Self-dual cyclic codes over <i>M₂(\mathbb{Z}₄)</i>

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