Z2Z2[u]-Cyclic and Constacyclic Codes

Abstract

Following the very recent studies on Z(2)Z(4)-additive codes, Z(2)Z(2)[u]-linear codes have been introduced by Aydogdu et al. In this paper, we introduce and study the algebraic structure of cyclic, constacyclic codes and their duals over the R-module Z(2)(alpha)R(beta) where R = Z(2)+uZ(2) = {0, 1, u, u + 1} is the ring with four elements and u(2) = 0. We determine the generating independent sets and the types and sizes of both such codes and their duals. Finally, we present a bound and an optimal family of codes attaining this bound and also give some illustrative examples of binary codes that have good parameters which are obtained from the cyclic codes in Z(2)(alpha)R(beta).