Linear Codes over the Ring Z4+uZ4+vZ4+wZ4+uvZ4+uwZ4+vwZ4+uvwZ4

We investigate linear codes over the ring (Formula presented), with conditions u² = u, v ² = v, w² = w, uv = vu, uw = wu and vw = wv. We first analyze the structure of the ring and then define linear codes over this ring. The Lee weight and the Gray map for these codes are defined and MacWilliams relations for complete, symmetrized, and Lee weight enumerators are derived. The Singleton bound as well as maximum distance separable codes are also considered. Furthermore, cyclic and quasi-cyclic codes are discussed, and as an application some new linear codes over Z4 with the highest known minimum Lee distance are also obtained.