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

Abstract

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 $\hat{I}¸$ and a derivation $\hat{I}{”}_{\hat{I}}¸,$ namely codes as modules over a skew polynomial ring $R[x;\hat{I}¸,\hat{I}{”}_{\hat{I}}¸],$ whose multiplication is defined using an automorphism $\hat{I}¸$ and a derivation $\hat{I}{”}_{\hat{I}}¸.$ We investigate the structures of a skew polynomial ring $R[x;\hat{I}¸,\hat{I}{”}_{\hat{I}}¸].$ We define $\hat{I}{”}_{\hat{I}}¸$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $\hat{I}{”}_{\hat{I}}¸$-cyclic codes as well as dual $\hat{I}{”}_{\hat{I}}¸$-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.

25 pages, Communications in Combinatorics and Optimization (accepted)