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

DNA Codes Over the Ring F₄[U]/〈U³〉