Cyclic codes over ${\mathbb{F}}_{2^m}[u]\mathrm{/}⟨u^k⟩$ 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}${2^m} +u$\mathbb{F}${2^m}+\ldots+u^{k-1}$\mathbb{F}${2^m}$($u^k=0$)where$k\in $\mathbb{Z}$^{+}$satisfies$k\geq 2$\mathrm{.}Foranyoddpositiveinteger$n$, it is known that cyclic codes over $R$ of length $2n$ are identified with ideals of the ring $R[x]\mathrm{/}⟨x^{2n}-1⟩$. 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)