Skew Generalized Polycyclic Codes with Derivations

TL;DR

This paper introduces a new class of skew generalized polycyclic codes over finite rings and fields, analyzing their structure, minimum distance bounds, and conditions for MDS properties, with practical examples.

Contribution

It constructs skew generalized polycyclic codes using iterated skew polynomial rings and derives their generator and parity check matrices, also improving the BCH bound.

Findings

Derived the structure of generator and parity check matrices.

Improved the BCH lower bound for minimum distance.

Provided conditions and examples for MDS codes.

Abstract

In this paper, we first consider the iterated skew polynomial ring $\mathscr{R}[z_1;\tau_1,\delta_{\tau_1}]$\\$[z_2;\tau_2,\delta_{\tau_2}]$, where $\mathscr{R}$ is a finite ring with unity. Then we use this structure for the construction of skew generalized polycyclic codes over the ring $\mathscr{R}$ and finite field $\mathbb{F}_q$, where $q=p^m$ for some positive integer $m$. Further, we derive the structure of the generator and parity check matrices for skew generalized polycyclic codes. Furthermore, we improve the Bose-Chaudhuri-Hocquenghem (BCH) lower bound for a minimum distance of skew generalized polycyclic codes with non-zero derivations over a finite field. Moreover, we find a sufficient condition for a code to be a maximum-distance-separable (MDS) code. In addition, we provide examples of MDS codes to show the importance of our results. A comparative summary of our work with other linear codes is also discussed.

Tables

q	n=	k	d	g such that	a such that
	deg f			g divides f	f= $x^n-a$
8	2	1	2	$x+w^6,x+w^2,x+w^5,x+w,x+w^4,x+1$,	1,w,$w^2$,$w^6$
				$x+w^3$	$w^3,w^4,w^5$
8	3	2	2	$x+w^6,x+w^2,x+w^5,x+w,x+w^4,x+1,$	1,w, $w^2$,$w^6$
				$x+w^3$	$w^3,w^4,w^5$
		1	3	$x^2+x+w,x^2+w^{3}x+1,x^2+wx+w^3$	$\nexists$
				$x^2+w^{4}x+w^2,x^2+w^{2}x+w^5,x^2+w^{2}x+w,$
				$x^2+w^{5}x+w^4,x^2+w^{5}x+w,x^2+x+w^4$
				$x^2+x+w^2,x^2+w^{3}x+w^3,x^2+w^{6}x+w^6$
				$x^2+w^{3}x+w,x^2+w^{6}x+w^2,x^2+wx+w^6$
				$x^2+w^6+1,x^2+wx+w^4,x^2+w^{4}x+w^5$
				$x^2+w^{4}x+w^3,x^2+w^{2}x+w^6,x^2+w^{5}x+w^5$
8	4	3	2	$x+w^6,x+w^2,x+w^5,x+w,x+w^4,x+1,$	$1,w,w^2,$
				$x+w^3$	$w^3,w^4,w^5,$
					$w^6$
	4			for codes of
				the type [4,1,4] and [4,2,3] $\nexists f$
8	5	4	2	$x+w^6,x+w^2,x+w^5,x+w,x+w^4,x+1,$	$1,w,w^2,$
				$x+w^3$	$w^3,w^4,w^5,$
					$w^6$
	5			for codes of
				the type [5,3,3], [5,1,5] and [5,2,4] $\nexists f$
8	6	5	2	$x+w^6,x+w^2,x+w^5,x+w,x+w^4,x+1,$	$1,w,w^2,w^6$
				$x+w^3$	$w^3,w^4,w^5,$
	6			for codes of
				the type [6,1,6], [6,4,3], [6,2,5] and [6,3,4] $\nexists f$
8	7			for codes of
				the type [7,2,6], [7,1,7], [7,5,3],
				[7,6,2], [7,3,5] and [7,4,4] always $\exists f$	1
11	2	1	2	$x+10$	$1$
11	3	2	2	$x+10$	$1$
		1	3	$x^2+9x+1$	$\nexists$
11	4	3	2	$x+10$	$1$
		2	3	$x^2+9x+1$	$\nexists$
		1	4	$x^3+8x^2+3x+10$	$\nexists$
11	5	4	2	$x+10$	$1$
		1	5	$x^4+7x^3+6x^2+7x+1$	$\nexists$
		2	4	$x^3+8x^2+3x+10$	$\nexists$
		3	3	$x^2+9x+1$	$\nexists$

q	n=	k	d	g such that	a such that
	deg f			g divides f	f= $x^n-a$
11	6	5	2	$x+10$	$1$
		1	6	$x^5+6x^4+10x^3+x^2+5x+10$	$\nexists$
		2	5	$x^4+7x^3+6x^2+7x+1$	$\nexists$
		3	4	$x^3+8x^2+3x+10$	$\nexists$
		4	3	$x^2+9x+1$	$\nexists$
11	7	6	2	$x+10$	$1$
		2	6	$x^5+6x^4+10x^3+x^2+5x+10$	$\nexists$
		3	5	$x^4+7x^3+6x^2+7x+1$	$\nexists$
		4	4	$x^3+8x^2+3x+10$	$\nexists$
		1	7	$x^6+5x^5+4x^4+2x^3+4x^2+5x+1$	$\nexists$
		5	3	$x^2+9x+1$	$\nexists$
11	8	7	2	$x+10$	$1$
		3	6	$x^5+6x^4+10x^3+x^2+5x+10$	$\nexists$
		4	5	$x^4+7x^3+6x^2+7x+1$	$\nexists$
		5	4	$x^3+8x^2+3x+10$	$\nexists$
11	8	2	7	$x^6+5x^5+4x^4+2x^3+4x^2+5x+1$	$\nexists$
		6	3	$x^2+9x+1$	$\nexists$
		1	8	$x^7+4x^6+10x^5+9x^4+2x^3+x^2+7x+10$	$\nexists$
11	9	8	2	$x+10$	$1$
		4	6	$x^5+6x^4+10x^3+x^2+5x+10$	$\nexists$
		5	5	$x^4+7x^3+6x^2+7x+1$	$\nexists$
		6	4	$x^3+8x^2+3x+10$	$\nexists$
		3	7	$x^6+5x^5+4x^4+2x^3+4x^2+5x+1$	$\nexists$
		7	3	$x^2+9x+1$	$\nexists$
		2	8	$x^7+4x^6+10x^5+9x^4+2x^3+x^2+7x+10$	$\nexists$
11	9	1	9	$x^8+3x^7+6x^6+10x^5+4x^4+10x^3$	$\nexists$
				$6x^2+3x+1$	$\nexists$
11	10	9	2	$x+10$	$1$
		5	6	$x^5+6x^4+10x^3+x^2+5x+10$	$\nexists$
		6	5	$x^4+7x^3+6x^2+7x+1$	$\nexists$
		7	4	$x^3+8x^2+3x+10$	$\nexists$
		4	7	$x^6+5x^5+4x^4+2x^3+4x^2+5x+1$	$\nexists$
		8	3	$x^2+9x+1$	$\nexists$
		3	8	$x^7+4x^6+10x^5+9x^4+2x^3+x^2+7x+10$	$\nexists$
		2	9	$x^8+3x^7+6x^6+10x^5+4x^4+10x^3+$	$\nexists$
				$6x^2+3x+1$	$\nexists$
		1	10	$x^9+2x^8+3x^7+4x^6+5x^5+6x^4+$	$\nexists$
				$7x^3+8x^2+9x+10$

