Self-Dual Skew Cyclic Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$

TL;DR

This paper investigates the existence and construction of Hermitian self-dual skew cyclic and negacyclic codes over a finite chain ring, establishing their properties and connections to skew quasi-twisted codes via Gray maps.

Contribution

It provides new conditions for the existence of self-dual skew cyclic codes over finite chain rings and extends algorithms for their construction.

Findings

Gray images of codes are equivalent to skew quasi-twisted codes.

Conditions for Hermitian self-duality over the ring are established.

An extended algorithm for constructing self-dual codes is proposed.

Abstract

In this paper, we give conditions for the existence of Hermitian self-dual $\Theta-$cyclic and $\Theta-$negacyclic codes over the finite chain ring $\mathbb{F}_q+u\mathbb{F}_q$. By defining a Gray map from $R=\mathbb{F}_q+u\mathbb{F}_q$ to $\mathbb{F}_{q}^{2}$, we prove that the Gray images of skew cyclic codes of odd length $n$ over $R$ with even characteristic are equivalent to skew quasi-twisted codes of length $2n$ over $\mathbb{F}_q$ of index $2$. We also extend an algorithm of Boucher and Ulmer \cite{BF3} to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over $\mathbb{F}_q+u\mathbb{F}_q$.

Equations

$$Aut(R)={\Theta }_{\theta ,\beta };\theta \in Aut({\mathbb{F}}_q)\text{ and }\beta \in {\mathbb{F}}_q^{\ast },$$

$$\Re =R[x,\Theta ]=\left\{a_0+a_{1}x+\dots +a_n{x}^n;a_i\in R\text{ and }n\in {\mathbb{N}}^{\ast }\right\},$$

$${\rho }_{\Theta ,\lambda }:R^n\to R^n,$$

$${\rho }_{\Theta ,\lambda }((a_0,a_1,\dots ,a_{n-1}))=(\Theta (\lambda a_{n-1}),\Theta (a_0),\dots ,\Theta (a_{n-2})).$$

$$G=\left[\begin{array}{ccccccc}{g}_0& \dots & g_{n-k-1}& 1& 0& \dots & 0\\ 0& \Theta (g_0)& \dots & \Theta (g_{n-k-1})& 1& \dots & 0\\ 0& \dots & \dots & \dots & {\Theta }^2(g_{n-k-1})& \dots & 0\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & ⋮\\ 0& \dots & 0& {\Theta }^{k-1}(g_0)& \dots & {\Theta }^{k-1}(g_{n-k-1})& 1\end{array}\right]$$

$$\varphi \left(\sum _{i=0}^t{a}_i{x}^i\right)=\sum _{i=0}^t\Theta (a_i)x^i.$$

$$g^{\perp }=\varphi (x^{deg(h(x))}\phi (h(x))),$$

$$\phi \left(\sum _{i=0}^t{a}_i{x}^i\right)=\sum _{i=0}^t{x}^{-i}{a}_i,$$

$$(\sum _{i=0}^{k-1}{g}_i{x}^i+x^k)({\Theta }^{-k-1}(g_0^{-1})+\sum _{i=1}^{k-1}{\Theta }^{i-k-1}(g_0^{-1}{g}_{k-i})x^i+x^k)=x^n-\lambda .$$

$$h=\sum _{i=0}^m{h}_i{X}^i\in \Re ,$$

$$h^{\ast }=\sum _{i=0}^m{X}^{m-i}{h}_i=\sum _{i=0}^m{\Theta }^i\left(h_{m-i}\right)X^i.$$

$$a_{1}e+a_{2}n=1⟹a_{1}e=1-a_{2}n.$$

$$x^{a_{1}e}c(x)={\Theta }^{a_{1}e}(c_0)x^{1+\ell n}+{\Theta }^{a_{1}e}(c_1)x^{2+\ell n}+\cdots +{\Theta }^{a_{1}e}(c_{n-1})x^{n+\ell n},$$

$$x^{a_{1}e}C(x)=\lambda c_{n-1}+c_{0}x+\cdots +c_{n-2}{x}^{n-2}\in C.$$

$$\left(\begin{array}[]{cc}c_{0,0},c_{0,1},\dots,c_{0,\ell-1},c_{1,0},c_{1,1},\dots,c_{1,\ell-1},\dots,\\ c_{s-1,0},c_{s-1,1},\dots,c_{s-1,\ell-1}\end{array}\right)\in C,$$

$$\left(\begin{array}[]{cc}\lambda\Theta(c_{s-1,0}),\lambda\Theta(c_{s-1,1}),\dots,\lambda\Theta(c_{s-1,\ell-1}),\Theta(c_{0,0}),\Theta(c_{0,1}),\dots,\Theta(c_{0,\ell-1}),\dots,\\ \Theta(c_{s-2,0}),\Theta(c_{s-2,1}),\dots,\Theta(c_{s-2,\ell-1})\end{array}\right)\in C.$$

$$\left(\begin{array}[]{cc}c_{0,0},c_{0,1},\dots,c_{0,\ell-1},c_{1,0},c_{1,1},\dots,c_{1,\ell-1},\dots,\\ c_{s-1,0},c_{s-1,1},\dots,c_{s-1,\ell-1}\end{array}\right)\in C.$$

$$\Theta^{\ell+jn}\left(\begin{array}[]{cc}c_{0,0},c_{0,1},\dots,c_{0,\ell-1},c_{1,0},c_{1,1},\dots,c_{1,\ell-1},\dots,\\ c_{s-1,0},c_{s-1,1},\dots,c_{s-1,\ell-1}\end{array}\right)$$

$$=\left(\begin{array}[]{cc}\Theta^{\ell+jn}(\lambda c_{s-1,0}),\Theta^{\ell+jn}(\lambda c_{s-1,1}),\dots,\Theta^{\ell+jn}(\lambda c_{s-1,\ell-1}),\Theta^{\ell+jn}(c_{0,0}),\dots,&\\ \Theta^{\ell+jn}(c_{0,\ell-1}),\dots,\Theta^{\ell+jn}(c_{s-2,0}),\Theta^{\ell+jn}(c_{s-2,1}),\dots,\Theta^{\ell+jn}(c_{s-2,\ell-1})\end{array}\right)$$

$$\Theta^{\ell+jn}\left(\begin{array}[]{cc}c_{0,0},c_{0,1},\dots,c_{0,\ell-1},c_{1,0},c_{1,1},\dots,c_{1,\ell-1},\dots,\\ c_{s-1,0},c_{s-1,1},\dots,c_{s-1,\ell-1}\end{array}\right)$$

$$=\left(\begin{array}[]{cc}\lambda c_{s-1,0},\lambda c_{s-1,1},\dots,\lambda c_{s-1,\ell-1},c_{0,0},\dots,&\\ c_{0,\ell-1},\dots,c_{s-2,0},c_{s-2,1},\dots,c_{s-2,\ell-1}\end{array}\right)\in C.$$

$$\Phi :R^n\to {\mathbb{F}}_q^{2n}$$

$$\Phi (a+ub)=(b,a+b),$$

$$w_L(a+ub)=w_H(b)+w_H(a+b),\text{mbox}fora,b\in {\mathbb{F}}_q^n.$$

$$\sigma (a^{(0)}\mid a^{(1)})=({\rho }_{\Theta ,\lambda }(a^{(0)})\mid {\rho }_{\Theta ,\lambda }(a^{(1)})),$$

