Z2-double cyclic codes

TL;DR

This paper studies the structure and polynomial representations of Z2-double cyclic codes, a class of binary linear codes invariant under cyclic shifts within two coordinate subsets, and characterizes their generators and duals.

Contribution

It provides a detailed structural analysis and explicit generator polynomials for Z2-double cyclic codes, advancing understanding of their algebraic properties.

Findings

Determined the generator polynomials of Z2-double cyclic codes.

Analyzed the polynomial representation and duals of these codes.

Established relations between the polynomial generators.

Abstract

A binary linear code $C$ is a $\mathbb{Z}_2$-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the $\mathbb{Z}_2[x]$-module $\mathbb{Z}_2[x]/(x^r-1)\times\mathbb{Z}_2[x]/(x^s-1).$ We determine the structure of $\mathbb{Z}_2$-double cyclic codes giving the generator polynomials of these codes. The related polynomial representation of $\mathbb{Z}_2$-double cyclic codes and its duals, and the relations between the polynomial generators of these codes are studied.