On self-duality and hulls of cyclic codes over $\frac{\mathbb{F}_{2^m}[u]}{\langle u^k\rangle}$ with oddly even length

Let $\mathbb{F}{2^m}$ be a finite field of $2^m$ elements, and $R=\mathbb{F}{2^{m}[u]/\langle} u^{k\rangle=\mathbb{F}{2m}+u\mathbb{F}{2m}+\ldots+u{k-1}\mathbb{F}_{2m}$} ($u^k=0$) where $k$ is an integer satisfying $k\geq 2$. For any odd positive integer $n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$ and a mass formula to count the number of these codes are given first. Then a generator matrix is provided for the self-dual and $2$-quasi-cyclic code of length $4n$ over $\mathbb{F}{2^m}$ derived by every self-dual cyclic code of length $2n$ over $\mathbb{F}{2^{m}+u\mathbb{F}_{2m}$} and a Gray map from $\mathbb{F}{2^{m}+u\mathbb{F}}{2^m}$ onto $\mathbb{F}{2^m}2$. Finally, the hull of each cyclic code with length $2n$ over $\mathbb{F}{2^{m}+u\mathbb{F}_{2m}$} is determined and all distinct self-orthogonal cyclic codes of length $2n$ over $\mathbb{F}{2^{m}+u\mathbb{F}}{2^m}$ are listed.