$$\begin{array}[]{ccl}\sigma(\Phi(r))&=&\sigma(b_{0},b_{1},\dots,b_{n-1},a_{0}+b_{0},a_{1}+b_{1},\dots,a_{n-1}+b_{n-1})\\ &=&\left(\begin{array}[]{cc}\Theta(\lambda b_{n-1}),\Theta(b_{0}),\dots,\Theta(b_{n-2}),\Theta(\lambda a_{n-1})+\Theta(\lambda b_{n-1}),&\\ \Theta(a_{0})+\Theta(b_{0}),\dots,\Theta(a_{n-2})+\Theta(b_{n-2}),\end{array}\right),\end{array}$$

$$\sigma \left(\Phi (C)\right)=\Phi \left({\rho }_{\Theta ,\lambda }(C)\right)=\Phi (C).$$

$$\sigma \left(\Phi (C)\right)=\Phi (C).$$

$$\Phi (C)=\sigma \left(\Phi (C)\right)=\Phi \left({\rho }_{\Theta ,\lambda }(C)\right),$$

$$\upsilon (c(x))=c(\lambda x).$$

Full text

Self-Dual Skew Cyclic Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$

Zineb Hebbache, Kenza Guenda, N. Tugba Özzaim, Mehmet Özen and T. Aaron Gulliver

E-mail: zinebhebache@gmail.com (Z. Hebbache), ken.guenda@gmail.com (K. Guenda)111Faculty of Mathematics, USTHB, Laboratory of Algebra and Number Theory, BP 32 El Alia, Bab Ezzouar, Algeria

E-mail: ozen@sakarya.edu.tr (M. Özen), tugbaozzaim@gmail.com (N.T. Özzaim)222Department of Mathematics, Sakarya University, Sakarya, Turkey

E-mail: agullive@ece.uvic.ca (T. Aaron Gulliver)333Department of Electrical and Computer Engineering, University of Victoria, PO Box 1700, STN CSC, Victoria, BC, Canada V8W 2Y2

Abstract

• In this paper, we give conditions for the existence of Hermitian self-dual $\Theta-$cyclic and $\Theta-$negacyclic codes over the finite chain ring $\mathbb{F}_{q}+u\mathbb{F}_{q}$. By defining a Gray map from $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ to $\mathbb{F}_{q}^{2}$, we prove that the Gray images of skew cyclic codes of odd length $n$ over $R$ with even characteristic are equivalent to skew quasi-twisted codes of length $2n$ over $\mathbb{F}_{q}$ of index $2$. We also extend an algorithm of Boucher and Ulmer [9] to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over $\mathbb{F}_{q}+u\mathbb{F}_{q}$.

Keywords: Finite chain ring, skew polynomial ring, self-dual skew codes, self-dual skew cyclic codes, complexity.

1 Introduction

Codes over finite rings have been studied for many years and interest in these codes has increased recently due to applications such as quantum and DNA systems by Nabil et al. [5]. Boucher et al. [10] generalized the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. Jitman et al. [17] defined skew constacyclic codes over finite chain rings, in particular over the ring $\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$. Recently, Ashraf et al. [2] studied the structural properties of skew cyclic codes over the finite semi-local ring $\mathbb{F}_{p^{m}}+v\mathbb{F}_{p^{m}}$ with $v^{2}=1$. An advantage of skew cyclic codes over $\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$ with $u^{2}=0$ is that many of the best linear codes over finite fields can be obtained using the Gray images of these codes.

Batoul et al. [4] gave conditions on the existence of self-dual codes derived from constacyclic codes over finite principal ideal rings. Further, Boucher et al. ([6], [9]) gave conditions on the existence of self-dual $\Theta-$cyclic and self-dual $\Theta-$negacyclic codes. In this paper, we give necessary and sufficient conditions on the existence of self-dual $\Theta-$cyclic codes and $\Theta-$negacyclic codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$ where $\Theta$ is an automorphism of $\mathbb{F}_{q}+u\mathbb{F}_{q}$. We also extend an algorithm of Boucher [9] to construct self-dual skew cyclic codes based on the least common multiples of polynomials over the ring $\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$ with $u^{2}=0$.

The remainder of this article is organized as follows. Some facts regarding Hermitian self-dual skew codes are recalled in Section 2. In Section 3, we prove that the Gray images of skew cyclic codes of odd length $n$ over $R$ are equivalent to skew quasi-twisted codes of length $2n$ over $\mathbb{F}_{q}$ with even characteristic using an approach by Amarra et al. [1]. In Section 4, necessary and sufficient conditions are given on the existence of Hermitian self-dual skew codes generated by skew binomials and skew trinomials when the characteristic of $R$ is odd. Further, sufficient conditions on the existence of Hermitian self-dual skew cyclic codes of length $n$ over $R$ are given using the order of $q$ modulo $n$. Section 5 gives an iterative construction of self-dual $\Theta-$cyclic codes over $R$ using the generator polynomials of self-dual $\Theta-$cyclic codes as the least common left multiple of skew polynomials.

2 Preliminaries

This section provides some useful results regarding skew constacyclic codes over the ring $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$. First, we recall some facts concerning $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$. This is a commutative ring and can be defined as the quotient ring $R=\frac{\mathbb{F}_{q}[u]}{\langle u^{2}\rangle}$ with $u^{2}=0$. This is an extension of degree $2$ of $\mathbb{F}_{q}$ so $|R|=q^{2}$. This ring is principal with a unique maximal ideal $\langle u\rangle$. To define skew polynomials over $R$, we first give the structure of the automorphism group of $R$ denoted by $Aut(R)$. For $\theta\in\mathbb{F}_{q}$, $Aut(R)$ is given by

[TABLE]

where for $a+ub\in R$, $\Theta_{\theta,\beta}(a+ub)=\theta(a)+\beta\theta(b)u$ [17, Corollary 2.1]. For simplicity, where no confusion arises, the subscripts $\theta$ and $\beta$ will be dropped.

Considering the finite chain ring $R$ and automorphism $\Theta$ of $R$ defined above, the set of formal polynomials

[TABLE]

forms a ring under the usual addition of polynomials where multiplication is defined using the rule $x\times a=\Theta(a)x$. This multiplication is extended to all elements in $\Re$ by associativity and distributivity. The ring $\Re$ is called a skew polynomial ring over $R$ and an element of $\Re$ is called a skew polynomial. It is easily seen that the ring $\Re$ is non-commutative unless $\Theta$ is the identity automorphism of $R$. According to [10, Section 1], $\Re$ is no longer left or right Euclidean, but left or right division can be defined for some elements. From [17, Section 2], the subring of the elements of $R$ that are fixed by $\Theta$ is $R^{\Theta}$.

** Proposition 1****.**

Let $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^{2}=0$, $\Theta$ be an automorphism of $R$ with order $e$ and $\Re=R[x,\Theta]$. Then the center $Z(\Re)$ of $\Re$ is $R^{\Theta}[x^{e}]$.

