Z2Z4-additive cyclic codes, generator polynomials and dual codes

TL;DR

This paper studies Z2Z4-additive cyclic codes, characterizing their structure through generator polynomials and describing how to determine dual codes, advancing algebraic coding theory.

Contribution

It provides a detailed algebraic framework for Z2Z4-additive cyclic codes, including their generator polynomials and dual code characterization.

Findings

Parameters expressed via generator polynomial degrees

Dual code generator polynomials derived from original code

Enhanced understanding of code structure and duality

Abstract

A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. These codes can be identified as submodules of the $\mathbb{Z}_4[x]$-module $\mathbb{Z}_2[x]/(x^\alpha-1)\times\mathbb{Z}_4[x]/(x^\beta-1)$. The parameters of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic code are stated in terms of the degrees of the generator polynomials of the code. The generator polynomials of the dual code of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic code are determined in terms of the generator polynomials of the code ${\cal C}$.