Equations

$$T(\bar{v})={\sigma }^{\prime }(\bar{v}).M+{\delta }^{\prime }(\bar{v}),$$

$$f=\sum _{a_1=0}^s\sum _{a_2=0}^l{r}_{a_1,a_2}{x}_1^{a_1}{x}_2^{a_2}\in B,$$

$${\Gamma }_f(\bar{c})=[c]\in B/Bf$$

$$c=\sum _{a_1=0}^{s-1}\sum _{a_2=0}^{l-1}{c}_{a_1,a_2}{x}_1^{a_1}{x}_2^{a_2},$$

$$\left(\begin{array}{cccc}{c}_{0,0}& c_{0,1}& \dots & c_{0,l-1}\\ c_{1,0}& c_{1,1}& \dots & c_{1,l-1}\\ ⋮& ⋮& \ddots & ⋮\\ c_{s-1,0}& c_{s-1,1}& \dots & c_{s-1,l-1}\end{array}\right).$$

$$M_1=\left(\begin{array}{cccc}0& 1& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 1\\ -r_{0,0}& -r_{0,1}& \dots & -r_{0,l-1}\end{array}\right)$$

$$M_2=\left(\begin{array}{cccc}0& 1& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 1\\ -r_{0,0}& -r_{1,0}& \dots & -r_{s-1,0}\end{array}\right).$$

$$d_{\mathcal{C}}=min\text{leavevmode}\{d(\bar{x},\bar{y})\text{leavevmode}|\text{leavevmode}\bar{x},\text{leavevmode}\bar{y}\in \mathcal{C},\bar{x}\ne \bar{y}\}.$$

$${\mathcal{B}}^{\prime }=$$

$$x_{1}g,$$

$$\mathbb{G}=\left(\begin{array}{c}\bar{g}\\ T_f(\bar{g})\\ ⋮\\ T_f^{k-1}(\bar{g})\end{array}\right)$$

$$\bar{g}=\left(\begin{array}{ccccccc}{g}_{0,0}& g_{0,1}& \dots & g_{0,k_2-1}& 0& \dots & 0\\ g_{1,0}& g_{1,1}& \dots & g_{1,k_2-1}& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ g_{k_1-1,0}& g_{k_1-1,1}& \dots & g_{k_1-1,k_2-1}& 0& \dots & 0\\ 0& 0& \dots & 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0& 0& \dots & 0\end{array}\right).$$

$$\mathbb{H}=\left(\begin{array}{c}\bar{h}\\ T_f(\bar{h})\\ ⋮\\ T_f^{k^{\prime }-1}(\bar{h})\end{array}\right)$$

$$\bar{a}=\left(\begin{array}{cccc}{a}_{0,0}& a_{0,1}& \dots & a_{0,l-1}\\ a_{1,0}& a_{1,1}& \dots & a_{1,l-1}\\ ⋮& ⋮& \ddots & ⋮\\ a_{s-1,0}& a_{s-1,1}& \dots & a_{s-1,l-1}\end{array}\right)$$

$$=\{\begin{array}{ll}{x}_1^i{x}_2^j{h}_1^{\prime }& \text{for 0≤i≤k1−1,0≤j≤l−1}\\ x_1^i{x}_2^j{h}_2^{\prime }& \text{for k1≤i≤s−1,0≤j≤k2−1}\end{array}$$

$$N_0^{\theta ,\delta }(\alpha )=1$$

$$N_{i+1}^{\theta ,\delta }(\alpha )=\theta (N_i^{\theta ,\delta }(\alpha ))\alpha +\delta (N_i^{\theta ,\delta }(\alpha )).$$

$$c=\sum _{i=0}^{n-1}{c}_i{x}^i,$$

$$x^n-f_{n-1}{x}^{n-1}-\cdots -f_{1}x-f_0=0,$$

$$x^n-f_{n-1}{x}^{n-1}-\cdots -f_{1}x=f_0.$$

$$(f_0{)}^{-1}(x^{n-1}-f_{n-1}{x}^{n-2}-\cdots -f_1)x=1,$$

$$(f_0^{-1}{x}^{n-1}-f_0^{-1}{f}_{n-1}{x}^{n-2}-\cdots -f_0^{-1}{f}_{2}x-f_0^{-1}{f}_1)x=1.$$

$$x^n-f_{n-1}{x}^{n-1}-\cdots -f_{1}x=f_0,$$

$$f_0^{-1}{x}^n-f_0^{-1}{f}_{n-1}{x}^{n-1}-\cdots -f_0^{-1}{f}_2{x}^2-f_0^{-1}{f}_{1}x=1.$$

$$r.c=1+\sum _{i=1}^{w-1}{c}_i{x}^{a_i}+\sum _{i=1}^{w-1}{d}_i{x}^{k_i}\in {\Omega }_f(\mathcal{C}),$$

$$w{t}_H(r.c)=w{t}_H(1+\sum _{i=1}^{w-1}{c}_i{x}^{a_i}+\sum _{i=1}^{w-1}{d}_i{x}^{k_i})=2w-1.$$

$$c$$

$$=b_{k_0}{x}^{k_0}(1+{\theta }^{-k_0}(b_{k_0}^{-1}{b}_{k_1})x^{k_1-k_0}-\delta ({\theta }^{-k_0}(b_{k_0}^{-1}{b}_{k_1}))x^{k_1}+\cdots$$

$$+{\theta }^{-k_0}(b_{k_0}^{-1}{b}_{k_{w-1}})x^{k_{w-1}-k_0}-\delta ({\theta }^{-k_0}(b_{k_0}^{-1}{b}_{k_{w-1}}))x^{k_{w-1}})$$

$$=b_{k_0}{x}^{k_0}(1+{\theta }^{-k_0}(b_{k_0}^{-1}{b}_{k_1})x^{k_1-k_0}+\cdots +{\theta }^{-k_0}(b_{k_0}^{-1}{b}_{k_{w-1}})x^{k_{w-1}-k_0}$$

$$-\delta ({\theta }^{-k_0}(b_{k_0}^{-1}{b}_{k_1}))x^{k_1}-\cdots -\delta ({\theta }^{-k_0}(b_{k_0}^{-1}{b}_{k_{w-1}}))x^{k_{w-1}})$$

$$x^{-k_0}{b}_{k_0}^{-1}c$$

Full text

Keywords: Finite fields; Skew polynomial ring; Cyclic codes; Skew cyclic codes; Pseudo-linear transformation.

Subclass: Primary: 12Y05; 16Z05. Secondary: 94B05; 94B35.

1 Introduction

Coding theory is mainly the study of the methods for efficient and as long as possible to preserve the accuracy of information during transmission from one place to another place. In coding theory, linear codes have been studying for the last six decades. Initially, linear codes were studied over the binary field. In $1970$, the study of codes over finite rings was initiated by Blake [1]. The cyclic codes are the most important class of linear codes from an implementation point of view and also playing a prominent role in the development of algebraic coding theory. In $1972$, Hartmann and Tzeng [7] studied the BCH lower bounds for the minimum distance of a cyclic code.

In $2007$, Boucher et al. [3] introduced skew cyclic codes which is a generalization of the cyclic codes over a non-commutative ring, namely skew polynomial ring $\mathbb{F}[x;\theta]$, where $\mathbb{F}$ is a field, and $\theta$ is an automorphism on $\mathbb{F}$. Besides the commutative case, skew polynomial rings also have many applications in the construction of algebraic codes with better parameters. In $2009$, Boucher and Ulmer [6] obtained some skew codes with Hamming distance more significant than the known linear codes with the same parameters. Again, in $2014$, they [5] revisited the linear codes by using the concept of the skew polynomial ring with derivation. Later on, in $2017$, Cuiti$\tilde{n}$o and Tironi [13] studied the structural properties of skew generalized cyclic (SGC) codes over finite fields and also obtained some BCH type lower bounds for their minimum distance.

The two-dimensional cyclic code was introduced by Ikai et al. [8] and Imai [10] in $1975$. It is a generalization of the cyclic code. Inspired by these works, we study structural properties of the two-dimensional skew generalized cyclic (SGC) codes and obtain the BCH lower bound for the minimum distance of a skew generalized cyclic code with non-zero derivation. The two-dimensional theory is beneficial for the analysis and generation of two-dimensional periodic arrays. It gives a construction method for the two-dimensional feedback shift register with a minimum number of storage devices that generates a given two-dimensional periodic array.

Presentation of the manuscript is as follows: Section $2$ contains some preliminaries while Section $3$ discusses SGC codes and their properties. In Section $4$, we give some BCH lower bounds which generalize the known result of [13] for non-zero derivation. To show the importance of our results, in Section $5$, we provide some examples of MDS codes in $\mathbb{F}_{q}^{n}$ and also MDS codes for SGC codes over $\mathbb{F}_{q}[x;\theta]$, respectively. Section $6$ concludes the work.

