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

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 $U\equiv$ $\left(\genfrac{}{}{0ex}{}{11}{11}\right)$. 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.

10 pages