Proof.

For any integer $i\in N$, the power $x^{ei}$ is also in the center $Z(\Re)$ of $\Re$. This follows from the fact that $e$ is the order of the automorphism $\Theta$ so we have that $x^{ei}(\alpha)=(\Theta^{e})^{i}(\alpha)x^{ei}=\alpha x^{ei}$ for any $\alpha\in R$. This implies that $f=\alpha_{0}+\alpha_{1}x^{e}+\alpha_{2}x^{2e}+\dots+\alpha_{s}x^{se}$ with $\alpha_{i}\in R^{\Theta}$ a central element. Conversely, for any $f\in Z(\Re)$ and $a\in R$, if $xf=fx$ and $af=fa,$ then $f\in R^{\Theta}[x^{e}]$. ∎

We restrict our study to the case where the length $n$ of codes is a multiple of the order of $\Theta$ and $\lambda$ is a unit in $R^{\Theta}.$ The following Proposition explains why the factors in the decomposition of the generator of a central monic polynomial into two monic polynomials always commute.

** Proposition 2****.**

Under the hypothesis of Proposition 1, if $f\in R^{\Theta}[x^{e}]$ is a monic polynomial which decomposes into a product of two monic polynomials $h$ and $g$ as $hg$ over $\Re=R[x,\Theta]$, then $hg=gf$ in $\Re$.

Proof.

Since $hg$ is a central element we have $(hg)h=h(hg)$. Therefore $h(gh-hg)=0$. Since the leading coefficient of $h$ is invertible, $h$ is not a zero divisor, so that $hg=gh$ in $\Re$. ∎

A code of length $n$ over $R$ is a nonempty subset of $R^{n}$. A code $C$ over $R$ is said to be linear if it is a submodule of the $R-$module $R^{n}$. In this paper, all codes are assumed to be linear unless otherwise stated.

Given an automorphism $\Theta$ of $R$ and a unit $\lambda$ in $R$, a code $C$ is said to be skew constacyclic, or specifically, $\Theta-\lambda-$constacyclic, if $C$ is closed under the $\Theta-\lambda-$constacyclic shift

[TABLE]

defined by

[TABLE]

In particular, such codes are called skew cyclic and skew negacyclic when $\lambda$ is $1$ and $-1$, respectively. When $\Theta$ is the identity automorphism, they become the classical constacyclic, cyclic and negacyclic codes.

When the order of $\Theta$ is $2$, the Hermitian dual code of a code $C$ of length $n$ over $R$ is defined using the Hermitian inner product $\langle u,v\rangle_{H}=\sum_{i=0}^{n-1}u_{i}\Theta(v_{i})$ for $u=(u_{0},u_{1},\dots,u_{n-1})$ and $v=(v_{0},v_{1},\dots,v_{n-1})$ in $R^{n}$ as $C^{\perp H}=\{v\in R^{n}|\left\langle v,c\right\rangle_{H}=0$ for all $c\in C\}$. A code $C$ is said to be Hermitian self-dual if $C=C^{\perp H}.$

Given a right divisor $g(x)=\sum_{i=0}^{n-k-1}g_{i}x^{i}+x^{n-k}$ of $x^{n}-\lambda$, a generator matrix of the $\Theta-\lambda-$constacyclic code $C$ generated by $g(x)$ is given by

[TABLE]

From [17], the ring automorphism $\phi$ on $\Re$ is defined as

[TABLE]

According to [17], we have the following results which characterize the skew constacyclic and Hermitian self-dual skew constacyclic codes when the order of $\Theta$ is $2$.

** Proposition 3****.**

*([17])

Let $C$ be a code of even length $n$ over $R,\lambda^{2}=1$ and let $h,g\in\Re$ such that $g$ is a monic right divisor of $x^{n}-\lambda$ and $h(x)=\frac{x^{n}-\lambda}{g(x)}$. Then the $\Theta-\lambda-$constacyclic code $C$ generated by $g(x)$ is a free $R-$module with $|C\arrowvert=|R\arrowvert^{n-deg(g(x))}$. Furthermore, $C$ is a $\Theta-\lambda-$constacyclic code generated by $g(x)$ if and only if $C^{\perp H}$ is a $\Theta-\lambda-$constacyclic code generated by*

[TABLE]

where $\varphi:\Re\longrightarrow\Re S^{-1}$ is the ring anti-monomorphism defined by

[TABLE]

with $S=\{x^{i};i\in\mathbb{N}\}$.

** Proposition 4****.**

*([17, Theorem 3.8])

Assume that the order of $\Theta$ is $2$, $\lambda^{2}=1$ and $n$ is an even integer $n=2k$. Let $g(x)=\sum_{i=0}^{k-1}g_{i}x^{i}+x^{k}$ be a right divisor of $x^{n}-\lambda$. Then the $\Theta-\lambda-$constacyclic code generated by $g(x)$ is Hermitian self-dual if and only if*

[TABLE]

This is called the self-dual skew condition.

** Definition 1****.**

*([7, Definition 2])

Let $R$ be a commutative finite ring. The skew reciprocal polynomial of*

[TABLE]

of degree $m$ is

[TABLE]

The left monic skew reciprocal polynomial of $h$ is $h^{\natural}=\frac{1}{\Theta^{m}(h_{0})}h^{\ast}$.

In the following, we focus on the relationship between skew $\lambda-$constacyclic codes, $\lambda-$constacyclic codes and $\lambda-$quasi-twisted codes over $R$.

** Proposition 5****.**

Let $C$ be a skew $\lambda-$constacyclic code of length $n$ and let $\Theta$ be an automorphism of $R$ with order $e$. If $\gcd(e,n)=1$, then $C$ is a constacyclic code of length $n$ over $R$.

Proof.

Let $C$ be a skew $\lambda-$constacyclicc code of length $n$ such that $\gcd(e,n)=1$ and let $\lambda$ be a unit of $R$ fixed by $\Theta$. We know that there exist integers $a_{1},a_{2}$ such that

[TABLE]

We may assume that $a_{2}$ is a negative integer, so we can write $a_{1}e=1+\ell n$ where $\ell>0$. Let $c(x)=c_{0}+c_{1}x+\dots+c_{n-1}x^{n-1}$ be a codeword in $C$. Note that for

[TABLE]

over the ring $\Re/\langle x^{n}-\lambda\rangle$, we have $x^{n}=\lambda$ for any $a\in R$ and $\Theta^{e}(a)=a$. This implies that

[TABLE]

Thus, $C$ is a constacyclic code of length $n$. ∎

From [14, Definition 2], we give the definition of $\Theta-\lambda-$quasi-twisted codes.

** Definition 2****.**

Let $C$ be a linear code of length $n$ over $R$ and $n=s\ell$. Then $R$ is said to be a $\Theta-\lambda-$quasi-twisted code if for any

[TABLE]

then

[TABLE]

If $\Theta$ is the identity map, we call $C$ a quasi-twisted code of index $\ell$ over $R$.

** Proposition 6****.**

Let $C$ be a skew $\lambda-$constacyclic code of length $n=s\ell$ over $R$ and $\Theta$ be an automorphism with order $e$. If $\gcd(e,n)=\ell$, then $C$ is a $\lambda-$quasi-twisted code of length $n$ with index $\ell$ over $R$.

