On DNA Codes Over the Non-Chain Ring ${\mathbb{Z}}_4+u{\mathbb{Z}}_4+u^2{\mathbb{Z}}_4$ with $u^3=1$

Abstract

In this paper, we present a novel design strategy of DNA codes with length $3n$ over the non-chain ring $R={\mathbb{Z}}_4+u{\mathbb{Z}}_4+u^2{\mathbb{Z}}_4$ with $64$ elements and $u^3=1$, where $n$ denotes the length of a code over $R$. We first study and analyze a distance conserving map defined over the ring $R$ into the length-$3$ DNA sequences. Then, we derive some conditions on the generator matrix of a linear code over $R$, which leads to a DNA code with reversible, reversible-complement, homopolymer $2$-run-length, and $\frac{w}{3n}$-GC-content constraints for integer $w$ ($0\le w\le 3n$). Finally, we propose a new construction of DNA codes using Reed-Muller type generator matrices. This allows us to obtain DNA codes with reversible, reversible-complement, homopolymer $2$-run-length, and $\frac{2}{3}$-GC-content constraints.

This paper has been presented in IEEE Information Theory Workshop (ITW) 2022, Mumbai, INDIA