2 Preliminaries

Let $R$ be a ring with unity and $\sigma$ an endomorphism of $R$. A map $\delta:R\rightarrow R$ is said to be a $\sigma$-derivation if $\delta(a+b)=\delta(a)+\delta(b)$ and $\delta(ab)=\sigma(a)\delta(b)+\delta(a)b$ for all $a,b\in R$. Let $\mathbb{F}_{q}$ be a Galois field with $q$ elements where $q=p^{m}$ for some prime $p$ and $m\geq 1$. Consider the Frobenius automorphism $\theta:\mathbb{F}_{q}\rightarrow\mathbb{F}_{q}$ defined by $\theta(a)=a^{p}$, of order $m$ which fixes the subfield $\mathbb{F}_{p}$.

Definition 2.1**.**

[9, 11, 12]** A pseudo-linear map $T:R^{n}\rightarrow R^{n}$ is an additive map defined by

[TABLE]

where $\bar{v}=(v_{1},v_{2},...,v_{n})\in R^{n}$, $\sigma^{\prime}(\bar{v})=(\sigma(v_{1}),\sigma(v_{2}),...,\sigma(v_{n}))\in R^{n}$, $M$ is an $n\times n$ matrix over $R$ and $\delta^{\prime}(\bar{v})=(\delta(v_{1}),\delta(v_{2}),...,\delta(v_{n}))\in R^{n}$. If $\delta=0$, then $T$ is called a semi-linear map.

Consider $R[x;\sigma,\delta]=\{a_{s}x^{s}+\cdots+a_{1}x+a_{0}\leavevmode\nobreak\ |\leavevmode\nobreak\ a_{i}\in R\leavevmode\nobreak\ and\leavevmode\nobreak\ s\in\mathbb{N}\}$. Then $R[x;\sigma,\delta]$ is a noncommutative ring under usual addition of polynomials and multiplication is defined with respect to $xr=\sigma(r)x+\delta(r)$ for any $r\in R$. This ring is also known as skew polynomial ring.

We consider the iterated skew polynomial ring over the ring $R$. Let $S=R[x_{1};\sigma_{1},\delta_{1}]$ be a skew polynomial ring over the ring $R$, where $\sigma_{1}$ is an automorphism and $\delta_{1}$ is a $\sigma_{1}$-derivation of $R$. If $\sigma_{2}$ is an endomorphism and $\delta_{2}$ is a $\sigma_{2}$-derivation of $S$, then the skew polynomial ring $B=S[x_{2};\sigma_{2},\delta_{2}]$ is called an iterated skew polynomial ring over $R$. It is clear that $B=R[x_{1};\sigma_{1},\delta_{1}][x_{2};\sigma_{2},\delta_{2}]$.

For brevity of notation, let $B_{i}=R[x_{1};\sigma_{1},\delta_{1}][x_{2};\sigma_{2},\delta_{2}]\cdots[x_{i};\sigma_{i},\delta_{i}]$ and finite sets $H=\{\sigma_{1},\sigma_{2},...,\sigma_{n}\}$ and $D=\{\delta_{1},\delta_{2},...,\delta_{n}\}$ where $\sigma_{i}$ is an endomorphism of $B_{i-1}$ and $\delta_{i}$ is the $\sigma_{i}$-derivation of $B_{i-1}$, for $i=1,2,...,n$. Then, by induction, $B_{i},2\leq i\leq n$ are the iterated skew polynomial rings over $R$. If $H$ has only identity automorphism, then $B_{i}$ is iterated skew polynomial ring of derivation type and if $D$ has only zero derivation, then $B_{i}$ is the iterated skew polynomial ring of automorphism type over $R$.

Let $\preceq$ be the lexicographical order on $\mathbb{Z}\times\mathbb{Z}$. Now, for any $(a,b),(a^{\prime},b^{\prime})\in\mathbb{Z}\times\mathbb{Z}$, $(a,b)\preceq(a^{\prime},b^{\prime})$ if and only if $a<a^{\prime}$ or $a=a^{\prime}$, $b\leq b^{\prime}$. Further, if $(a,b)\leq(a^{\prime},b^{\prime})$ and $(a,b)\neq(a^{\prime},b^{\prime})$, then we write $(a,b)\prec(a^{\prime},b^{\prime})$ and $\preceq$ is a total order on $\mathbb{Z}\times\mathbb{Z}$. For any polynomial $f\in R[x_{1};\sigma_{1},\delta_{1}][x_{2};\sigma_{2},\delta_{2}]$, we define

$\mathcal{V}_{f}=\{(a_{1},a_{2})\leavevmode\nobreak\ |\leavevmode\nobreak\ f$ contains a term $mx_{1}^{a_{1}}x_{2}^{a_{2}},\leavevmode\nobreak\ m\in R,\leavevmode\nobreak\ m\neq 0\}$ .

Recall that the lex-degree of $f$, denoted by $lexdeg(f)$, is the greatest element of $\mathcal{V}_{f}$ with respect to the total order $\preceq$ on $\mathcal{V}_{f}$. The lexicographical order $\preceq$ on $R[x_{1};\sigma_{1},\delta_{1}][x_{2};\sigma_{2},\delta_{2}]$ is defined as $f\preceq g$ if and only if $lexdeg(f)\preceq lexdeg(g),$ for any $f,g\in R[x_{1};\sigma_{1},\delta_{1}][x_{2};\sigma_{2},\delta_{2}]$. The term of the polynomial $f$ corresponding to its lex-degree is called the lex-leading term and its coefficient is known as lex-leading coefficient.

Now, we present some basic results for iterated skew polynomial ring.

Lemma 2.1**.**

[15, Lemma 3.6]** Let $R$ be a ring and $\delta,$ a derivation on $R$. Let $S=R[x;\delta]$ be the skew polynomial ring of derivation type over $R$ and $\delta_{1}$ be another derivation of $R$. Then $\delta_{1}$ can be extended to a derivation of $S$ by $\delta_{1}(x)=0$ if and only if $\delta_{1}$ commutes with $\delta$.

Theorem 2.1**.**

[15, Theorem 3.7]** Let $R$ be a ring and $D=\{\delta_{1},\delta_{2},...,\delta_{n}\},$ a finite set of derivations of $R$. Let $B_{i}$ be the set of all polynomials in indeterminates $x_{1},x_{2},\cdots,x_{i}$ with coefficients in $R$, for $i=0,1,2,\cdots,n$, where $B_{0}=R$. In $B_{n}$, addition is usual and multiplication is defined with respect to $x_{i}r=rx_{i}+\delta_{i}(r)$ for all $r$ in $R$, $x_{i}x_{j}=x_{j}x_{i}$ for $i,j=1,2,...,n$. Then $B_{i}$ is a skew polynomial ring (of derivation type) over $B_{i-1}$, for all $i=1,2,...,n$ if and only if $\delta_{i}\delta_{j}=\delta_{j}\delta_{i}$, for all $i,j=1,2,...,n$.

Definition 2.2**.**

[8]** A binary two-dimensional code of area $sl$ is the set of $s\times l$ arrays over $\mathbb{F}_{2}$, called codewords or code-arrays. A two-dimensional code $\mathcal{C}$ is said to be linear if and only if $\mathcal{C}$ forms a subspace of the $sl$-dimensional space of the $s\times l$ arrays over $\mathbb{F}_{2}$. A two-dimensional cyclic code is defined as a two dimensional linear code such that for each code-array $C$, all the arrays obtained by permuting the columns or the rows of $C$ cyclically are also code-arrays.

We fix a monic polynomial

[TABLE]

where $r_{a_{1},a_{2}}\in R$.

Define a map $\Gamma_{f}:R^{n}\rightarrow B/Bf$ by

[TABLE]

for all $\bar{c}\in R^{n}$, where class $[c]\in B/Bf$ and

[TABLE]

$c_{a_{1},a_{2}}\in R^{n}$, $n=sl$ and $\bar{c}$ is an $s\times l$-arrays, i.e., $\bar{c}\in R^{n}$ is written as

[TABLE]

Clearly, $\Gamma_{f}$ is an $R$-linear isomorphism.