Proof.

Let $n=s\ell$ and

[TABLE]

Since $\gcd(e,n)=\ell,$ there exist integers $a_{1},a_{2}$ such that $a_{1}e+a_{2}n=\ell$. Therefore $a_{1}e=\ell-a_{2}n=\ell+jn$ where $j$ is a nonnegative integer. Furthermore, let $\lambda$ be a unit of $R$ fixed by $\Theta$ and

[TABLE]

[TABLE]

Since the order of $\Theta$ is $e$, $\Theta^{\ell+jn}(a)=\Theta^{a_{1}e}(a)=a$ for any $a\in R$ which implies that

[TABLE]

[TABLE]

Thus, $C$ is a $\lambda-$quasi-twisted code of length $n$ with index $\ell$ over $R$. ∎

3 Gray Images of Skew Cyclic Codes with Odd Length

In this section, we give a characterization of the Gray images of skew cyclic codes of odd length $n$ over $R$ with even characteristic, where $\Theta=\Theta_{\theta,1}$ is defined as in the previous section. Let $\lambda=1+u$ be a unit of $R$. Then we have that $\lambda^{n}=1$ if $n$ is even and $\lambda^{n}=\lambda$ if $n$ is odd. Thus, we only consider the properties of skew $\lambda-$constacyclic codes of odd length in this section.

We know that every element of $R$ can be expressed as $a+ub$ where $a,b\in\mathbb{F}_{q}$. According to Ling et al. [20], we have the following Gray map

[TABLE]

[TABLE]

where $a,b\in\mathbb{F}_{q}^{n}$. Following [23], the Lee weight is defined as the Hamming weight of the Gray image

[TABLE]

The Lee distance of $x,y\in R^{n}$ is defined as $w_{L}(x-y)$. Thus, the Gray map $\Phi$ is a linear isometry from ($R^{n}$, Lee distance) to ($\mathbb{F}_{q}^{2n}$, Hamming distance).

Let $\sigma:\mathbb{F}_{q}^{2n}\rightarrow\mathbb{F}_{q}^{2n}$ be the skew quasi-twisted shift operator defined by

[TABLE]

where $a^{(0)},a^{(1)}\in\mathbb{F}_{q}^{n},\mid$ is vector concatenation and $\rho_{\Theta,\lambda}$ is the skew constacyclic shift operator as defined in the previous section. A linear code $C$ of length $2n$ over $\mathbb{F}_{q}$ is said to be skew quasi-twisted of index $2$ if $\sigma(C)=C$.

** Proposition 7****.**

With the previous notation, we have $\Phi\circ\rho_{\lambda,\Theta}=\sigma\circ\Phi$.

Proof.

Let $r=(r_{0},r_{1},\dots,r_{n-1})\in R^{n}$ where $r_{i}=a_{i}+ub_{i},0\leq i\leq n-1$. Then we can write $\Phi(r)=(b_{i},a_{i}+ub_{i})$, so that

$\Phi(\rho_{\Theta,\lambda}(r))=(\Theta(\lambda b_{n-1}),\Theta(b_{0}),\dots,\Theta(b_{n-2}),\Theta(\lambda a_{n-1})+\Theta(\lambda b_{n-1})$,

$\Theta(a_{0})+\Theta(b_{0}),\dots,\Theta(a_{n-2})+\Theta(b_{n-2}))$.

On the other hand

[TABLE]

and the result follows. ∎

** Theorem 1****.**

Let $C$ be a code of length $n$ over $R$ and $\Theta:=\Theta_{\theta,1}$ where $\theta$ is an automorphism of $\mathbb{F}_{q}$. Then $C$ is a skew $\lambda-$constacyclic code of length $n$ over $R$ if and only if $\Phi(C)$ is a skew quasi-twisted code of length $2n$ over $\mathbb{F}_{q}$ of index $2$.

Proof.

If $C$ is a $\Theta-\lambda-$constacyclic code, then $\rho_{\Theta,\lambda}(C)=C$. We have $\Phi\left(\rho_{\Theta,\lambda}(C)\right)=\Phi\left(C\right)$, and from Proposition 7

[TABLE]

Hence, $\Phi(C)$ is a skew quasi-twisted code of index $2$. Conversely, if $\Phi(C)$ is a skew quasi-twisted code of index $2$, then

[TABLE]

Proposition 7 gives that

[TABLE]

and since $\Phi$ is injective, it follows that $\rho_{\Theta,\lambda}(C)=C$. ∎

** Theorem 2****.**

Define $\upsilon:\Re/\langle x^{n}-1\rangle\rightarrow\Re/\langle x^{n}-\lambda\rangle$ as

[TABLE]

If $n$ is odd, then $\upsilon$ is a left $R-$module isomorphism.

Proof.

The proof is straightforward starting from the fact that if $n$ is odd

[TABLE]

if and only if $c(\lambda x)\equiv b(\lambda x)\bmod(x^{n}-\lambda)$. ∎

Immediate consequences of this theorem are given as follow

** Corollary 1****.**

$I$* is an ideal of $\Re/\langle x^{n}-1\rangle$ if and only if $\upsilon(I)$ is an ideal of $\Re/\langle x^{n}-\lambda\rangle$.*

** Corollary 2****.**

Let $\mu$ be the permutation of $R^{n}$ with $n$ odd such $\mu(c_{0},c_{1},\dots,c_{n-1})=(c_{0},(1+u)c_{1},\dots,(1+u)^{n-1}c_{n-1})$ and $C$ be a non-empty subset of $R^{n}$. Then $C$ is a skew cyclic code of length $n$ if and only if $\mu(C)$ is a skew $\lambda-$constacyclic code of length $n$ over $R$.

We introduce the following permutation of $\mathbb{F}_{q}^{2n}$ from [19] which is called the Nechaev permutation. This will be useful in studying cyclic codes over $\mathbb{F}_{q}$.

** Definition 3****.**

Let $n$ be an odd integer and $\psi$ be the permutation of $\{0,1,2,\dots,2n-1\}$ given by

[TABLE]

The Nechaev permutation is the permutation $\varrho$ of $\mathbb{F}_{q}^{2n}$ defined by

[TABLE]

** Proposition 8****.**

$\Phi\circ\mu=\varrho\circ\Phi$.

Proof.

Let $(r_{0},r_{1},\dots,r_{n-1})\in R^{n}$ where $r_{i}=a_{i}+ub_{i},0\leq i\leq n-1.$ From

[TABLE]

It follow that

[TABLE]

On the other hand, since

[TABLE]

[TABLE]

Therefore $(\Phi\mu)(r)=(\varrho\Phi)(r)$ and so $\Phi\mu=\varrho\Phi$. ∎

** Corollary 3****.**

The Gray image of a skew cyclic code of length $n$ over $R$ is equivalent to a skew quasi-twisted code of length $2n$ over $\mathbb{F}_{q}$ of index $2$.

Proof.

