Some Results on Linear Codes over $\mathbb{Z}_4+v\mathbb{Z}_4$

In this paper, we study the linear codes over the commutative ring R = Z4 + vZ4, where v2 = v. We define the Gray weight of the elements of R and give a Gray map from Rn to Z2n 4 , which lead to the MacWillams identity of the linear code over R. Some useful results on self-dual code over R are given. Furthermore, the relationship between some complex unimodular lattices and Hermitian self-dual codes over R is given. Furthermore, the existing conditions of MDS codes over R is given, and the results show that there are no non-trivial MDS codes over R. Structural properties of cyclic codes are also discussed in this paper. As a special class of cyclic codes, quadratic residue and their extension codes over R are considered.