  We investigate linear codes over the ring $\mathbb{Z}_4 + u\mathbb{Z}_4 +
v\mathbb{Z}_4 + w\mathbb{Z}_4 + uv\mathbb{Z}_4 + uw\mathbb{Z}_4 +
vw\mathbb{Z}_4 + uvw\mathbb{Z}_4$, with conditions $u^2=u$, $v^2=v$, $w^2=w$,
$uv=vu$, $uw=wu$ and $vw=wv.$ We first analyze the structure of the ring and
then define linear codes over this ring. Lee weight and Gray map for these
codes are defined and MacWilliams relations for complete, symmetrized, and Lee
weight enumerators are obtained. The Singleton bound as well as maximum
distance separable codes are also considered. Furthermore, cyclic and
quasi-cyclic codes are discussed, and some examples are also provided.
