Reversible Codes and Its Application to Reversible DNA Codes over $F_{4^k}$

TL;DR

This paper generalizes coterm polynomials to construct optimal reversible codes and applies these concepts to develop reversible DNA codes over the finite field extension $F_{4^k}$, enhancing coding theory and DNA computing applications.

Contribution

It introduces generalized $m$-quasi-reciprocal polynomials for constructing reversible codes and develops a new mapping for reversible DNA codes over $F_{4^k}$.

Findings

Constructed optimal reversible codes using generalized polynomials.

Developed a mapping from DNA bases to $F_{4^k}$ elements.

Created reversible DNA codes over $F_{4^k}$ with specific polynomial methods.

Abstract

Coterm polynomials are introduced by Oztas et al. [a novel approach for constructing reversible codes and applications to DNA codes over the ring $F_2[u]/(u^{2k}-1)$, Finite Fields and Their Applications 46 (2017).pp. 217-234.], which generate reversible codes. In this paper, we generalize the coterm polynomials and construct some reversible codes which are optimal codes by using $m$-quasi-reciprocal polynomials. Moreover, we give a map from DNA $k$-bases to the elements of $F_{4^k}$, and construct reversible DNA codes over $F_{4^k}$ by DNA-$m$-quasi-reciprocal polynomials.