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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 ChineseRemainder 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 acyclic code over $R^{n}$ is a linear code over $\mathbb{Z}_4$. Furthermore, westudy the MacWilliams identities of linear codes over $R$ and give the thegenerator polynomials of cyclic codes over $R$. Finally, we discuss someproperties of MDS codes over $R$.

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