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 skew polynomial ring $R[x;\theta,\Delta_{\theta}],$ whose multiplication is defined using an automorphism $\theta$ and a derivation $\Delta_{\theta}.$ We investigate the structures of a skew polynomial ring $R[x;\theta,\Delta_{\theta}].$ We define $\Delta_{\theta}$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $\Delta_{\theta}$-cyclic codes as well as dual $\Delta_{\theta}$-cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.

翻译：本文研究环$R:=\mathbb{Z}_4+v\mathbb{Z}_4$（其中$v^2=v$）上的一类斜循环码，该环配备自同构$\theta$与导子$\Delta_\theta$，即作为斜多项式环$R[x;\theta,\Delta_{\theta}]$上的模的码，其乘法由自同构$\theta$和导子$\Delta_{\theta}$定义。我们探讨了斜多项式环$R[x;\theta,\Delta_{\theta}]$的结构，将$\Delta_{\theta}$-循环码定义为循环码概念的推广，并推导了$\Delta_{\theta}$-循环码及其对偶$\Delta_{\theta}$-循环码的性质。作为应用，通过Plotkin和构造、Gray映射以及这些码的剩余码与挠码，获得了一些具有良好参数的$\mathbb{Z}_4$上新线性码。