Skew cyclic codes over $\mathbb{Z}_4+v\mathbb{Z}_4$ with derivation: structural properties and computational results
Djoko Suprijanto, Hopein Christofen Tang
TL;DR
This work extends cyclic codes to skew cyclic codes with a derivation over the ring $R=\mathbb{Z}_4+v\mathbb{Z}_4$, developing the algebraic framework of $R[x;\theta,\Delta_{\theta}]$ and introducing $\Delta_{\theta}$-cyclic codes as left submodules of $R_{n,\Delta_{\theta}}$. It derives structural results, including generator and parity-check constructions, and shows that for even length the duals of free codes are free; odd length yields cyclic behavior while even length yields quasi-cyclic index $2$. The paper further translates these ring-theoretic codes into practical linear codes over $\mathbb{Z}_4$ via Gray maps, residue, and torsion codes, and demonstrates the production of many new codes with good parameters using Plotkin sums and related methods, validated by computational experiments in Python and Magma. Overall, it broadens the landscape of $\mathbb{Z}_4$-based coding by leveraging derivations in skew polynomial rings to obtain high-quality codes with potential applications in communication and data storage.
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 $θ$ and a derivation $Δ_θ,$ namely codes as modules over a skew polynomial ring $R[x;θ,Δ_θ],$ whose multiplication is defined using an automorphism $θ$ and a derivation $Δ_θ.$ We investigate the structures of a skew polynomial ring $R[x;θ,Δ_θ].$ We define $Δ_θ$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $Δ_θ$-cyclic codes as well as dual $Δ_θ$-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.