Reversible cyclic Codes Over $\mathbb F_q+ u \mathbb F_q+ u^2\mathbb F_q+ u^3\mathbb F_q$ and applications to DNA codes

Abstract

In this work, we investigate the structure of reversible and reversible-complement cyclic codes of length $n$ over the ring $\mathbf{R}=\mathbb F_q+ u F_q+ u^2\mathbb F_q+ u^3\mathbb F_q$, $ u^4=0$, when $n\text{ and }q$ are coprime. We give the necessary and sufficient conditions for the cyclic codes of length $n$ over $\mathbf{R}$ to be reversible and reversible-compliment cyclic codes, when $(n,q)=1$ over $\mathbf{R}$. We also studied the dual cyclic codes over $\mathbf{R}$ when $(n,q)=1$ and obtain the reversibility condition for dual cyclic codes. Additionally, we explore cyclic DNA codes over nucleotide 4-base pair. First, we establish a one-to-one correspondence between $\mathbf{R_{1}}$, where $\mathbf{R_{1}}=\mathbb F_4+ u F_4+ u^2\mathbb F_4+ u^3\mathbb F_4$, $ u^4=0$ and 4-mers and then cyclic DNA codes constructed as the images of reversible-complement cyclic codes over $\mathbf{R_{1}}$.