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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 thegenerator matrix of a linear code over $R$, which leads to a DNA code withreversible, 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 typegenerator matrices. This allows us to obtain DNA codes with reversible,reversible-complement, homopolymer $2$-run-length, and $\frac{2}{3}$-GC-contentconstraints.

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