On the structure of Z2Z2[u3]-linear and cyclic codes

Abstract

Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides Z(2)Z(4)-additive codes, Z(2)Z(2)[u]-linear codes are introduced as a new member of such families. In this paper, we are interested in a new family of such mixed alphabet codes, i.e., codes over Z(2)Z(2) [u(3)] where Z(2) [U-3] = {0, 1, u,1 + u, u(2), 1 + U-2, U + u(2),1 u u(2)} is an 8-element ring with u3 = 0. We study and determine the algebraic structures of linear and cyclic codes defined over this family. First, we introduce Z(2)Z(2)[u3]-linear codes and give standard forms of generator and parity-check matrices and later we present generators of both cyclic codes and their duals over Z(2)Z(2)[u(3)]. Further, we present some examples of optimal binary codes which are obtained through Gray images of Z(2)Z(2)[u(3)]-cyclic codes. (C) 2017 Published by Elsevier Inc.