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Let $R=\mathbb{Z}_{4}[u]/\langleu^k\rangle=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+\ldots+u^{k-1}\mathbb{Z}_{4}$($u^k=0$) where $k\in \mathbb{Z}^{+}$ satisfies $k\geq 2$. For any odd positiveinteger $n$, it is known that cyclic codes over $R$ of length $n$ areidentified with ideals of the ring $R[x]/\langle x^{n}-1\rangle$. In thispaper, 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 isgiven. Then a formula to calculate the number of cyclic codes over $R$ oflength $n$ is obtained. Precisely, the dual code of each cyclic code andself-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 $R=\mathbb{Z}_{4} [u]/\langleu^4\rangle$.

Cyclic codes over $\mathbb{Z}_4[u]/\langle u^k\rangle$ of odd length