From Proposition 2, a code $C$ of length $n$ over $R$ is skew cyclic if and only if $\mu(C)$ is a skew $\lambda-$constacyclic code. By Theorem 1, this is true if and only if $\Phi(\mu(C))$ is a skew quasi-twisted code of index $2$ over $\mathbb{F}_{q}$, i.e. if and only if $\varrho(\Phi(C))$ is a skew quasi-twisted code of index $2$ over $\mathbb{F}_{q}$. ∎

4 Hermitian Self-Dual Skew Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$

In this section, we give necessary and sufficient conditions for the existence of Hermitian self-dual $\Theta-$cyclic and self-dual $\Theta-$negacyclic codes.

** Proposition 9****.**

Let $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ be a finite ring with $q=p^{m}$, $p$ odd prime number of the residue field $\mathbb{F}_{q}$ and $\Theta\in\text{Aut}(R)$. Assume that the order of $\Theta$ is $2$, $\lambda\in R$ such that $\lambda^{2}=1$ and $n$ is even, denoted by $n=2k$.

*If $k$ is even, then there exist Hermitian self-dual *$\Theta-$cyclic codes over $R$ if and only if $p^{m}\equiv 1\bmod 4$. 2. 2.

*If $k$ is odd, then there exist Hermitian self-dual *$\Theta-$negacyclic codes over $R$.

Proof.

Assume that the order of $\Theta$ is $2$, $\lambda^{2}=1$ and $n$ is even denoted by $n=2k$. Let $g(x)=\sum_{i=0}^{k-1}g_{i}x^{i}+x^{k}$ be a right divisor of $x^{n}-\lambda$. Now suppose that there is a Hermitian self-dual $\Theta-\lambda-$constacyclic code generated by $g(x)$. Then by Proposition 4, $g(x)$ satisfies (4), so $-\lambda=g_{0}\Theta^{-k-1}(g_{0}^{-1})$. Since $\lambda$ is fixed by $\Theta$, it follows that $\lambda=-\Theta^{k+1}(g_{0})g_{0}^{-1}$. As the order of $\Theta$ is $2$, we have the following.

If $k$ is even then $\lambda=-\Theta(g_{0})g_{0}^{-1}=\frac{-\Theta(g_{0})}{g_{0}}=\frac{\frac{-1}{g_{0}}}{g_{0}}=\frac{-1}{g_{0}^{2}}$ and from [13, Lemma 4.2] we have that $-1$ is a square in $R$ if and only if $p^{m}\equiv 1\bmod 4$. Then there exists $g_{0}\in R$ such that $g_{0}^{2}=-1$, so $\lambda=\frac{-\Theta(g_{0})}{g_{0}}=\frac{-1}{g_{0}^{2}}=1$. This implies that there exists a Hermitian self-dual $\Theta-$cyclic code over $R$ if and only if $p^{m}\equiv 1\bmod 4$ and $k$ is even. 2. 2.

If $k$ is odd then $\lambda=-\Theta^{k+1}(g_{0})g_{0}^{-1}=-g_{0}g_{0}^{-1}=-1,$ so there exists a Hermitian self-dual $\Theta-$negacyclic code over $R$.

∎

** Proposition 10****.**

Let $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ be a finite ring with $q=p^{m}$, $p$ prime number of the residue field $\mathbb{F}_{q}$ and $\Theta\in\text{Aut}(R)$. Assume that the order of $\Theta$ is $2$, $\lambda\in R$ such that $\lambda^{2}=1$ and $n$ is even denoted by $n=2k$. Let $g(x)=\sum_{i=0}^{k-1}g_{i}x^{i}+x^{k}$ be a right divisor of $x^{n}-\lambda$ with $g_{0}$ is a unit in $R$. If $p$ is even, then there exist Hermitian self-dual $\Theta-$cyclic codes over $R$ for any integer $k$.

Proof.

Suppose that there is a Hermitian self-dual $\Theta-\lambda-$constacyclic code generated by $g(x)$. Then by Proposition 4, $g(x)$ satisfies (4), so $-\lambda=g_{0}\Theta^{-k-1}(g_{0}^{-1})$. Since $\lambda$ is fixed by $\Theta$, it follow that $\lambda=-\Theta^{k+1}(g_{0})g_{0}^{-1}$. Further, if $p$ has even characteristic then $\lambda=\Theta^{k+1}(g_{0})g_{0}^{-1}$. As the order of $\Theta$ is $2$ we have the following.

If $k$ is even then $\lambda=\Theta(g_{0})g_{0}^{-1}=\frac{1}{g_{0}}g_{0}^{-1}=\frac{1}{g_{0}^{2}}$ as $g_{0}$ is a unit in $R$, so $\lambda=\frac{1}{g_{0}^{2}}=1$. This implies that there exists a Hermitian self-dual $\Theta-$cyclic code over $R$. 2. 2.

If $k$ is odd then $\lambda=\Theta^{k+1}(g_{0})g_{0}^{-1}=g_{0}g_{0}^{-1}=1$, so there exists a Hermitian self-dual $\Theta-$cyclic code over $R$.

∎

We now consider the existence of Hermitian self-dual $\Theta-$cyclic and self-dual $\Theta-$negacyclic codes over $R$ of odd characteristic generated by skew binomials and skew trinomials.

** Proposition 11****.**

Let $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ be a finite ring with $q=p^{m}$, $p$ an odd prime number and $q\equiv 1\bmod 4$. Consider $r\in\mathbb{N}$ and $\Theta\in\text{Aut}(R)$. Assume that the order of $\Theta$ is $2$ and $\theta\in\text{Aut}(\mathbb{F}_{q})$ defined by $\theta:x\mapsto x^{p^{r}}$ and $n$ is even denoted by $n=2k$.

*There are no Hermitian self-dual *$\Theta-$*cyclic or self-dual *$\Theta-$negacyclic codes over $R$ generated by a skew trinomial. 2. 2.

*There are Hermitian self-dual *$\Theta-$*cyclic and self-dual *$\Theta-$negacyclic codes over $R$ generated by a skew binomial under the following conditions.

1. (i)

*There exists a Hermitian self-dual *$\Theta-$cyclic code over $R$ if and only if $p\equiv 3\bmod 4$, $k$ and $m$ are even and $r$ is odd. 2. (ii)

*There exists a Hermitian self-dual *$\Theta-$negacyclic code over $R$ if and only if $p\equiv 1\bmod 4$ or $p\equiv 3\bmod 4$, $k$ is odd and $m$ and $r$ are even.

Proof.

Consider $g=g_{0}+g_{j}x^{j}+x^{k}\in\Re$ and let $\Theta=\Theta_{\theta,1}$ be an automorphism of $R$ as defined in Section 2. The $(\Theta,\lambda)-$constacyclic code generated by $g(x)$ is Hermitian self-dual if and only if $g$ satisfies (4), i.e.

[TABLE]

and from Proposition 9 we have

[TABLE]

Considering this skew polynomial relation, one obtains the equivalent conditions

[TABLE]

If $g_{j}\neq 0$, this system of equations has no solution, so there is no Hermitian self-dual $\Theta-$code over $R$ generated by a skew trinomial. For part 2, if $g_{j}=0$, $g(x)=g_{0}+x^{k}\in\Re$, then the skew binomial $g$ is a skew reciprocal polynomial that generates a self-dual skew $\lambda-$constacyclic code if and only if $g$ satisfies (4), i.e.

