Generator matrix for two-dimensional cyclic codes of arbitrary length

TL;DR

This paper introduces a new algebraic method to analyze two-dimensional cyclic codes of arbitrary length over finite fields, enabling the explicit construction of generator polynomials and matrices.

Contribution

The paper presents a novel approach to determine generator polynomials and matrices for two-dimensional cyclic codes of any length over finite fields.

Findings

Derived generator polynomials for 2D cyclic codes.

Established a method to obtain generator matrices.

Applicable to codes of arbitrary length.

Abstract

Two-dimensional cyclic codes of length $n=\ell s$ over the finite field $\mathbb{F}$ are ideals of the polynomial ring $\frac{\mathbb{F}[x,y]}{< x^{s}-1,y^{\ell}-1 >}$. The aim of this paper, is to present a novel method to study the algebraic structure of two-dimensional cyclic codes of any length $n=\ell s$ over the finite field $\mathbb{F}$. By using this method, we find the generator polynomials for ideals of $\frac{\mathbb{F}[x,y]}{< x^{s}-1,y^{\ell}-1 >}$ corresponding to two dimensional cyclic codes. These polynomials will be applied to obtain the generator matrix for two- dimensional cyclic codes.