Definition 2.3**.**

Let $\mathcal{C}\subseteq R^{n}$ be a non-empty set. Then

$\mathcal{C}$* is a $R$-linear code if $\mathcal{C}$ is a submodule of the left $R$-module $R^{n}$.* 2. 2.

suppose $\mathcal{C}$ is a linear code over $R$ of length $n=sl$ in which each codeword is viewed as an $s\times l$ array, i.e., $\bar{c}\in\mathcal{C}$ is written as

[TABLE]

If $\mathcal{C}$ is closed under row $\sigma_{1}$-shift and column $\sigma_{2}$-shift of codewords, then $\mathcal{C}$ is a $2$-dimensional skew-cyclic code of size $s\times l$ over $R$ under $\sigma_{1}$ and $\sigma_{2}$. 3. 3.

$\mathcal{C}$* is an $(M_{i},\sigma_{i},\delta_{i})$-skew code for $i=1,2$ if $\mathcal{C}$ is an $R$-linear code such that $T*\mathcal{C}\subseteq\mathcal{C}$, where $T$ is given in equation $(1)$ and $T*\mathcal{C}=\{T(\bar{v})\leavevmode\nobreak\ |\leavevmode\nobreak\ \bar{v}\in\mathcal{C}\}$, $M_{1}$ is of order $l\times l$ and $M_{2}$ is of order $s\times s$. Moreover, $T(v_{a_{1},0},v_{a_{1},1},\cdots,v_{a_{1},l-1})=\sigma_{1}(v_{a_{1},0},v_{a_{1},1},\dots,v_{a_{1},l-1}).M_{1}+\delta_{1}(v_{a_{1},0},v_{a_{1},1},\\ \dots,v_{a_{1},l-1})$, $0\leq a_{1}\leq s-1$ and $T(v_{0,a_{2}},v_{1,a_{2}},\dots,v_{s-1,a_{2}})=\sigma_{2}(v_{0,a_{2}},v_{1,a_{2}},\\ \dots,v_{s-1,a_{2}}).M_{2}+\delta_{2}(v_{0,a_{2}},v_{1,a_{2}},\dots,v_{s-1,a_{2}})$, $0\leq a_{2}\leq l-1$, where *

[TABLE]

and

[TABLE] 4. 4.

$\mathcal{C}$* is an $(f,\sigma_{1},\sigma_{2},\delta_{1},\delta_{2})$-skew code if $\mathcal{C}$ is an $R$-linear code such that $T_{f}*\mathcal{C}\subseteq\mathcal{C}$, where $T_{f}$ is given in equation $(1),$ $M_{1}=(M_{1})_{f}$ is of order $l\times l,$ $M_{2}=(M_{2})_{f}$ is of order $s\times s$ defined as above and $T_{f}*\mathcal{C}=\{T_{f}(\bar{v})\leavevmode\nobreak\ |\leavevmode\nobreak\ \bar{v}\in\mathcal{C}\}$.*

Definition 2.4**.**

Let $\bar{c}\in\mathcal{C}\subseteq\mathbb{F}_{q}^{n}$. The Hamming weight $wt_{H}(\bar{c})$ of a codeword $\bar{c}=(c_{0},c_{1},...,c_{n-1})\in\mathcal{C}$ is the number of non-zero components. For any two codewords $\bar{c}$ and $\bar{c^{\prime}}$ of $\mathcal{C},$ the Hamming distance is defined as $d_{H}(\bar{c},\bar{c^{\prime}})=wt(\bar{c}-\bar{c^{\prime}})$. The Hamming distance for the code $\mathcal{C}$ is

[TABLE]

Let $\bar{c}=(c_{0},c_{1},...,c_{n-1})$ and $\bar{c^{\prime}}=(c^{\prime}_{0},c^{\prime}_{1},\dots,c^{\prime}_{n-1})$ be two elements of $\mathcal{C}$. Then the Euclidean inner product of $\bar{c}$ and $\bar{c^{\prime}}$ in $\mathbb{F}_{q}^{n}$ is $\bar{c}\cdot\bar{c^{\prime}}=\sum_{i=0}^{n-1}c_{i}c^{\prime}_{i}$. The dual code of $\mathcal{C}$ is $\mathcal{C}^{\perp}=\{\bar{c}\in\mathbb{F}_{q}^{n}\leavevmode\nobreak\ |\leavevmode\nobreak\ \bar{c}\cdot\bar{c}^{\prime}=0,\leavevmode\nobreak\ for\leavevmode\nobreak\ all\leavevmode\nobreak\ \bar{c}^{\prime}\in\mathcal{C}\}$. A code $\mathcal{C}$ is called self-orthogonal if $\mathcal{C}\subseteq\mathcal{C}^{\perp}$ and self dual if $\mathcal{C}=\mathcal{C}^{\perp}$.

3 Skew generalized cyclic (SGC) codes and its properties

In this section, we discuss some algebraic properties of two-dimensional skew generalized cyclic codes. In $2017$, Cuiti$\tilde{n}$o and Tironi [13] studied the structural properties of skew generalized cyclic codes over $B=R[x;\sigma,\delta]$. We generalize these results over $B=R[x_{1};\sigma_{1},\delta_{1}][x_{2};\sigma_{2},\delta_{2}]$.

Remark 3.1**.**

[13]** Let $T$ be any pseudo-linear transformation on $R^{n}$. If $p=\sum_{i=0}^{m}p_{i}x^{i}$, then $p(T)=\sum_{i=0}^{m}p_{i}T^{i}$ is not in general a pseudo-linear transformation. Also, $\Gamma_{f}(p(T_{f})(\bar{v}))=[pv]$ for every $p\in B$ and $\bar{v}\in R^{n}$ such that $\Gamma_{f}(\bar{v})=[v]$. This implies that the map $\Gamma_{f}$ is a left $B$-module isomorphism when $B/Bf$ is a $B$-module with the product $p[v]=[pv]$ for any $p,v\in B$ and $R^{n}$ is endowed with the left $B$-module structure given by the left action of $T_{f}$, i.e., $t.\bar{v}=T_{f}(\bar{v})$ for any $\bar{v}\in R^{n}$.

Based on above facts, we have the following characterization of $(f,\sigma_{1},\sigma_{2},\delta_{1},\\ \delta_{2})$-skew code.

Theorem 3.1**.**

Let $\mathcal{C}$ be a non-empty subset of $R^{n}$. Then $\mathcal{C}$ is an $(f,\sigma_{1},\sigma_{2},\delta_{1},\delta_{2})$-skew code if and only if $\Gamma_{f}(\mathcal{C})$ is a left $B$-submodule of $B/Bf$.

Now, we introduce the definition of skew generalized cyclic $(SGC)$ code over $B$.

Definition 3.1**.**

Let $\mathcal{C}\subseteq R^{n}$ be a nonempty set. Then $\mathcal{C}$ is said to be a $(f,\sigma_{1},\sigma_{2},\\ \delta_{1},\delta_{2})$-skew generalized cyclic code if $\Gamma_{f}(\mathcal{C})=Bg/Bf$, where $g\in S[x_{2};\sigma_{2},\delta_{2}]$ is a monic polynomial such that $f\in Bg$. The polynomial $g\in B$ is called the generator polynomial of $\mathcal{C}$ and we write $\mathcal{C}=(g)_{n,q}^{\sigma_{1},\sigma_{2},\delta_{1},\delta_{2}}$.