[TABLE]

This skew polynomial relation gives the equivalent condition

[TABLE]

Let $g_{0}=a_{0}+ub_{0}\in R$ with $a_{0},b_{0}\in\mathbb{F}_{q}$. If $g_{0}^{2}=-1$, we have $(a_{0}+ub_{0})(a_{0}+ub_{0})=a_{0}^{2}+2ua_{0}b_{0}=-1$ which implies that $a_{0}^{2}=-1$ and $2ua_{0}b_{0}=0$ so $b_{0}=0$ and then $\Theta(g_{0})=\theta(g_{0})$. We have the following two cases.

There exists $g_{0}\in R$ such that $g_{0}+\Theta(g_{0})=g_{0}^{2}+1=0$ if and only if $p\equiv 3\bmod 4$, $k$ and $m$ are even, and $r$ is odd. Suppose that $p\equiv 3\bmod 4$, $m\equiv 0\bmod 2$ and $r\equiv 1\bmod 2$. Then $-1$ is a square in $R$ and one can consider $g_{0}\in R$ such that $g_{0}^{2}=-1$. As $r$ is odd, $p^{r}\equiv 3\bmod 4$ so $p^{r}-1\equiv 2\bmod 4$ and $\frac{p^{r}-1}{2}\equiv 1\bmod 2$. Therefore

[TABLE]

Conversely, consider $g_{0}$ in $R$ such that $g_{0}+\Theta(g_{0})=g_{0}^{2}+1=0$ and suppose that $p\equiv 1\bmod 4$. Then $-1$ is a square in $\mathbb{F}_{p}$ so $g_{0}$ belongs to $\mathbb{F}_{p}$ and $g_{0}$ is left fixed by $\theta$. The equality $g_{0}+\Theta(g_{0})=g_{0}+\theta(g_{0})=0$ implies that $2g_{0}=0$, which is impossible as $p$ is odd. Therefore $p\equiv 3\bmod 4$ and as $-1$ is a square in $R$, $m$ must be even. Then $g_{0}+\Theta(g_{0})=g_{0}^{2}+1=0$ which gives

[TABLE]

so $\frac{p^{r}-1}{2}$ is odd and $p^{r}-1\equiv 2\bmod 4$. As $p\equiv 3\bmod 4$, $r$ must be odd. 2. 2.

There exists $g_{0}$ in $R$ such that $g_{0}-\Theta(g_{0})=g_{0}^{2}+1=0$ if and only if $p\equiv 1\bmod 4$ or $p\equiv 3\bmod 4$, $k$ is odd, and $m$ and $r$ are even. If $p\equiv 1\bmod 4$ then $g_{0}\in\mathbb{F}_{p}$ is left fixed by $\theta$. Thus, $g_{0}-\Theta(g_{0})=g_{0}-\theta(g_{0})=0$. If $p\equiv 3\bmod 4$ and $r\times m\equiv 0\bmod 2$, then $-1$ has a square root in $R$ and $p^{r}-1\equiv 0\bmod 4$, so $\frac{p^{r}-1}{2}\equiv 0\bmod 2$. Consider $g_{0}\in R$ such that $g_{0}^{2}=-1$ so then

[TABLE]

Conversely, consider $g_{0}\in R$ such that $g_{0}-\Theta(g_{0})=g_{0}^{2}+1=0$. Therefore $-1$ is a square in $R$ and either $p\equiv 1\bmod 4$ or $p\equiv 3\bmod 4$ and $m\equiv 0\bmod 2$. If $p\equiv 3\bmod 4$ and $r\equiv 1\bmod 2$ then $p^{r}-1\equiv 2\bmod 4$, so $\frac{p^{r}-1}{2}\equiv 1\bmod 2$. Thus $g_{0}^{p^{r}-1}=-1$, which contradicts the hypothesis $g_{0}-\Theta(g_{0})=g_{0}-\theta(g_{0})=0$ because $R$ has odd characteristic. Therefore, $p\equiv 1\bmod 4$ or $p\equiv 3\bmod 4$ and $r\times m\equiv 0\bmod 2$.

∎

In the following theorem, we give sufficient conditions for the existence of Hermitian self-dual skew cyclic codes of length $n$ over $R$ based on the order of $q$ modulo $n$ where $q$ is odd. Let $n$ be a positive integer, and denote by $ord_{n}(p^{m})$ the multiplicative order of $p^{m}$ modulo $n$ which is the smallest integer $j$ such that $(p^{m})^{j}\equiv 1\bmod n$. Further, let $\mbox{lcm}(a,b)$ be the least common multiple of $a$ and $b$.

** Theorem 3****.**

Let $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $q=p^{m}$ odd, $\Theta\in\text{Aut}(R)$ and $n$ be an even integer denoted by $n=2k$ with $k$ odd. If $ord_{n}(p^{m})$ is even, then there are no non-trivial $\Theta-$cyclic Hermitian self-dual codes of length $n$ over $R$.

Proof.

Let $n$ be an even integer denoted by $n=2k$ with $(2,k)=1$ and $(2,p^{m})=(k,p^{m})=1$ such that $k$ and $p^{m}$ are odd. Then $ord_{n}(p^{m})=\mbox{lcm}(ord_{2}(p^{m}),ord_{k}(p^{m}))$ is even, and thus $ord_{2}(p^{m})$ or $ord_{k}(p^{m})$ must be even. Suppose that $ord_{k}(p^{m})$ is even. Then there exist $1\leq i\leq ord_{k}(p^{m})$ such that $(p^{m})^{i}\equiv-1\bmod k$, and so $q^{i}(n-2)\equiv-(n-2)\bmod n$ with $i\leq ord_{k}(p^{m})$. Thus, there is at least one class that is reversible, $C_{n-2}$. Then from [4, Theorem 4.5], there are no non-trivial $\Theta-$cyclic Hermitian self-dual codes. ∎

5 Construction of Self-Dual $\Theta-$Cyclic Codes over $R$

Before giving the construction of self-dual $\Theta-$cyclic codes, we provide some results which will be useful later. The following lemma is easy to prove as $\mathbb{F}_{q}$ can be considered as a sub-ring of $R$. This result is similar to [16, Remark 3.2].

** Lemma 1****.**

Let $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^{2}=0$ and residue field $\mathbb{F}_{q}$ of characteristic $p$. Consider $s,t\in\mathbb{N}$ and let $k=t\times p^{s}$ be an integer such that $\gcd(t,p)=1$. Then the factorization of $x^{n}-1$ into polynomials over $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ is the same as the factorization over $\mathbb{F}_{q}$.

Recall that the center of $\mathbb{F}_{q}[x,\theta]$ is the commutative polynomial ring $Z(\mathbb{F}_{q}[x,\theta])=(\mathbb{F}_{q})^{\theta}\left[x^{\lvert\theta\rvert}\right]$ where $(\mathbb{F}_{q})^{\theta}$ is the fixed field of $\theta$ and $\lvert\theta\rvert$ is the order of $\theta$. Using Lemma 1, we prove the following proposition.

