$\mathbb{Z}_{q}(\mathbb{Z}_{q}+u\mathbb{Z}_{q})-$ 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 $\mathbb{Z}_{q}R,$ also we determine their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic codes over $\mathbb{Z}_{q}R$ we obtained some new linear codes over $\mathbb{Z}_{4}$. Finally, we have generalized these codes to double skew constacyclic codes over $\mathbb{Z}_{q}R$.