Some results of linear codes over the ring ${\mathbb{Z}}_4+u{\mathbb{Z}}_4+v{\mathbb{Z}}_4+uv{\mathbb{Z}}_4$

Abstract

In this paper, we mainly study the theory of linear codes over the ring $R={\mathbb{Z}}_4+u{\mathbb{Z}}_4+v{\mathbb{Z}}_4+uv{\mathbb{Z}}_4$. By the Chinese Remainder Theorem, we have $R$ is isomorphic to the direct sum of four rings ${\mathbb{Z}}_4$. We define a Gray map $\Phi$ from $R^n$ to ${\mathbb{Z}}_4^{4n}$, which is a distance preserving map. The Gray image of a cyclic code over $R^n$ is a linear code over ${\mathbb{Z}}_4$. Furthermore, we study the MacWilliams identities of linear codes over $R$ and give the the generator polynomials of cyclic codes over $R$. Finally, we discuss some properties of MDS codes over $R$.