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

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