Some results on linear codes over the ring $$\mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4$$Z4+uZ4+vZ4+uvZ4

In this paper, we mainly study the theory of linear codes over the ring $$R =\mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4$$R=Z4+uZ4+vZ4+uvZ4. By using the Chinese Remainder Theorem, we prove that R is isomorphic to a direct sum of four rings. We define a Gray map $$\Phi $$Φ from $$R^{n}$$Rn to $$\mathbb {Z}_4^{4n}$$Z44n, which is a distance preserving map. The Gray image of a cyclic code over R is a linear code over $$\mathbb {Z}_4$$Z4. 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.