  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.
