A class of constacyclic codes over ${\mathbb{F}}_{p^m}[u]\mathrm{/}<u^2>$

Let $p$ be an odd prime, and let $m$ be a positive integer satisfying $p^m \equiv 3~($\text{mod }$4).$Let$$\mathbb{F}${p^m}$bethefinitefieldwith$p^m$elements,andlet$R=$\mathbb{F}${p^m}[u]/\left<u^2\right>$ be the finite commutative chain ring with unity. In this paper, we determine all constacyclic codes of length $4p^s$ over $R$ and their dual codes, where $s$ is a positive integer. We also determine their sizes and list some isodual constacyclic codes of length $4p^s$ over $R\mathrm{.}$