The cyclic codes of length $5p^s$ over $\mathbb{F}_{p^m}+u \mathbb{F}_{p^m}$ and their dual codes

Abstract

Let $p$ be a prime integer and $m$ be an integer such that $p \equiv 2\pmod 5$ or $p \equiv \pmod 5$ and $m$ is odd. We classify explicitly the cyclic codes of length $5p^s$ over $R= \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ with $u^2=0$ and we compute completely their dual codes.