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

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