Matrix-product structure of constacyclic codes over finite chain rings ${\mathbb{F}}_{p^m}[u]\mathrm{/}⟨u^e⟩$

Let $m,e$ be positive integers, $p$ a prime number, ${\mathbb{F}}_{p^m}$ be a finite field of $p^m$ elements and $R={\mathbb{F}}_{p^m}[u]\mathrm{/}⟨u^e⟩$ which is a finite chain ring. For any $\omega \in R^{\times }$ and positive integers $k,n$ satisfying $\mathrm{g}\mathrm{c}\mathrm{d}(p,n)=1$, we prove that any $(1+\omega u)$-constacycliccodeoflength$p^kn$over$R$ is monomially equivalent to a matrix-product code of a nested sequence of $p^k$ cyclic codes with length $n$ over $R$ and a $p^k\times p^k$ matrix $A_{p^k}$ over ${\mathbb{F}}_p$. Using the matrix-product structures, we give an iterative construction of every $(1+\omega u)$-constacyclic code by $(1+\omega u)$-constacyclic codes of shorter lengths over $R$.