DNA codes from $(\text{\textbaro}, \mathfrak{d}, \gamma)$-constacyclic codes over $\mathbb{Z}_4+\omega\mathbb{Z}_4$

TL;DR

This paper introduces a new method for constructing DNA codes using $(\text{|o}, \mathfrak{d}, \gamma)$-constacyclic codes over a specific ring extension of \(\mathbb{Z}_4\), providing criteria for reverse and reverse-complement constraints and generating optimal codes.

Contribution

It develops a novel framework for DNA code construction from linear codes over a non-chain extension of \(\mathbb{Z}_4\), including generator determination, reverse constraints, and new optimal code examples.

Findings

Constructed DNA codes with better parameters than existing codes.

Established necessary and sufficient conditions for reverse-complement codes.

Generated new classical codes with optimal parameters.

Abstract

This work introduces a novel approach to constructing DNA codes from linear codes over a non-chain extension of $\mathbb{Z}_4$. We study $(\text{\textbaro},\mathfrak{d}, \gamma)$-constacyclic codes over the ring $\mathfrak{R}=\mathbb{Z}_4+\omega\mathbb{Z}_4, \omega^2=\omega,$ with an $\mathfrak{R}$-automorphism $\text{\textbaro}$ and a $\text{\textbaro}$-derivation $\mathfrak{d}$ over $\mathfrak{R}.$ Further, we determine the generators of the $(\text{\textbaro},\mathfrak{d}, \gamma)$-constacyclic codes over the ring $\mathfrak{R}$ of any arbitrary length and establish the reverse constraint for these codes. Besides the necessary and sufficient criterion to derive reverse-complement codes, we present a construction to obtain DNA codes from these reversible codes. Moreover, we use another construction on the $(\text{\textbaro},\mathfrak{d},\gamma)$-constacyclic codes to generate additional optimal and new classical codes. Finally, we provide several examples of $(\text{\textbaro},\mathfrak{d}, \gamma)$ constacyclic codes and construct DNA codes from established results. The parameters of these linear codes over $\mathbb{Z}_4$ are better and optimal according to the codes available at \cite{z4codes}.