Reversible and Reversible-Complement Double Cyclic Codes over F4+vF4 and its Application to DNA Codes

In this article, we study the algebraic structure of double cyclic codes of length $(m, n)$ over $\mathbb{F}_4$ and we give a necessary and sufficient condition for a double cyclic code over $\mathbb{F}_4$ to be reversible. Also, we determine the algebraic structure of double cyclic codes of length $(m, n)$ over $\mathbb{F}4+v\mathbb{F}4$ with $v^2=v$, satisfying the reverse constraint and the reverse-complement constraint. Then we establish a one-to-one correspondence $ψ$ between the 16 DNA double pairs $S{D{16}} $ and the 16 elements of the finite ring $\mathbb{F}_4+v\mathbb{F}_4$. We also discuss the GC-content of DNA double cyclic codes.

💡 Research Summary

This paper investigates the algebraic structure of double cyclic codes of length (m, n) over the finite field F₄ and over the ring R = F₄ + vF₄ with the idempotent relation v² = v, and it explores their applications to DNA coding.

First, the authors recall that a double cyclic code C ⊂ F₄^m × F₄^n can be identified with a submodule of A_{m,n}=F₄