Let $\mathcal{C}=(g)_{n,q}^{\sigma_{1},\sigma_{2},\delta_{1},\delta_{2}}$ be an $(f,\sigma_{1},\sigma_{2},\delta_{1},\delta_{2})$-skew generalized cyclic $(SGC)$ code with $g=\sum_{a_{1}=0}^{k_{1}}\sum_{a_{2}=0}^{k_{2}}g_{a_{1},a_{2}}x_{1}^{a_{1}}x_{2}^{a_{2}}\in B$. We know that $\mathcal{C}$ is a free left $R$-module of dimension $k=(s-k_{1})(l-k_{2})=n-sk_{2}-lk_{1}+k_{1}k_{2}$ with basis

[TABLE]

where $(k_{1},k_{2})=lexdeg\leavevmode\nobreak\ (g)$ and if $[g]=\Gamma_{f}(\bar{g})$. Then generator matrix of $\mathcal{C}$ is

[TABLE]

where $g=\sum_{a_{1}=0}^{k_{1}}\sum_{a_{2}=0}^{k_{2}}g_{a_{1},a_{2}}x_{1}^{a_{1}}x_{2}^{a_{2}}\in B$, i.e., $\bar{g}$ is an $s\times l$ arrays defined as

[TABLE]

It is well-known that $dim(\mathcal{C})+dim(\mathcal{C}^{\perp})=sl$. Therefore, $dim(\mathcal{C}^{\perp})=k^{\prime}=sk_{2}+lk_{1}-k_{1}k_{2}$. Now, we have the following proposition for the dual code $\mathcal{C}^{\perp}$ of $(f,\sigma_{1},\sigma_{2},\delta_{1},\delta_{2})$-skew generalized cyclic code $\mathcal{C}\subseteq\mathbb{F}_{q}^{n}$.

Proposition 3.1**.**

Let $f\in B$, as in equation $(2)$, $T_{f}$ be its associated pseudo-linear transformation given in equation $(3)$ and $R=\mathbb{F}_{q}^{n}$. If $f=hg=gh^{\prime}$ for some monic skew polynomials $g,h,h^{\prime}\in B$ and $r_{0,0}\neq 0$, then linearly independent columns of the matrix

[TABLE]

form a basis of $\mathcal{C}^{\perp}$, where $\Gamma_{f}(\bar{h})=[h^{\prime}]$, $lexdeg(h^{\prime})=(a,b)$ and $\mathcal{C}=(g)_{n,q}^{\sigma_{1},\sigma_{2},\delta_{1},\delta_{2}}$.

Proof.

Let $\bar{a}\in\mathcal{C}=(g)_{n,q}^{\sigma_{1},\sigma_{2},\delta_{1},\delta_{2}}$, where

[TABLE]

and $\Gamma_{f}(\bar{h^{\prime}})=[h^{\prime}]$. So, $\Gamma_{f}(\bar{a})=[a]=[\alpha g]$, for some $\alpha,a\in B$. Then $\Gamma_{f}(a(T_{f})(\bar{h^{\prime}}))\\ =[ah^{\prime}]=[(\alpha g)h^{\prime}]=[\alpha(gh^{\prime})]=[\alpha f]=[0]\in B/Bf$. This implies $a(T_{f})(\bar{h^{\prime}})=\bar{0}$. Therefore, $\bar{0}=a(T_{f})(\bar{h^{\prime}})=(\sum_{a_{1}=0}^{s-1}\sum_{a_{2}=0}^{l-1}a_{a_{1},a_{2}}x_{1}^{a_{1}}x_{2}^{a_{2}})\\ (T_{f})(\bar{h^{\prime}})=\sum_{a_{1}=0}^{s-1}\sum_{a_{2}=0}^{l-1}a_{a_{1},a_{2}}T_{f}^{a_{1}}T_{f}^{a_{2}}(\bar{h^{\prime}})$. This shows that $\bar{a}.\mathbb{H}=0$ for any $\bar{a}\in\mathcal{C}$. Further, we note that $\Gamma_{f}(T_{f}^{k^{\prime}}(\bar{h^{\prime}}))=\Gamma_{f}(T_{f}^{i+j}(\bar{h^{\prime}}))=x_{1}^{i}x_{2}^{j}h^{\prime}$

[TABLE]

where lexdeg$(h_{1}^{\prime})=(a,0)$ and lexdeg$(h_{2}^{\prime})=(0,b)$. Hence, $\{\bar{h^{\prime}},T_{f}(\bar{h^{\prime}}),\dots,T_{f}^{k^{\prime}-1}\\ (\bar{h^{\prime}})\}$ are linearly independent. ∎

Also, when $R$ is commutative, the following theorem gives the idea of the dual code $\mathcal{C}^{\perp}$ of $(M_{i},\sigma_{i},\delta_{i})$-skew generalized cyclic code $\mathcal{C}\subseteq\mathbb{F}_{q}^{n}$ for $i=1,2$.

Theorem 3.2**.**

Suppose $R$ is commutative and $\delta_{i}:R\rightarrow R$ is a $\sigma_{i}$-derivation, where $\sigma_{i}\in Aut(R)$ for $i=1,2$. If $\mathcal{C}\subseteq R^{n}$ is an $(M_{i},\sigma_{i},\delta_{i})$-skew generalized cyclic $(SGC)$ code, then $\mathcal{C}^{\perp}$ is a $((M_{i}^{t})_{\sigma_{i}^{-1}},\sigma_{i}^{-1},\delta_{i}^{\prime\prime})$-skew code with $\sigma_{i}^{-1}$-derivation $\delta_{i}^{\prime\prime}=-\sigma_{i}^{-1}\delta_{i}$ for $i=1,2$, where for a matrix $E=[(a_{mn})],$ we denote $E_{\sigma_{i}^{-1}}=[\sigma_{i}^{-1}(a_{mn})]$ and $E^{t}$ its transpose matrix.

Proof.

We define two pseudo-linear maps $T_{i}$ and $T_{i}^{\prime}$ by $T_{i}(\bar{v})=\sigma_{i}(\bar{v})\cdot\leavevmode\nobreak\ M_{i}+\delta_{i}(\bar{v})$ and $T_{i}^{\prime}(\bar{v})=\sigma_{i}^{-1}(\bar{v})\cdot\leavevmode\nobreak\ (M_{i}^{t})_{\sigma_{i}^{-1}}+\delta_{i}^{\prime\prime}(\bar{v})$ for $i=1,2$ for every $\bar{v}\in R^{n}$, respectively. For any $\bar{a}\in\mathcal{C}$ and $\bar{b}\in\mathcal{C}^{\perp}$, we have $0=\bar{b}\cdot\leavevmode\nobreak\ T_{i}^{t}(\bar{a})=\bar{b}\cdot\leavevmode\nobreak\ (\sigma_{i}(\bar{a})\cdot\leavevmode\nobreak\ M_{i})^{t}+\bar{b}\cdot\leavevmode\nobreak\ (\delta_{i}(\bar{a}))^{t}=\bar{b}\cdot\leavevmode\nobreak\ (\sigma_{i}(\bar{a})\cdot\leavevmode\nobreak\ M_{i})^{t}-\delta_{i}(\bar{b})\cdot\leavevmode\nobreak\ (\sigma_{i}(\bar{a}))^{t}.$ This implies $0=\bar{b}\cdot\leavevmode\nobreak\ M_{i}^{t}\cdot\leavevmode\nobreak\ (\sigma_{i}(\bar{a}))^{t}-\delta_{i}(\bar{b})\cdot\leavevmode\nobreak\ (\sigma(\bar{a}))^{t}=(\bar{b}\cdot\leavevmode\nobreak\ M_{i}^{t}-\delta_{i}(\bar{b}))\cdot\leavevmode\nobreak\ (\sigma_{i}(\bar{a}))^{t}$, i.e., $(\bar{b}\cdot\leavevmode\nobreak\ M_{i}^{t}-\delta_{i}(\bar{b}))\cdot\leavevmode\nobreak\ (\sigma_{i}(\bar{a}))^{t}=0$, for $i=1,2$. Therefore, $0=\sigma_{i}^{-1}(0)=\sigma_{i}^{-1}(\bar{b}\cdot\leavevmode\nobreak\ M_{i}^{t}-\delta_{i}(\bar{b}))\cdot\leavevmode\nobreak\ (\bar{a})^{t}=(\sigma_{i}^{-1}(\bar{b})\cdot\leavevmode\nobreak\ (M_{i}^{t})_{\sigma_{i}^{-1}}-\sigma_{i}^{-1}\delta_{i}(\bar{b}))\cdot\leavevmode\nobreak\ (\bar{a})^{t}=(\sigma_{i}^{-1}(\bar{b})\cdot\leavevmode\nobreak\ (M_{i}^{t})_{\sigma_{i}^{-1}}+\delta_{i}^{\prime\prime}(\bar{b}))\cdot\leavevmode\nobreak\ (\bar{a})^{t}=T_{i}^{\prime}(\bar{b})\cdot\leavevmode\nobreak\ (\bar{a})^{t}$, i.e., $T_{i}^{\prime}(\bar{b})\cdot\leavevmode\nobreak\ (\bar{a})^{t}=0$ for all $\bar{a}\in\mathcal{C}$, $\bar{b}\in\mathcal{C}^{\perp}$ and $i=1,2$. ∎

