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In this work, we study a class of skew cyclic codes over the ring$R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $\theta$and a derivation $\Delta_\theta,$ namely codes as modules over a skewpolynomial ring $R[x;\theta,\Delta_{\theta}],$ whose multiplication is definedusing an automorphism $\theta$ and a derivation $\Delta_{\theta}.$ Weinvestigate the structures of a skew polynomial ring$R[x;\theta,\Delta_{\theta}].$ We define $\Delta_{\theta}$-cyclic codes as ageneralization of the notion of cyclic codes. The properties of$\Delta_{\theta}$-cyclic codes as well as dual $\Delta_{\theta}$-cyclic codesare derived. As an application, some new linear codes over $\mathbb{Z}_4$ withgood parameters are obtained by Plotkin sum construction, also via a Gray mapas well as residue and torsion codes of these codes.

Skew cyclic codes over $\mathbb{Z}_4+v\mathbb{Z}_4$ with derivation: structural properties and computational results