Self-dual cyclic codes over $M_2(\mathbb{Z}_4)$

TL;DR

This paper explores the algebraic structure and properties of self-dual cyclic codes over the matrix ring $M_2(\mathbb{Z}_4)$, introducing new code constructions and characterizations over this ring for the first time.

Contribution

It provides the first detailed analysis of cyclic and self-dual cyclic codes over the matrix ring $M_2(\mathbb{Z}_4)$, including generator and dual code descriptions.

Findings

Derived generators for cyclic codes over $M_2(\mathbb{Z}_4)$

Characterized dual codes within the ring structure

Presented examples illustrating code constructions

Abstract

In this paper, we study the codes over the matrix ring over $\mathbb{Z}_4$, which is perhaps the first time the ring structure $M_2(\mathbb{Z}_4)$ is considered as a code alphabet. This ring is isomorphic to $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, where $w$ is a root of the irreducible polynomial $x^2+x+1 \in \mathbb{Z}_2[x]$ and $U\equiv$ ${11}\choose{11}$. We first discuss the structure of the ring $M_2(\mathbb{Z}_4)$ and then focus on algebraic structure of cyclic codes and self-dual cyclic codes over $M_2(\mathbb{Z}_4)$. We obtain the generators of the cyclic codes and their dual codes. Few examples are given at the end of the paper.