DNA Codes over the Ring $\mathbb{Z}_4 + w\mathbb{Z}_4$
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 reversecomplement constraints. The technical contributions of this paper are twofold. Considering the Generalized Gau distance, Sphere Packinglike bound, GVlike bound, Singleton like bound and Plotkinlike bound are established over the rings $\mathcal{R}_\theta$. In addition to this, optimal class of codes are provided with respect to Singletonlike bound and Plotkinlike bound. Moreover, the construction of family of DNA codes is proposed that satisfies reverse and reversecomplement constraints using the ReedMuller type codes over the rings $\mathcal{R}_\theta$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.09089
 Bibcode:
 2021arXiv211009089A
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Emerging Technologies;
 Mathematics  Rings and Algebras
 EPrint:
 32 pages