4 BCH lower bounds for the minimum distance of a SGC code with non-zero derivation

Let $\mathbb{F}_{q}$ be the Galois field, where $q=p^{m},\leavevmode\nobreak\ p$ a prime, and $m\in\mathbb{Z}_{+}$. In $2017$, Cuiti$\tilde{n}$o and Tironi [13] investigated the BCH type lower bounds for the minimum distance of a skew generalized cyclic (SGC) code for $\mathbb{F}_{q}[x;\theta]$, where $\theta$ is an automorphism.

Following the norm defined in [5], which generalizes the classical notion of the norm of a field element, for $i\in\mathbb{N},N_{i}^{\theta,\delta}(\alpha)$ is recursively defined as

[TABLE]

and

[TABLE]

In particular, if $\delta=0$, we get the classical norm $N_{i}^{\theta,\delta}(\alpha)=\alpha\theta(\alpha)\dots\theta^{i-1}(\alpha).$

In this section, we generalize the BCH lower bound given in [13] for the minimum distance of the SGC codes of the ring $R=\mathbb{F}_{q}[x;\theta,\delta]$ under certain conditions, where $\delta$ is a $\theta$-derivation.

Now, we consider Frobenius automorphism $\theta:\mathbb{F}_{q}\rightarrow\mathbb{F}_{q}$ by $\theta(a)=a^{p}$, where $a\in\mathbb{F}_{q}$ and let $f=x^{n}-f_{n-1}x^{n-1}-\cdots-f_{1}x-f_{0}$, where $f_{n-1},\dots,f_{1},f_{0}\in\mathbb{F}_{q}$ and $f_{0}\neq 0$. Again, consider a map $\Omega_{f}:\mathbb{F}_{q}^{n}\rightarrow R/Rf$ by $\Omega_{f}(\bar{c})=[c]\in R/Rf$ for all $\bar{c}\in\mathbb{F}_{q}^{n}$, where

[TABLE]

$c_{i}\in\mathbb{F}_{q}$, i.e., $\Omega_{f}(\bar{c})=\Omega_{f}((c_{0},c_{1},\dots,c_{n-1}))=[c]\in R/Rf$ for all $(c_{0},c_{1},\dots,\\ c_{n-1})\in\mathbb{F}_{q}^{n}$. The map $\Omega_{f}$ is an $\mathbb{F}_{q}$-linear isomorphism between $\mathbb{F}_{q}$-modules. Now, we provide the following two lemmas that will be used later in the proof of the main result.

Lemma 4.1**.**

For any non-zero $\theta$-derivation $\delta$, in $R/Rf$ we have $\alpha x=1$ and $x\beta=1+\delta(\beta)$, where $\alpha=f_{0}^{-1}x^{n-1}-f_{0}^{-1}f_{n-1}x^{n-2}-\cdots-f_{0}^{-1}f_{2}x-f_{0}^{-1}f_{1}$ and $\beta=\theta^{-1}(f_{0}^{-1})x^{n-1}-\theta^{-1}(f_{0}^{-1}f_{n-1})x^{n-2}-\cdots-\theta^{-1}(f_{0}^{-1}f_{2})x-\theta^{-1}(f_{0}^{-1}f_{1})$.

Proof.

Note that in $R/Rf,$ we have the following equivalences:

[TABLE]

i.e.,

[TABLE]

As $f_{0}\neq 0\in\mathbb{F}_{q}$, so

[TABLE]

i.e.,

[TABLE]

Hence, $\alpha x=1$.

Moreover,

[TABLE]

i.e.,

[TABLE]

This implies $f_{0}^{-1}xx^{n-1}-f_{0}^{-1}f_{n-1}xx^{n-2}-\cdots-f_{0}^{-1}f_{1}x=1$, i.e., $x\theta^{-1}(f_{0}^{-1})x^{n-1}-x\theta^{-1}(f_{0}^{-1}f_{n-1})x^{n-2}-\cdots-x\theta^{-1}(f_{0}^{-1}f_{1})=1+\delta(\theta^{-1}(f_{0}^{-1}))x^{n-1}-\delta(\theta^{-1}(f_{0}^{-1}f_{n-1}))\\ x^{n-2}-\cdots-\delta(\theta^{-1}(f_{0}^{-1}f_{1}))$. This gives $x(\theta^{-1}(f_{0}^{-1})x^{n-1}-\theta^{-1}(f_{0}^{-1}f_{n-1})x^{n-2}-\cdots-\theta^{-1}(f_{0}^{-1}f_{1}))=1+\delta(\theta^{-1}(f_{0}^{-1}))x^{n-1}-\delta(\theta^{-1}(f_{0}^{-1}f_{n-1}))x^{n-2}-\cdots-\delta(\theta^{-1}(f_{0}^{-1}f_{1}))$.

Hence, $x\beta=1+\delta(\beta)$ where $\beta=\theta^{-1}(f_{0}^{-1})x^{n-1}-\theta^{-1}(f_{0}^{-1}f_{n-1})x^{n-2}-\cdots-\theta^{-1}(f_{0}^{-1}f_{2})x-\theta^{-1}(f_{0}^{-1}f_{1})$.

Further, if $\theta=id.$, then $\beta=f_{0}^{-1}x^{n-1}-f_{0}^{-1}f_{n-1}x^{n-2}-\cdots-f_{0}^{-1}f_{2}x-f_{0}^{-1}f_{1}=\alpha$. ∎

Lemma 4.2**.**

*Let $\mathcal{C}\subseteq\mathbb{F}_{q}{{}^{n}}$ be a $SGC$ code, and consider a polynomial $c\in\Omega_{f}(\mathcal{C})$ with $wt_{H}(c)=w$. Then there exists $r\in R$ such that

[TABLE]

where $c_{i},\leavevmode\nobreak\ d_{i}\in\mathbb{F}_{q}^{*}$ and $a_{i},\leavevmode\nobreak\ k_{i}\in\mathbb{N}$ with $a_{i},\leavevmode\nobreak\ k_{i}\leq n-1$ for $i=1,\dots,w-1$. Also,

[TABLE]

Proof.

Since $wt_{H}(c)=w$, we can write $c=b_{k_{0}}x^{k_{0}}+b_{k_{1}}x^{k_{1}}+\cdots+b_{k_{w-1}}x^{k_{w-1}}$, where $b_{k_{j}}\in\mathbb{F}_{q}$, $k_{0}<\dots<k_{w-1}$ and $b_{k_{j}}\neq 0$, for $j=0,\dots,w-1$. Hence,

[TABLE]

In Lemma 4.1, we have $\alpha.x=1$. This implies $x^{-k_{0}}=\alpha^{k_{0}}$.

 Therefore, $\alpha^{k_{0}}b_{k_{0}}^{-1}c=1+\theta^{-k_{0}}(b_{k_{0}}^{-1}b_{k_{1}})x^{k_{1}-k_{0}}+...+\theta^{-k_{0}}(b_{k_{0}}^{-1}b_{k_{w-1}}x^{k_{w-1}-k_{0}})\\ -\delta(\theta^{-k_{0}}(b_{k_{0}}^{-1}b_{k_{1}}))x^{k_{1}}-\cdots-\delta(\theta^{-k_{0}}(b_{k_{0}}^{-1}b_{k_{w-1}}))x^{k_{w-1}}\in\Omega_{f}(\mathcal{C})\subseteq R/Rf$.

