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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 andthen define linear codes over this ring. Lee weight and Gray map for thesecodes are defined and MacWilliams relations for complete, symmetrized, and Leeweight enumerators are obtained. The Singleton bound as well as maximumdistance separable codes are also considered. Furthermore, cyclic andquasi-cyclic codes are discussed, and some examples are also provided.

Linear codes over the ring $\mathbb{Z}_4 + u\mathbb{Z}_4 + v\mathbb{Z}_4  + w\mathbb{Z}_4 + uv\mathbb{Z}_4 + uw\mathbb{Z}_4 + vw\mathbb{Z}_4 +  uvw\mathbb{Z}_4$