On the structure of Z(2)Z(2)[u(3)]-linear and cyclic codes

Aydoğdu İ., Siap I., Ten-Valls R.

FINITE FIELDS AND THEIR APPLICATIONS, vol.48, pp.241-260, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 48
  • Publication Date: 2017
  • Doi Number: 10.1016/j.ffa.2017.03.001
  • Journal Name: FINITE FIELDS AND THEIR APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.241-260
  • Keywords: Linear codes, Cyclic codes, Z(2)Z(2)[u(3)]-linear cyclic codes, Duality
  • Yıldız Technical University Affiliated: Yes

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.