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

TL;DR

This paper explores skew cyclic codes over a specific ring with derivation, analyzing their structure and properties, and constructs new linear codes over  with good parameters using algebraic methods.

Contribution

It introduces elta-yclic codes over , studies their algebraic properties, and provides new linear code constructions with improved parameters.

Findings

Structural properties of skew polynomial rings are characterized.

Dual elta-yclic codes are explicitly described.

New linear codes over  with good parameters are constructed.

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 $\theta$ 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.