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Coterm polynomials are introduced by Oztas et al. [a novel approach forconstructing 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 thecoterm polynomials and construct some reversible codes which are optimal codesby 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.

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