  Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$ and $s$ a
positive integer. Using properties for Kronecker product of matrices and
calculation for linear equations over $\mathbb{F}_{2^m}$, an efficient method
for the construction of all distinct self-dual cyclic codes with length $2^s$
over the finite chain ring $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ is
provided. On that basis, an explicit representation for every self-dual cyclic
code of length $2^s$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ and an exact
formula to count the number of all these self-dual cyclic codes are given.
