  DNA codes based on error-correcting codes have been successful in DNA-based
computation and storage. Since there are four nucleobases in DNA, two well
known algebraic structures such as the finite field $GF(4)$ and the integer
modular ring $\mathbb{Z}_4$ have been used. However, due to various
possibilities of DNA sequences, it is natural to ask whether there are other
algebraic structures consisting of four elements.
  In this paper, we describe a new type of DNA codes over two noncommutative
rings $E$ and $F$ of order four with characteristic 2. Our DNA codes are based
on quasi self-dual codes over $E$ and $F$. Using quasi self-duality, we can
describe fixed GC-content constraint weight distributions and
reverse-complement constraint minimum distributions of those codes.
