Reversible double cyclic codes over a chain ring

2026-06-04

AI Summary

This study investigates double cyclic codes of length $(\gamma, \delta)$ over the finite chain ring $\mathbb{F}_q + u\mathbb{F}_q$ (where $u^2 = 0$) and their duals, with a focus on applications to DNA coding. By leveraging the algebraic structure of cyclic codes over finite chain rings, the work establishes, for the first time, necessary and sufficient conditions for reversibility and reverse-complementarity of such codes and explicitly determines their minimal generating sets. Building on these theoretical results, several optimal DNA codes are constructed over $\mathbb{F}_4 + u\mathbb{F}_4$, demonstrating the efficacy of the proposed approach. This research not only extends the applicability of algebraic coding theory to bioinformatics but also provides concrete examples of optimal codes that substantiate the theoretical findings.

Abstract

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.

Problem

• double cyclic codes
• reversible codes
• reversible-complement codes
• DNA codes
• chain ring

Innovation

• double cyclic codes
• reversible codes
• DNA codes
• chain ring
• minimal spanning set