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 $R<sup>{n}$ to $\mathbb{Z}_4<sup>{4n}$, which is a distance preserving map. The Gray image of a cyclic code over $R<sup>{n}$ 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 $R$.