** Proposition 12****.**

Let $\Theta\in\text{Aut}(R)$ and $\Re=R[x;\Theta]$, and assume that the order of $\theta$ is $2$. Consider $s,t\in\mathbb{N}$ and let $n=2k=2\times t\times p^{s}$ be an integer such that $\gcd(t,p)=1$. Then $x^{n}-1$ has the following decomposition in $\Re$

[TABLE]

where the $f_{i}(x^{2})=f_{i}^{\natural}(x^{2})$ are pairwise coprime polynomials in $(\mathbb{F}_{q})^{\theta}\left[x^{2}\right]=(\mathbb{F}_{q})^{\theta}\left[y\right]$ which are divisors of $y^{t}-1$.

Proof.

Let $\Theta\in\text{Aut}(R)$ and $\theta\in$ Aut$\left(\mathbb{F}_{q}\right)$ of order $2$. Consider $s,t\in\mathbb{N}$ and let $k=t\times p^{s}$ be an integer. Then, from [9, Lemma 27], we have that

[TABLE]

factors in $(\mathbb{F}_{q})^{\theta}\left[y\right]$ as a product $f_{1}(y)\dots f_{l}(y)$ of pairwise coprime polynomials of minimal degree such that $f_{i}(y)=f_{i}^{\natural}(y)$. Furthermore, we have that

[TABLE]

and according to Lemma 1, if $\gcd(t,p)=1$ then

[TABLE]

where the $f_{i}(y)=f_{i}^{\natural}(y)$ are pairwise coprime polynomials in $(\mathbb{F}_{q})^{\theta}\left[x^{2}\right]=(\mathbb{F}_{q})^{\theta}\left[y\right]$ which are divisors of $y^{t}-1$. ∎

We require the following proposition.

** Proposition 13****.**

Let $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ be a finite ring with $q=p^{m}$ and $p$ a prime number of the residue field $\mathbb{F}_{q}$. Assume that the order of $\Theta$ is $2$ and $n$ is an even integer denoted by $n=2k$ such that $k$ is even. Let $g(x)=\sum_{i=0}^{k-1}g_{i}x^{i}+x^{k}$ be a right divisor of $x^{n}-1$ with $g_{0}$ a unit in $R$. Then the $\Theta-$cyclic code generated by $g(x)$ is Hermitian self-dual if and only if

[TABLE]

Proof.

Let $g(x)=g_{0}+g_{1}x+\ldots+g_{k-1}x^{k-1}+x^{k}$ be a right divisor of $x^{n}-1$. Then by Definition 1, the reciprocal of $g(x)$ is the polynomial $g^{\ast}(x)=1+\Theta(g_{k-1})x+\dots+\Theta^{k-1}(g_{1})x^{k-1}+\Theta^{k}(g_{0})x^{k}$, and the left monic skew reciprocal polynomial of $g(x)$ is

[TABLE]

As the order of $\Theta$ is $2$ and from Proposition 9 Hermitian self-dual skew cyclic codes exist if $k$ is even, we have that

[TABLE]

Similarly, from Proposition 4 there exists a Hermitian self-dual skew cyclic code generated by a skew polynomial $g(x)$ if and only if

[TABLE]

Then from (9), if $g_{0}$ is a unit in $R$ (4) is equivalent to $gg^{\natural}=x^{n}-1$, so from Lemma 2 $gg^{\natural}=g^{\natural}g=x^{n}-1$. Thus, (4) equivalent to

[TABLE]

∎

** Remark 1****.**

From [15, Section 2], the greatest common right divisor of $f$ and $g,$ denoted by $\mbox{gcrd}(f,g)$, is the unique monic polynomial $w\in\mathbb{F}_{q}[x,\theta]$ of highest degree such that there exist $u,v\in\mathbb{F}_{q}[x,\theta]$ with $f=uw$ and $g=vw$. The least common right multiple of $f$ and $g$, denoted by $h=\mbox{lcrm}(f,g)$, is the unique monic polynomial $w\in\mathbb{F}_{q}[x,\theta]$ of lowest degree such that there exist $u,v\in\mathbb{F}_{q}[x,\theta]$ with $h=uf$ and $h=vg$.

Before giving the construction algorithm for self-dual $\Theta-$cyclic codes we provide the following lemma.

** Lemma 2****.**

Let $f,g\in\Re$ such that $r=deg(f)$ and $s=deg(g)$ with $r\geq s$. If the coefficients of $f,g$ are invertible, then the computation of $\mbox{gcrd}(f,g)$ and $\mbox{lcrm}(f,g)$ in $\Re$ is the same as in $\mathbb{F}_{q}[x,\theta]$.

Proof.

Consider $f(x),g(x)\in\Re$ such that $f(x)=a_{0}+a_{1}x+\dots+a_{r}x^{r}$ and $g(x)=b_{0}+b_{1}x+\dots+b_{s}x^{s}$. From [17] we have that for $s\leq r$, $f(x)-a_{r}\Theta^{r-s}(b_{s}^{-1})x^{r-s}g(x)$ has degree less than that of $f(x)$. Then iterating the above procedure by subtracting further left multiples of $g(x)$ from the result until the degree is less than the degree of $g(x)$, we obtain skew polynomials $q(x)$ and $r(x)$ such that

[TABLE]

Note that $q(x)$ and $r(x)$ are unique. Further, the division in $\Re$ is the same as in the ring $\mathbb{F}_{q}[x,\theta]$ and from Remark  1, for $\mbox{gcrd}(f,g)$ and $\mbox{lcrm}(f,g)$ to be unique with $f,g\in\Re$, the coefficients of $f$ and $g$ must be invertible. Therefore, if the coefficients of $f,g$ are invertible then $\mbox{gcrd}(f,g)$ and $\mbox{lcrm}(f,g)$ can be obtained using the same procedure as for $\mathbb{F}_{q}[x,\theta]$. ∎

In the following algorithm, the computation of $g\in\Re$ such that $g_{0}$ is a unit in $R$ with the property $g^{\natural}g=x^{2k}-1$ (Proposition 13), is replaced with the computation of polynomials $g_{i}$ such that $g_{i}^{\natural}g_{i}=f_{i}^{p^{s}}(x^{2})$.

** Remark 2****.**

According to Lemma 2, $\mbox{lclm}(g^{\natural}_{1},\dots,g^{\natural}_{l})$ in step $6$ of Algorithm 1 can be computed using the extended Euclidean algorithm in [15, Section 2]. Thus, this algorithm can be executed for $k$ odd for generator polynomials s$g=lclm(g^{\natural}_{1},\dots,g^{\natural}_{l})$ with coefficient which are invertible.

5.1 Algorithm Complexity

In this subsection, the complexity of Algorithm 1 is analyzed. From Proposition 12 we have that if $\gcd(t,p)=1$ with $n=2k=2\times t\times p^{s}$, then $x^{n}-1$ has decomposition in $\Re$ given by

[TABLE]

where the $f_{i}=f_{i}^{\natural}$ are pairwise coprime polynomials in $(\mathbb{F}_{q})^{\theta}\subset\mathbb{F}_{q}[x,\theta]$. Then the complexity of the skew factorization in $\Re$ is the complexity of the skew factorization in $\mathbb{F}_{q}[x,\theta]$ which is given in the following theorem.

