DNA Codes Over the Ring F₄[U]/〈U³〉

In this paper, we develop the method for constructing DNA codes of odd length over the finite chain ring $R=\mathbb {F}_{4}[u]/\langle u^{3}\rangle $ , which plays an important role in genetics, bioengineering and DNA computing. By using a Gray map $\eta $ from $R$ to $\mathbb {F}_{4}$ , we give a one-to-one correspondence between 64 DNA codons of the alphabet $\{A, T, G,C\}^{3}$ and the 64 elements of the chain ring $R$ . We also establish a map $\Theta $ from $R^{n}$ to $\{A, T, G,C\}^{3n}$ . Then we provide a necessary and sufficient condition for cyclic codes of odd length over the chain ring $R$ to be reversible. Finally, when $C$ is a reversible cyclic code over $R$ , we derive a necessary and sufficient condition for $\Theta (C)$ to be a DNA code.