  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.