Now, put $r=\alpha^{k_{0}}b_{k_{0}}^{-1}$, $c_{j}=\theta^{-k_{0}}(b_{k_{0}}^{-1}b_{k_{j}})$, $d_{j}=-\delta(\theta^{-k_{0}}(b_{k_{0}}^{-1}b_{k_{j}}))=-\delta(c_{j})$ and $a_{j}=k_{j}-k_{0}$. Further,

[TABLE]

∎

Inspired by the work done in [13, 7], the following results give the lower bounds on the minimum Hamming distance for the SGC codes in $\mathbb{F}_{q}[x;\theta,\delta]$.

Theorem 4.1**.**

Let $\mathcal{C}=(g)_{n,q}^{\theta,\delta}$ be a SGC code. Suppose $\exists$ $\beta\in\bar{\mathbb{F}_{q}}$, the algebraic closure of $\mathbb{F}_{q}$ and $l,m\in\mathbb{Z}_{+}\cup\{0\}$, such that $g(\beta_{i}^{l+mi})=0$, where $\beta_{i}=N_{l+mi}^{\theta,\delta}(\beta)$ for $i=0,\dots,\bigtriangleup-2$. If $N_{i}(\beta^{m})\neq 1$, $N_{l+mi-1}(N_{i}(\beta^{m}))=1$ and $N_{i}^{\theta,\delta}(N_{l+mi}^{\theta,\delta}(\beta))=N_{l+mi}^{\theta,\delta}(N_{i}^{\theta,\delta}(\beta))$ for all $i=1,\dots,n-1$, then $d_{\mathcal{C}}\geqslant\bigtriangleup$.

Proof.

If possible, let there exists a polynomial $c\in\Omega_{f}(\mathcal{C})$ with $wt_{H}(c)=w<\bigtriangleup$. Then from Lemma 4.2, $c$ can be represented as

[TABLE]

Hence, we have

[TABLE]

where $c^{\prime}_{i},\leavevmode\nobreak\ d_{i}\in\mathbb{F}_{q}^{*}$ and $a_{i},k_{i}\in\mathbb{Z}_{+}$ with $a_{i},\leavevmode\nobreak\ k_{i}<n$ and $c_{i}=c^{\prime}_{i}b_{k_{0}}-\delta(c^{\prime}_{i})$, for $i=1,\dots,w-1$.

Define $L_{i}=N_{a_{i}}^{\theta,\delta}(\beta_{i})$ and $P_{j}=\sum_{i=1}^{w-1}c_{i}L_{i}^{j}=c(\beta_{i}^{j})-1$ and consider a polynomial $p\in\mathbb{F}_{q}[t]$ given by

[TABLE]

Then $L_{i}^{m}=N_{a_{i}}^{\theta,\delta}(\beta_{i}^{m})=N_{a_{i}}^{\theta,\delta}(N_{l+mi}^{\theta,\delta}(\beta^{m}))=N_{l+mi}^{\theta,\delta}(N_{a_{i}}^{\theta,\delta}(\beta^{m})).$

If $N_{l+mi}^{\theta,\delta}(N_{a_{i}}^{\theta,\delta}(\beta^{m}))=1$, then

[TABLE]

This implies $N_{a_{i}}^{\theta,\delta}(\beta^{m})=1$, which contradicts the assumption.

So, $L_{i}^{m}=N_{l+mi}^{\theta,\delta}(N_{a_{i}}^{\theta,\delta}(\beta^{m}))\neq 1$. Hence, $1$ is not a root of polynomial $p(t)$, i.e., $p(1)\neq 1$.

Next, we have

[TABLE]

i.e.,

[TABLE]

Since $c\in\Omega_{f}(\mathcal{C})=Rg/Rf$, so $c(\beta_{i}^{l+mi})=0$, for $i=0,\dots,\bigtriangleup-2$. Then $P_{l+mi}=c(\beta_{i}^{l+mi})-1=-1$. From equation (5), we have $-1+p_{1}(-1)+\cdots+p_{w-2}(-1)+p_{w-1}(-1)=0$. This implies $p(1)=0$, which contradicts the assumption. Therefore, $d_{\mathcal{C}}\geqslant\bigtriangleup$. ∎

The following result is a generalization of the Theorem 4.1 and gives a better BCH lower bound for the minimum distance of SGC code over $\mathbb{F}_{q}$.

Theorem 4.2**.**

Let $\mathcal{C}=(g)_{n,q}^{\theta,\delta}$ be a SGC code. Suppose there exists $\beta\in\bar{\mathbb{F}_{q}}$ and $l,m_{1},m_{2}\in\mathbb{Z}_{+}\cup\{0\}$ such that $(m_{1},m_{2})\neq(0,0)$, $g(\beta_{i_{1}}^{l+m_{1}i_{1}+m_{2}i_{2}})=0$, where $\beta_{i_{1}}=N_{l+m_{1}i_{1}+m_{2}i_{2}}^{\theta,\delta}(\beta)$ for $i_{1}=0,\dots,\bigtriangleup-2$ and $i_{2}=0,\dots,s$. If $N_{i}(\beta^{m_{j}})\neq 1$, $N_{l+m_{1}i_{1}+m_{2}i_{2}-1}(N_{i}(\beta^{m_{j}}))=1$ and $N_{i}^{\theta,\delta}(N_{l+m_{1}i_{1}+m_{2}i_{2}}^{\theta,\delta}(\beta))=N_{l+m_{1}i_{1}+m_{2}i_{2}}^{\theta,\delta}(N_{i}^{\theta,\delta}(\beta))$ for all $i=1,\dots,n-1$, $j=1,2$, then $d_{\mathcal{C}}\geqslant\bigtriangleup+s$.

Proof.

From Theorem 4.1, we have $d_{\mathcal{C}}\geqslant\bigtriangleup$. If possible, suppose there exists a polynomial $c\in\Omega_{f}(\mathcal{C})$ with $wt_{H}(c)=w$ s.t. $\bigtriangleup\leq w<\bigtriangleup+s$. Then from equation (5),

[TABLE]

where $c_{i}\in\mathbb{F}_{q}^{*}$ and $a_{i}\in\mathbb{Z}_{+}$ with $a_{i}<n$, for $i=1,\dots,w-1$.

Now, define $L_{i}=N_{a_{i}}^{\theta,\delta}(\beta_{i})$ and $P_{j}=\sum_{i=1}^{w-1}c_{i}L_{i}^{j}=c(\beta_{i_{1}}^{j})-1$ and consider the polynomials $p,q\in\mathbb{F}_{q}[t]$ defined by

[TABLE]

Let $r=pq\in\mathbb{F}_{q}[t]$ and $L_{i_{j}}^{m_{j}}=N_{a_{i_{j}}}^{\theta,\delta}(\beta_{i_{1}}^{m_{j}})=N_{a_{i_{j}}}^{\theta,\delta}(N_{l+m_{1}i_{1}+m_{2}i_{2}}^{\theta,\delta}(\beta^{m_{j}}))=N_{l+m_{1}i_{1}+m_{2}i_{2}}^{\theta,\delta}(N_{a_{i_{j}}}^{\theta,\delta}(\beta^{m_{j}}))$. If $N_{l+m_{1}i_{1}+m_{2}i_{2}}^{\theta,\delta}(N_{a_{i_{j}}}^{\theta,\delta}(\beta^{m_{j}}))=1,$ then

[TABLE]

This implies $N_{a_{i_{j}}}^{\theta,\delta}(\beta^{m_{j}})=1$, a contradiction. Therefore, $L_{i_{j}}^{m_{j}}\neq 1$, for $j=1,2$. Since $L_{i_{1}}^{m_{1}}$ and $L_{i_{2}}^{m_{2}}$ are roots of $p,q$, respectively. So, $1$ is not a root of $r=pq$. Also,

[TABLE]

i.e.,

[TABLE]

Since $c\in\Omega_{f}(\mathcal{C})$ and $c(\beta_{i_{1}}^{l+m_{1}i_{1}+m_{2}i_{2}})=0$, therefore $P_{l+m_{1}i_{1}+m_{2}i_{2}}=\\ c(\beta_{i_{1}}^{l+m_{1}i_{1}+m_{2}i_{2}})-1=-1$, for $i_{1}=0,\dots,\bigtriangleup-2$, $i_{2}=0,\dots,s$ and $\bigtriangleup\leq w<\bigtriangleup+s$. Hence, from above equations, we have $(-1-p_{1}+\cdots+p_{\bigtriangleup-3}-p_{\bigtriangleup-2})(1+q_{1}+\cdots+q_{w-\bigtriangleup}-q_{w-\bigtriangleup+1})=0$. This implies $r(1)=0$, which contradicts the assumption. Thus, $w\geqslant\bigtriangleup+s$, i.e., $d_{\mathcal{C}}\geqslant\bigtriangleup+s$. ∎

In view of Theorems 4.1, we give an extension of Theorem 4.2, which can be proved inductively.