** Theorem 4****.**

*([12, Theorem 2.4.2])

The skew factorization algorithm factors a skew polynomial of degree $n$ in $\mathbb{F}_{q}[x,\theta]$ with complexity*

[TABLE]

operations for $\epsilon>0$. Here, $F(n,\mathbb{F}_{q})$ denotes the complexity of the factorization of a (commutative) polynomial of degree $n$ over the finite field $\mathbb{F}_{q}$.

From Theorem 4 we have the following result.

** Theorem 5****.**

Let $R=\mathbb{F}_{q}+u\mathbb{F}_{q}$ be a finite ring with $q=p^{m}$ and $p$ odd prime number of the residue field $\mathbb{F}_{q}$. Assume that the order of $\Theta$ is $2$ and $n$ is an even integer denoted by $n=2k$. Let $g(x)=\sum_{i=0}^{k-1}g_{i}x^{i}+x^{k}$ be a right divisor of $x^{n}-1$. Then the complexity of Algorithm 1 in

[TABLE]

operations.

Proof.

We now consider the complexity of the steps of the algorithm. In the following $n=2k$ and $k=t\times p^{s}$.

Computing the integers $s$ and $t$ takes $\tilde{\mathcal{O}}(1)$ operation in $R$. 2. 2.

As $\gcd(t,p)=1$, according to Proposition 12 and Theorem 4, factoring the skew polynomial $x^{n}-1$ in $\Re$ takes

[TABLE]

operations. 3. 3.

Before determining the complexity of obtaining $g_{i}^{\natural}g_{i}$, we outline the process. Recall that for $g_{i}=\sum_{i=0}^{k}g_{i}x^{i}\in\Re$ we have $g_{i}^{\natural}=\frac{1}{\Theta^{k}(g_{0})}\sum_{i=0}^{k}\Theta^{i}(g_{k-i})x^{i}$ and their product is

[TABLE]

Let $g_{k-i}=a_{k-i}+ub_{k-i}\in\Re$ so then $\Theta^{i}(g_{k-i})=\theta(a_{k-i})+u\theta(b_{k-i})$ where $a,b\in\mathbb{F}_{q}$ and $\theta\in\text{Aut}(\mathbb{F}_{q})$. From [12, (3.1.1)], $\Theta^{i}(g_{k-i})$ has complexity $\tilde{\mathcal{O}}(k2^{2})$ because $\theta(a_{k-i})$ and $\theta(b_{k-i})$ can be computed in $\tilde{\mathcal{O}}(k2^{2})$ operations so $\Theta^{i}(g_{k-i})\Theta^{i}(g_{i})$ can be computed in $\tilde{\mathcal{O}}(k^{2}2^{4})$ operations where $2$ is the order of $\Theta$ and $k$ is the degree of the skew polynomial $g$. Once we have the coefficients, it remains to compute the product which is done in $O(k^{2})$ operations in $R$ so the complexity of the skew product $g_{i}^{\natural}g_{i}$ is $\tilde{\mathcal{O}}(k^{2}2^{4}+k^{2}2)$ operations. 4. 4.

The coefficients of $g_{i}$ are invertible so determining $\mbox{lclm}(g^{\natural}_{1},\dots g^{\natural}_{r})$ over $R[x,\Theta]$ is the same as over $\mathbb{F}_{q}[x,\theta]$. Thus from [12], $\mbox{lclm}(g^{\natural}_{1},\dots,g^{\natural}_{i})$ can be computed in $\tilde{\mathcal{O}}(k2^{2})$ operations where $2$ is the order of $\Theta$ and $k$ is the degree of the skew polynomial $g$.

Combining the complexities of the steps gives the desired result. ∎

5.2 Computational Results

To obtain examples using Algorithm 1, $x^{n}-1=f_{1}^{p^{s}}(x^{2})\dots f_{l}^{p^{s}}(x^{2})$ must be factored where the $f_{i}$ are pairwise coprime polynomials in $\mathbb{F}_{q}[x^{2}]=Z(\mathbb{F}_{q}[x,\theta])$ satisfying $f_{i}(x^{2})=f_{i}^{\natural}(x^{2})$, $i\in\{1,2,\dots,l\}$. Furthermore, Gröbner basis computation is used to obtain the polynomials $f_{i}$ to compute the sets $G_{i}=\{g\in\Re;g^{\natural}g=f_{i}\}$. Then Lemma 2 can be used to compute the codes generated by $g^{\natural}=\mbox{lclm}(g^{\natural}_{1},g^{\natural}_{2},\dots,g^{\natural}_{l}),$ for all $g_{i}\in G_{i}$. The skew polynomials $g_{i}\in G_{i}$ are used to obtain the skew polynomial $g^{\natural}$ which generates a self-dual $\Theta-$cyclic code over $R[x,\Theta]$ of length $n$.

** Remark 3****.**

A right factor of degree $n-k$ of $x^{n}-1$ generates a linear code with parameters $(n,k)$. If $\Theta$ is not the identity, then the factorization of polynomials in the skew polynomial ring $R[x,\Theta]$ is not unique, and in general $x^{n}-1$ has many more factors in $R[x,\Theta]$ compared to the factorization in the usual polynomial ring. The minimum distance of a code, denoted by $d_{min}$, can be calculated using MAGMA [11]. There can be many codes with the same parameters $(n,k,d_{min})$.

** Example 1****.**

Consider $\mathbb{F}_{4}+u\mathbb{F}_{4}$ where $\mathbb{F}_{4}=\mathbb{F}_{2}(a)$, $\Theta$ the Frobenius automorphism $(\alpha\mapsto\alpha^{2})$ and $\Re=(\mathbb{F}_{4}+u\mathbb{F}_{4})[x,\Theta]$, so then

[TABLE]

where

[TABLE]

The $f_{i}(x)$ are pairwise coprime polynomials in $\mathbb{F}_{2}[x^{2}]=Z(\mathbb{F}_{2}[x,\theta])$ satisfying $f_{i}(x^{2})=f_{i}^{\natural}(x^{2})$, $i\in\{1,\dots,2\}$. Gröbner basis computation is used to obtain the polynomials $f_{i}$ and then the sets $G_{1}=\{g\in\Re,g^{\natural}g=x^{2}+1\}$ and $G_{2}=\{g\in\Re,g^{\natural}g=x^{4}+x^{2}+1\}$ are obtained which gives

[TABLE]

The codes generated by $g=\mbox{lclm}(g_{1}^{\natural},g_{2}^{\natural})$ for $g_{i}\in G_{i}$ can be obtained using Lemma  2. Further, from Remark 2, if $k$ is an odd integer, then $g=\mbox{lclm}(g_{1}^{\natural},g_{2}^{\natural})$ can be chosen as the generator polynomial which has coefficients that are units in $R$. Then the skew polynomials

[TABLE]

give the skew polynomial

[TABLE]

which generates a self-dual code over $\mathbb{F}_{4}+u\mathbb{F}_{4}$ of length $6$ and $d_{min}=4$. The generator matrix of this code can be obtained from the generator polynomial using (1) as

[TABLE]

Then using the Gray map (5) gives the matrix

[TABLE]

which generates a $[12,6,2]$ linear code over $GF(4)$.