Skew cyclic codes over $$\mathbb {Z}_{4}+u\mathbb {Z}_{4}+v\mathbb {Z}_{4}$$

In this paper, we study the skew-cyclic codes (also called $$\varvec{\theta }$$ -cyclic codes) over the ring $$\varvec{S}=\varvec{\mathbb {Z}}_{{\textbf {4}}}+\varvec{u}\mathbb {Z}_{{\textbf {4}}}+v\mathbb {Z}_{{\textbf {4}}}$$ , where $$\varvec{u}^2=v^{{\textbf {2}}}=\varvec{u}\varvec{v}=\varvec{v}\varvec{u}={\textbf {0}}$$ . Some structural properties of the skew polynomial ring $$\varvec{S}[\varvec{x},\varvec{\theta }]$$ , where $$\varvec{\theta }$$ is an automorphism of $$\varvec{S}$$ are discussed and the elements of $$\varvec{S}^{\varvec{\theta }}$$ , the subring of $$\varvec{S}$$ fixed by $$\varvec{\theta }$$ , are determined. Skew cyclic codes over $$\varvec{S}$$ are viewed as left $$\varvec{S}[\varvec{x},\varvec{\theta }]$$ -submodules. Generator and parity-check matrices of a free $$\varvec{\theta }$$ -cyclic code of even length over $$\varvec{S}$$ are determined and a Gray map on $$\varvec{S}$$ is used to obtain the $$\mathbb {Z}_{{\textbf {4}}}$$ -images. We show that the Gray image of a free skew cyclic code over $$\varvec{S}$$ is a free linear code over $$\mathbb {Z}_{{\textbf {4}}}$$ . Furthermore, these codes are generalized to double skew-cyclic codes. We obtained new linear codes over $$\mathbb {Z}_{{\textbf {4}}}$$ from Gray images of double skew-cyclic codes over $$\varvec{S}$$ .