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

Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides Z2Z4-additive codes, Z2Z2[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 Z2Z2[u3] where Z2[u3]={0,1,u,1+u,u2,1+u2,u+u2,1+u+u2} 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 Z2Z2[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 Z2Z2[u3]. Further, we present some examples of optimal binary codes which are obtained through Gray images of Z2Z2[u3]-cyclic codes.