Theorem 4.3**.**

Let $\mathcal{C}=(g)_{n,q}^{\theta,\delta}$ be a SGC code. Suppose there exists $\beta\in\bar{\mathbb{F}_{q}}$ and $l,m_{1},m_{2},\dots m_{r}\in\mathbb{Z}_{+}\cup\{0\}$ such that $(m_{1},m_{2}\dots m_{r})\neq(0,0)$, $g(\beta_{i_{1}}^{l+m_{1}i_{1}+\cdots+m_{r}i_{r}})\\ =0$, where $\beta_{i_{1}}=N_{l+m_{1}i_{1}+\cdots+m_{r}i_{r}}^{\theta,\delta}(\beta)$ for $i_{1}=0,\dots,\bigtriangleup-2$ and $i_{k}=0,\dots,s_{k}$ and $k=2,3,\dots,r$. If $N_{i}(\beta^{m_{j}})\neq 1$, $N_{l+m_{1}i_{1}+m_{2}i_{2}+\cdots+m_{r}i_{r}-1}(N_{i}(\beta^{m_{j}}))=1$ and $N_{i}^{\theta,\delta}(N_{l+m_{1}i_{1}+\cdots+m_{r}i_{r}}^{\theta,\delta}(\beta))=N_{l+m_{1}i_{1}+\cdots+m_{r}i_{r}}^{\theta,\delta}(N_{i}^{\theta,\delta}(\beta))$ for all $i=1,\dots,\\ n-1$, $j=1,2,\dots,r$, then $d_{\mathcal{C}}\geqslant\bigtriangleup+\sum_{k=2}^{r}s_{k}$.

Corollary 4.1**.**

Let $\mathcal{C}=(g)_{n,q}^{\theta,\delta}$ with $q\geq n+1$. If $\beta\in\bar{\mathbb{F}_{q}}$ and $l,c_{1},c_{2},\dots,c_{r}\in\mathbb{Z}_{+}\cup\{0\}$ such that $(c_{1},c_{2},\dots,c_{r})\neq(0,0,\dots,0)$,

[TABLE]

$N_{i}(\beta^{c_{j}})\neq 1$, $N_{l+c_{1}i_{1}+c_{2}i_{2}+\cdots+c_{r}i_{r}}(N_{i}(\beta^{c_{j}}))=1$ and $N_{i}^{\theta,\delta}(N_{l+c_{1}i_{1}+\cdots+c_{r}i_{r}}^{\theta,\delta}(\beta))\\ =N_{l+c_{1}i_{1}+\cdots+c_{r}i_{r}}^{\theta,\delta}(N_{i}^{\theta,\delta}(\beta))$ for all $i=1,\dots,n-1$, $j=1,2,\dots,r$, then $\mathcal{C}$ is a Maximum Distance Separable (MDS) code.

Proof.

Since $deg(g)\leq n-k$, and the singleton bound is given by $d_{\mathcal{C}}\leq n-dim(\mathcal{C})+1=n-(n-deg(g))+1\leq n-k+1$. From Theorem 4.3, it follows that $d_{\mathcal{C}}\geq n-k+1$. This implies $d_{\mathcal{C}}=n-k+1.$ ∎

5 Examples

The following table will give the examples of MDS codes in $\mathbb{F}^{n}_{q}$ for $q=8,\leavevmode\nobreak\ 11$ with $n\leq q-1\leq 11$. All computations are carried out by using Magma software [2].

Now, we give some examples of MDS (Maximum Distance Separable) generalized cyclic (GC) code for commutative case $(\theta=id)$ as well as non-commutative case. All computations are carried out by using Magma software [2].

Example 5.1**.**

Consider $f=x^{5}+wx^{4}+x^{3}+wx^{2}+1\in\mathbb{F}_{8}[x;\theta]$. Here, we obtain $21$ MDS skew generalized cyclic codes with parameters $[5,4,2]_{8}$, $[5,1,5]_{8}$, and $[5,3,3]_{8}$. If $\theta=id$, then we get only $2$ MDS GC codes with parameters $[5,4,2]_{8}$ and $[5,3,3]_{8}$ with generator polynomials $x+w^{6}$ and $x^{2}+w^{3}x+w^{4}$, respectively.

Example 5.2**.**

Consider $f=x^{5}+x^{4}+x^{3}+wx^{2}+1\in\mathbb{F}_{9}[x;\theta]$. Here, we obtain $16$ MDS skew generalized cyclic codes with parameters $[5,2,4]_{9}$ and $[5,3,3]_{9}$. If $\theta=id$, then we get only $1$ MDS GC codes with parameters $[5,3,3]_{9}$ and generator polynomial $x^{2}+w^{6}x+w^{7}$.

Example 5.3**.**

Consider $f=x^{3}+x^{2}+x+1\in\mathbb{F}_{9}[x;\theta]$. Then we obtain $80$ MDS skew generalized cyclic codes with parameters $[3,2,2]_{9}$ and $[3,1,3]_{9}$. If $\theta=id$, then we have only $3$ MDS GC codes with parameter $[3,2,2]_{5}$ and generator polynomials $x+1$, $x+w^{2}$ and $x+w^{6}$.

Example 5.4**.**

Consider $f=wx^{5}+x^{4}+x^{3}+x^{2}+1\in\mathbb{F}_{8}[x;\theta]$. Here, we obtain $21$ MDS skew generalized cyclic codes with parameters $[5,2,4]_{8}$, $[5,4,2]_{8}$ and $[5,1,5]_{8}$. If $\theta=id$, then we get only $2$ MDS GC codes with parameters $[5,4,2]_{8}$ and $[5,1,5]_{8}$ and generator polynomials $x+4$ and $x^{4}+2x^{3}+3x^{2}+4x+4,$ respectively.

Example 5.5**.**

Consider $f=wx^{5}+x^{4}+x^{3}+x^{2}+1\in\mathbb{F}_{9}[x;\theta]$. Here, we obtain $24$ MDS skew generalized cyclic codes with parameters $[5,2,4]_{9}$, $[5,4,2]_{9}$, $[5,3,3]_{9}$ and $[5,1,5]_{9}$. If $\theta=id$, then we get only $1$ MDS GC codes with parameters $[5,4,2]_{9}$ and generator polynomial $x+w^{7}$.

6 Conclusion

In this paper, we considered the skew generalized cyclic (SGC) codes over $R[x_{1};\sigma_{1},\delta_{1}][x_{2};\sigma_{2},\delta_{2}]$ that are invariant under pseudo-linear transformation of $R^{n}$, where $R$ is a ring with unity or a finite field $\mathbb{F}_{q}$. Further, for non-zero derivation, we studied the BCH lower bounds for the minimum distance of a SGC code over $\mathbb{F}_{q}[x;\theta,\delta]$. Finally, we provide some examples of MDS code as applications of our results.

Acknowledgements

The authors are thankful to the Department of Science and Technology (DST), Govt. of India (Ref No.- DST/INSPIRE/03/2016/001445) for financial support and Indian Institute of Technology Patna for providing the research facilities.