In this paper, we study the structure of double cyclic codes of length $(γ,δ)$ over $\mathbb F_q+u\mathbb F_q, u^2=0$. We also study the dual of double cyclic code of length $(γ,δ)$ and give a minimal spanning set of double cyclic codes. Moreover, we study the necessary and sufficient conditions for a double cyclic code to be reversible and reversible-complement double cyclic code and with the help of these codes, we constructed DNA codes over $\mathbb F_4+u\mathbb F_4, u^2=0$. We also constructed some optimal codes to support our results.

翻译：本文研究了域$\mathbb F_q+u\mathbb F_q(u^2=0)$上长度为$(γ,δ)$的双循环码的结构。我们同时研究了长度为$(γ,δ)$的双循环码的对偶码，并给出了双循环码的最小生成集。此外，我们探讨了双循环码成为可逆码和可逆补双循环码的充要条件，并借助这些码在$\mathbb F_4+u\mathbb F_4(u^2=0)$上构造了DNA码。我们还构造了一些最优码来支撑我们的结论。