DNA Codes over the Ring $\mathbb{Z}_4 + w\mathbb{Z}_4$

TL;DR

This paper generalizes the construction and analysis of DNA codes over rings 4 + w4, introducing new bounds, code classes, and mappings, and explores their properties including duality and reverse constraints.

Contribution

It extends the Gau map and distance to all rings 4 + w4, establishes isometries, and constructs optimal DNA codes satisfying reverse and reverse-complement constraints.

Findings

Established bounds like Sphere Packing, GV, Singleton, and Plotkin over the rings.

Constructed optimal codes with respect to these bounds.

Proposed DNA code families satisfying reverse and reverse-complement constraints.

Abstract

In this present work, we generalize the study of construction of DNA codes over the rings $\mathcal{R}_\theta=\mathbb{Z}_4+w\mathbb{Z}_4$, $w^2 = \theta $ for $\theta \in \mathbb{Z}_4+w\mathbb{Z}_4$. Rigorous study along with characterization of the ring structures is presented. We extend the Gau map and Gau distance, defined in \cite{DKBG}, over all the $16$ rings $\mathcal{R}_\theta$. Furthermore, an isometry between the codes over the rings $\mathcal{R}_\theta$ and the analogous DNA codes is established in general. Brief study of dual and self dual codes over the rings is given including the construction of special class of self dual codes that satisfy reverse and reverse-complement constraints. The technical contributions of this paper are twofold. Considering the Generalized Gau distance, Sphere Packing-like bound, GV-like bound, Singleton like bound and Plotkin-like bound are established over the rings $\mathcal{R}_\theta$. In addition to this, optimal class of codes are provided with respect to Singleton-like bound and Plotkin-like bound. Moreover, the construction of family of DNA codes is proposed that satisfies reverse and reverse-complement constraints using the Reed-Muller type codes over the rings $\mathcal{R}_\theta$.