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Let $\mathbb{F}_{2^m}$ be a finite field of characteristic $2$ and$R=\mathbb{F}_{2^m}[u]/\langle u^k\rangle=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$ ($u^k=0$) where $k\in\mathbb{Z}^{+}$ satisfies $k\geq 2$. For any odd positive integer $n$, it isknown that cyclic codes over $R$ of length $2n$ are identified with ideals ofthe ring $R[x]/\langle x^{2n}-1\rangle$. In this paper, an explicitrepresentation for each cyclic code over $R$ of length $2n$ is provided and aformula to count the number of codewords in each code is given. Then a formulato calculate the number of cyclic codes over $R$ of length $2n$ is obtained.Moreover, the dual code of each cyclic code and self-dual cyclic codes over $R$of length $2n$ are investigated. (AAECC-1522)

Cyclic codes over $\mathbb{F}_{2^m}[u]/\langle u^k\rangle$ of oddly even length