skew cyclic codes over $mathbb{z}_4+vmathbb{z}_4$ with derivation: structural properties and computational results

In this work, we study a class of skew cyclic codes over the ring $r:=mathbb{z}_4+vmathbb{z}_4,$ 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.