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.
翻译:在本文中,我们提出了一个DNA代码的新设计战略,其长度在非链环上为3美元(Rämathbb*4+u\mathbb*4+u2\mathbb}4美元,其中元素64美元,元素64美元,元3=1美元,其中元3=1美元表示代码的长度超过美元。我们首先研究并分析在环上定义的远距离保护地图,以3美元的DNA序列计算。然后,我们从线性代码的生成方格中得出一些条件,其值超过$,导致DNA代码具有可逆、可逆和可逆的组合,同质聚合物2美元运行长,和美元(frac{w}3美元-GC含量限制值为$0leq wleq 3n美元)。最后,我们提议使用Reed-Muller型发电机矩阵来构建新的DNA代码。这使我们能够获得DNA代码,并具有可逆、可逆、可逆、可逆合成、可逆的聚合物2美元运行长度和可逆的制约值3美元。