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

TL;DR

This paper introduces a new method for designing DNA codes over a specific non-chain ring, ensuring various biological constraints, using Reed-Muller type matrices for improved code properties.

Contribution

It presents a novel design strategy for DNA codes over a non-chain ring, including a distance-preserving map and Reed-Muller based constructions for multiple biological constraints.

Findings

Successfully constructed DNA codes with multiple biological constraints.

Derived conditions on generator matrices for desired code properties.

Achieved codes with 2/3 GC-content using Reed-Muller matrices.

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 $w$ ($0\leq w\leq 3n$). 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.