Two Dimensional $\left( \alpha,\beta \right) $-Constacyclic Codes of arbitrary length over a Finite Field

TL;DR

This paper explores the algebraic structure of two-dimensional $(eta,eta)$-constacyclic codes over finite fields, providing conditions for self-duality and examples of special code types.

Contribution

It characterizes the algebraic structure of two-dimensional $(eta,eta)$-constacyclic codes and their duals, and establishes conditions for self-duality.

Findings

Necessary and sufficient conditions for self-duality of these codes.

Self-dual codes cannot exist when $ ext{gcd}(s,q)=1$ for certain cases.

Examples of self-dual, isodual, MDS, and quasi-twisted codes provided.

Abstract

In this paper we characterize the algebraic structure of two-dimensional $(\alpha,\beta )$-constacyclic codes of arbitrary length $s.\ell$ and of their duals. For $\alpha,\beta \in \{1,-1\}$, we give necessary and sufficient conditions for a two-dimensional $(\alpha,\beta )$-constacyclic code to be self-dual. We also show that a two-dimensional $(\alpha,1 )$-constacyclic code $\mathcal{C}$ of length $n=s.\ell$ can not be self-dual if $\gcd(s,q)= 1$. Finally, we give some examples of self-dual, isodual, MDS and quasi-twisted codes corresponding to two-dimensional $(\alpha,\beta )$-constacyclic codes.