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Let $p$ be an odd prime number, $\mathbb{F}_{p^m}$ be a finite field ofcardinality $p^m$ and $s$ a positive integer. Using some combinatorialidentities, we obtain certain properties for Kronecker product of matrices over$\mathbb{F}_p$ with a specific type. On that basis, we give an explicitrepresentation and enumeration for all distinct self-dual cyclic codes oflength $p^s$ over the finite chain ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$$(u^2=0)$. Moreover, We provide an efficient method to construct everyself-dual cyclic code of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$precisely.

An efficient method to construct self-dual cyclic codes of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$