Linear codes over the ring Z_4 + uZ_4 + vZ_4 + wZ_4 + uvZ_4 + uwZ_4 + vwZ_4 + uvwZ_4

04/25/2019

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by   Bustomi, et al.

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We investigate linear codes over the ring Z_4 + uZ_4 + vZ_4 + wZ_4 + uvZ_4 + uwZ_4 + vwZ_4 + uvwZ_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.