In this paper, we mainly study the theory of linear codes over the ring $R =mathbb{Z}_4+umathbb{Z}_4+vmathbb{Z}_4+uvmathbb{Z}_4$. 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 $R$.
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