On Linear Codes over ${\mathbb{Z}}_4+v{\mathbb{Z}}_4$

Linear codes are considered over the ring ${\mathbb{Z}}_4+v{\mathbb{Z}}_4$, where $v^2=v$. Gray weight, Gray maps for linear codes are defined and MacWilliams identity for the Gray weight enumerator is given. Self-dual codes, construction of Euclidean isodual codes, unimodular complex lattices, MDS codes and MGDS codes over ${\mathbb{Z}}_4+v{\mathbb{Z}}_4$ are studied. Cyclic codes and quadratic residue codes are also considered. Finally, some examples for illustrating the main work are given.