${\mathbb{Z}}_q({\mathbb{Z}}_q+u{\mathbb{Z}}_q)$-Linear Skew Constacyclic Codes

Abstract

In this paper, we study skew constacyclic codes over the ring ${\mathbb{Z}}_{q}R$ where $R={\mathbb{Z}}_q+u{\mathbb{Z}}_q$, $q=p^s$ for a prime $p$ and $u^2=0$. We give the definition of these codes as subsets of the ring ${\mathbb{Z}}_q^{\alpha }{R}^{\beta }$. Some structural properties of the skew polynomial ring $R[x,\theta ]$ are discussed, where $\theta$ is an automorphism of $R$. We describe the generator polynomials of skew constacyclic codes over $R$ and ${\mathbb{Z}}_{q}R$. Using Gray images of skew constacyclic codes over ${\mathbb{Z}}_{q}R$ we obtained some new linear codes over ${\mathbb{Z}}_4$. Further, we have generalized these codes to double skew constacyclic codes over ${\mathbb{Z}}_{q}R$.