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

TL;DR

This paper explores the structure and properties of linear codes over a specific ring constructed from $\

Contribution

It introduces a Gray map for codes over the ring, studies cyclic codes, and derives MacWilliams identities and properties of MDS codes over this ring.

Findings

Gray map preserves distance between codes over R and Z4.

Cyclic codes over R map to linear codes over Z4.

Properties of MDS codes over R are discussed.

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 $R$.