Structure of codes over the ring \(Z_{3}[v]/\langle v^{3}-v\rangle \)
In this paper, we study the structure of linear codes over the non chain ring \(Z_{3}[v]/\langle v^{3}-v\rangle \). In order to study the codes, we first study the structure of this ring via a distance preserving Gray map which also induces a relation between codes over this ring and ternary codes. Further, the algebraic structure of cyclic and dual codes is also studied. A MacWilliams type Identity between the Gray weight enumerators of the original code and its dual is established. In all cases examples that illustrate the theorems and lemmas are provided. Also, a BCH-type bound and an example that attains this bound is presented.
linear codes non chain ring bch-type bound cyclic codes macwilliams identity