JOURNAL OF APPLIED MATHEMATICS AND COMPUTING, 2021 (Journal Indexed in SCI)
Z(2)Z(4)-additive codes have been defined as a subgroup of Zr-2 xZ(4)(s) in [6] where Z(2), Z(4) are the rings of integers modulo 2 and 4 respectively and r and s are positive integers. In this study, we define a family of codes over the set Z(2)[(xi) over bar](r) x Z(4)[xi](s) where xi is a root of a monic basic primitive polynomial in Z(4)[x]. We give the standard form of the generator and parity-check matrices of codes over Z(2)[(xi) over bar](r) x Z(4)[xi](s) and also we introduce skew cyclic codes and their spanning sets. Moreover, we study the Gray images of codes over both Z(4)[xi] and Z(2)[(xi) over bar](r) x Z(4)[xi](s) with respect to homogeneous weight and give the necessary and sufficient condition for their Gray images to be a linear code. We further present some examples of optimal codes which are actually Gray images of skew cyclic codes.