  In this paper, we study the theory for constructing DNA cyclic codes of odd
length over $\Z_4[u]/\langle u^2 \rangle$ which play an important role in DNA
computing. Cyclic codes of odd length over $\Z_4 + u \Z_4$ satisfy the reverse
constraint and the reverse-complement constraint are studied in this paper. The
structure and existence of such codes are also studied. The paper concludes
with some DNA example obtained via the family of cyclic codes.
