monic polynomials of $\mathbb{F}_{q}[x,\theta]$ such that $x^{n}-\theta^{-k_{i}}(\lambda_{i})=g_{i}h_{i}$ in $\mathbb{F}_{q}[x,\theta].$ Moreover $h^{*}(x)$ is a right divisor of $x^{n}-\lambda^{*}$ in $R_{q}[x,\Theta_{\theta}].$

Now we characterize self-dual skew $\Theta_{\theta}$-$\lambda$-constacyclic codes of length $n$ over $R_{q}.$

Theorem 5**.**

Let $C$ be a linear code of even length $n$ over $R_{q}$ and $\lambda$ be a unit of $R_{q}.$ If $C$ is a self-dual skew $\Theta_{\theta}$-$\lambda$-constacyclic code of length $n$ over $R_{q}$ then either $C$ is skew $\Theta_{\theta}$-cyclic or skew $\Theta_{\theta}$-negacyclic.

**Proof. **Let $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\cdots\oplus\eta_{q-1}C_{q-1}$ be a skew $\Theta_{\theta}$-$\lambda$-constacyclic code of length $n$ over $R_{q}.$ It is follows from Proposition 3.2 that $C$ is self-dual code over $R_{q}$ if and only if $C_{0},C_{1},\cdots,C_{q-1}$ are self-dual codes over $\mathbb{F}_{q}.$ From Corollary 2, $C$ is a skew $\Theta_{\theta}$-$\lambda$-constacyclic code of length $n$ over $R_{q}$ if and only if $C_{i}$ is a skew $\theta$-$\lambda_{i}$-constacyclic code of length $n$ over $\mathbb{F}_{q}$; for all $0\leq i\leq q-1.$ But the only self-dual skew $\theta$-constacyclic codes over $\mathbb{F}_{q}$ are self-dual skew $\theta$-cyclic codes and self-dual skew $\theta$-negacyclic codes [12, Proposition 5]. Moreover, from [7, Proposition 3], there cannot exist both a self-dual skew $\theta$-cyclic code and a self-dual skew $\theta$-negacyclic code with the same dimension over $\mathbb{F}_{q}.$ Then all $C_{i}$ are skew $\theta$-cyclic codes or skew $\theta$-negacyclic codes of length $n.$ Whence $C$ is a self-dual skew $\Theta_{\theta}$-cyclic code or a self-dual skew $\Theta_{\theta}$-negacyclic code of length $n$ over $R_{q}.$ ∎

The following result gives necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over a finite field [7, Propositions 4 and 5].

Proposition 4.1**.**

Let $\mathbb{F}_{q}$ be a finite field with characteristic $p,$ which is an odd prime and $q=p^{r}.$ Let $\theta$ be an $\mathbb{F}_{p^{s}}$-automorphism of $\mathbb{F}_{q},$ where $s$ divides $r.$

1. (1)

If $q\equiv 1\mod 4$, then:

• $i)$

there exists a self-dual skew cyclic code of dimension $k$ over $\mathbb{F}_{q}$ if and only if $p\equiv 3\mod 4$, $r$ is even and $s\cdot k$ is odd.

• $ii)$

there exists a self-dual skew negacyclic code of dimension $k$ over $\mathbb{F}_{q}$ if and only if $p\equiv 1\mod 4$ or $p\equiv 3\mod 4$, $r$ is even and $s\cdot k$ is even. 2. (2)

If $q\equiv 3\mod 4$, then:

• $i)$

There does not exist a self-dual skew cyclic code of dimension $k$ over $\mathbb{F}_{q}.$

• $ii)$

There exists a self-dual skew negacyclic code of dimension k over $\mathbb{F}_{q}$ if and only if $k\equiv 0\mod 2^{\mu-1}$, where $\mu\geq 2$ is the biggest integer such that $2^{\mu}$ divides $p+1.$

From Theorem 5, the previous result is extend naturally to the ring $R_{q}.$

Proposition 4.2**.**

Let $\mathbb{F}_{q}$ be a finite field with characteristic $p,$ which is an odd prime and $q=p^{r}.$ Let $\theta$ be an $\mathbb{F}_{p^{s}}$-automorphism of $\mathbb{F}_{q},$ where $s$ divides $r.$

1. (1)

If $q\equiv 1\mod 4$, then:

• $i)$

there exists a self-dual skew $\Theta_{\theta}$-cyclic code of dimension $k$ over $R_{q}$ if and only if $p\equiv 3\mod 4$, $r$ is even and $s\cdot k$ is odd.

