Construction and enumeration for self-dual cyclic codes of even length over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$

Let ${\mathbb{F}}_{2^m}$ be a finite field of cardinality $2^m$, $R={\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ $(u^2=0)$ and $s,n$ be positive integers such that $n$ is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring $R$ of length $2^{s}n$ and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and $2$-quasi-cyclic codes over ${\mathbb{F}}_{2^m}$ of length $2^{s+1}n$ can be obtained from self-dual cyclic code over $R$ of length $2^{s}n$ and by a Gray map preserving orthogonality and distances from $R$ onto ${\mathbb{F}}_{2^m}^2$.