  Let $R=\mathbb{Z}_{4}[u]/\langle
u^k\rangle=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+\ldots+u^{k-1}\mathbb{Z}_{4}$
($u^k=0$) where $k\in \mathbb{Z}^{+}$ satisfies $k\geq 2$. For any odd positive
integer $n$, it is known that cyclic codes over $R$ of length $n$ are
identified with ideals of the ring $R[x]/\langle x^{n}-1\rangle$. In this
paper, an explicit representation for each cyclic code over $R$ of length $n$
is provided and a formula to count the number of codewords in each code is
given. Then a formula to calculate the number of cyclic codes over $R$ of
length $n$ is obtained. Precisely, the dual code of each cyclic code and
self-dual cyclic codes over $R$ of length $n$ are investigated. When $k=4$,
some optimal quasi-cyclic codes over $\mathbb{Z}_{4}$ of length $28$ and index
$4$ are obtained from cyclic codes over $R=\mathbb{Z}_{4} [u]/\langle
u^4\rangle$.