• $ii)$

there exists a self-dual skew $\Theta_{\theta}$-negacyclic code of dimension $k$ over $R_{q}$ if and only if $p\equiv 1\mod 4$ or $p\equiv 3\mod 4$, $r$ is even and $s\cdot k$ is even. 2. (2)

If $q\equiv 3\mod 4$, then:

• $i)$

There does not exist a self-dual skew $\Theta_{\theta}$-cyclic code of dimension $k$ over $R_{q}.$

• $ii)$

There exists a self-dual skew $\Theta_{\theta}$-negacyclic code of dimension k over $R_{q}$ if and only if $k\equiv 0\mod 2^{\mu-1}$, where $\mu\geq 2$ is the biggest integer such that $2^{\mu}$ divides $p+1.$

Example 4**.**

Let $\mathbb{F}_{4}=\{0,1,\alpha,\alpha^{2}\}$ be a finite field with $\alpha^{2}+\alpha+1=0.$ Let $\theta$ be the Frobenius automorphism $(a\longmapsto a^{2})$ over $\mathbb{F}_{4}.$ We use the technique given in [12] to construct self-dual skew-$\theta$-cyclic codes of length 6 over $\mathbb{F}_{4}.$ The factorization of $y^{3}-1$ into irreducible monic polynomials over $\mathbb{F}_{2}$ is given by: $y^{3}-1=(y+1)(y^{2}+y+1),$ and we set $f_{1}(y):=y+1$ and $f_{2}(y):=y^{2}+y+1.$ We have also $f_{1}^{\natural}(y)=f_{1}(y)$ and $f_{2}^{\natural}(y)=f_{2}(y).$

Let $\mathcal{G}_{1}:=\{g\in\mathbb{F}_{4}[x,\theta]:g^{\natural}g=x^{2}+1\}$ and $\mathcal{G}_{2}:=\{g\in\mathbb{F}_{4}[x,\theta]:g^{\natural}g=x^{4}+x^{2}+1\}.$ We compute $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ by using Gr bner basis and Magma language system [6]:

[TABLE]

The polynomial $g_{1}=\texttt{lclm}(x+1,x^{2}+x+1)=x^{3}+1$ generates a $[6,3,2]$-self-dual skew $\theta$-cyclic code and $g_{2}=\texttt{lclm}(x+1,x^{2}+\alpha^{2})=x^{3}+\alpha x^{2}+\alpha x+1$; $g_{3}=\texttt{lclm}(x+1,x^{2}+\alpha)=x^{3}+\alpha^{2}x^{2}+\alpha^{2}x+1$ generate $[6,3,3]$-self-dual skew $\theta$-cyclic codes over $\mathbb{F}_{4}.$

Let $\eta_{0}=\overline{v}^{3}+1;\eta_{1}=\overline{v}^{3}+\overline{v}+1;\eta_{2}=\overline{v}^{3}+\alpha\overline{v}^{2}+\alpha^{2}\overline{v};\eta_{4}=\overline{v}^{3}+\alpha^{2}\overline{v}^{2}+\alpha\overline{v}$ be the complete set of primitive pairwise orthogonal idempotents of $R_{4}.$ Let $C=\eta_{0}C_{0}\oplus\eta_{1}C_{1}\oplus\eta_{2}C_{2}\oplus\eta_{3}C_{3}$ with $C_{0},C_{1},C_{2},C_{3}$ be the skew $\theta$-cyclic codes over $\mathbb{F}_{4}$ generated by $g_{1},g_{1},g_{2},g_{3}$ respectively. According to Corollary 3, $C$ is a $[6,3,2]$-self-dual skew $\Theta_{\theta}$-cyclic code over $R_{4}$ generated by $g=\eta_{0}g_{1}+\eta_{1}g_{1}+\eta_{2}g_{2}+\eta_{3}g_{3}=x^{3}+(\overline{v}^{3}+\overline{v}^{2})x^{2}+(\overline{v}^{3}+\overline{v}^{2})x+1.$ A generator matrix of $C$ is given by:

[TABLE]

5. Conclusion

In this work, skew constacyclic codes are investigated over the non-chain ring $R_{q}.$ The automorphism group of this ring is given. We defined a linear Gray map and established a connection between skew constacyclic codes of length $n$ over $R_{q}$ and skew multi-twisted codes of length $qn$ over $\mathbb{F}_{q}.$ We determined generator polynomials for skew constacyclic codes over $R_{q}$ and for their dual codes. We also characterized self-dual skew constacyclic codes and provided necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over $R_{q}.$