Repeated-root constacyclic codes over the finite chain ring ${\mathbb{F}}_{{\mathbf{p}}^{\mathbf{m}}}[\mathbf{u}]\mathbf{/}⟨{\mathbf{u}}^{\mathbf{3}}⟩$

Let $\mathcal{R}={\mathbb{F}}_{p^m}[u]\mathrm{/}⟨u^3⟩$ be the finite commutative chain ring with unity, where $p$ is a prime, $m$ is a positive integer and ${\mathbb{F}}_{p^m}$ is the finite field with $p^m$ elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over $\mathcal{R},$ their sizes and their dual codes. As an application, we list some isodual constacyclic codes over $\mathcal{R}\mathrm{.}$ We also determine Hamming distances, RT distances, and RT weight distributions of some repeated-root constacyclic codes over $\mathcal{R}\mathrm{.}$