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

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 $\mathbb{Z}_{q}R$.