Identifying cyclic and (1 + 2v)-constacyclic codes over Z?[v]/ <v³ – 1> with Z?-linear codes

Abstract

This paper studies cyclic and (1 + 2v)-constacyclic codes over the ring Z?[v]/<v³ – 1>. By introducing three different Gray maps, we show that the Gray images of cyclic codes are quasi-cyclic codes over Z? and that of (1 + 2v)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over Z?. Moreover, we show that the Gray image of skew (1 + 2v)-constacyclic code is a quasi-cyclic code over Z?.