DNA Cyclic Codes Over The Ring $ \F_2[u,v]/\langle u^2-1,v^3-v,uv-vu \rangle$

TL;DR

This paper explores the structure of cyclic DNA codes over a specific ring, establishing links to codons and analyzing reverse constraints, with implications for DNA computing.

Contribution

It introduces a Gray map linking ring elements to codons and studies reverse and reverse-complement constraints for DNA codes over the ring.

Findings

Established a Gray map from ring R to R1 for DNA code analysis.

Analyzed reverse and reverse-complement constraints over R and R1.

Provided examples of DNA codes obtained via the Gray map.

Abstract

In this paper, we mainly study the some structure of cyclic DNA codes of odd length over the ring $R = \F_2[u,v]/\langle u^2-1,v^3-v,uv-vu \rangle$ which play an important role in DNA computing. We established a direct link between the element of ring $R$ and 64 codons by introducing a Gray map from $R$ to $R_1 = F_2 + uF_2, u^2 = 1$ where $R_1$ is the ring of four elements. The reverse constrain and the reverse-complement constraint codes over $R$ and $R_1$ are studied in this paper. Binary image of the cyclic codes over R also study. The paper concludes with some example on DNA codes obtained via gray map.