Some results on linear codes over the ring $$mathbb {Z}_4+umathbb {Z}_4+vmathbb {Z}_4+uvmathbb {Z}_4$$

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 using the Chinese Remainder Theorem, we prove that R is isomorphic to a direct sum of four rings. 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 is a linear code over (mathbb {Z}_4). We also discuss some properties of MDS codes over R. Furthermore, we study the MacWilliams identities of linear codes over R and give the generator polynomials of cyclic codes over R.