  In this paper, we give a matrix construction method for designing DNA codes
that come from group matrix rings. We show that with our construction one can
obtain reversible $G^k$-codes of length $kn,$ where $k, n \in \mathbb{N},$ over
the finite commutative Frobenius ring $R.$ We employ our construction method to
obtain many DNA codes over $\mathbb{F}_4$ that satisfy the Hamming distance,
reverse, reverse-complement and the fixed GC-content constraints. Moreover, we
improve many lower bounds on the sizes of some known DNA codes and we also give
new lower bounds on the sizes of some DNA codes of lengths $48, 56, 60, 64$ and
$72$ for some fixed values of the Hamming distance $d.$
