On ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}[\xi ]$-skew cyclic codes

Abstract

${\mathbb {Z}}_2{\mathbb {Z}}_{4}$-additive codes have been defined as a subgroup of ${\mathbb {Z}}_2^{r}\times {\mathbb {Z}}_4^{s}$ in [6] where ${\mathbb {Z}}_2$, ${\mathbb {Z}}_{4}$ 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 ${\mathbb {Z}}_2[{\bar{\xi }}]^{r}\times {\mathbb {Z}}_4[\xi ]^{s}$ where $\xi$ is a root of a monic basic primitive polynomial in ${\mathbb {Z}}_{4}[x]$. We give the standard form of the generator and parity-check matrices of codes over ${\mathbb {Z}}_2[{\bar{\xi }}]^{r}\times {\mathbb {Z}}_4[\xi ]^{s}$ and also we introduce skew cyclic codes and their spanning sets. Moreover, we study the Gray images of codes over both ${{\mathbb {Z}}}_4[\xi ]$ and ${{\mathbb {Z}}_{2}[{\bar{\xi }}]^r\times {{\mathbb {Z}}_{4}[\xi ]^s}}$ 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.