Cyclic and Quasi-Cyclic DNA Codes

TL;DR

This paper explores the construction and properties of cyclic and quasi-cyclic DNA codes over specific rings, generalizing previous results and establishing relationships between Gau and Hamming distances for these codes.

Contribution

It generalizes the construction of DNA codes over sixteen rings and relates Gau and Hamming distances, advancing coding theory over algebraic structures.

Findings

Established a relationship between Gau and Hamming distances for codes over sixteen rings.

Provided methods to construct DNA codes with combinatorial constraints.

Determined upper bounds for Gau distances of self-dual codes.

Abstract

In this paper, we discuss DNA codes that are cyclic or quasi-cyclic over $\Z_{4}+\omega \Z_{4}$, where $\omega^{2}=2+2\omega$ along with methods to construct these with combinatorial constraints. We also generalize results obtained for the ring $\Z_{4}+\omega \Z_{4}$, where $\omega^{2}=2+2\omega$, and some other rings to the sixteen rings $R_{\theta}=\Z_{4}+\omega \Z_{4}$, where $\omega^{2}=\theta\in \Z_{4}+\omega \Z_{4}$, using the generalized Gau map and Gau distance in \cite{3}. We determine a relationship between the Gau distance and Hamming distance for linear codes over the sixteen rings $R_{\theta}$ which enables us to attain an upper boundary for the Gau distance of free codes that are self-dual over the rings $R_{\theta}$.