On Z2Z4[\xi]-Skew Cyclic Codes

TL;DR

This paper introduces a new family of skew cyclic codes over a combined ring structure involving Z2 and Z4 with a primitive polynomial, providing generator and parity-check matrix forms.

Contribution

It defines skew cyclic codes over Z2^{r}[ar{\xi}] x Z4^{s}[\xi], extending previous Z2Z4-additive codes and detailing their algebraic structure and spanning sets.

Findings

Standard forms of generator matrices derived

Parity-check matrices formulated

Introduction of skew cyclic codes over new ring structure

Abstract

Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and s positive integers. In this study, we define a new family of codes over the set Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] where \xi is the root of a monic basic primitive polynomial in Z4[x]. We give the standard form of the generator and parity-check matrices of codes over Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] and also we introduce skew cyclic codes and their spanning sets over this set.