A class of repeated-root constacyclic codes over ${\mathbb{F}}_{p^m}[u]\mathrm{/}⟨u^e⟩$ of Type $2$

Let ${\mathbb{F}}_{p^m}$ be a finite field of cardinality $p^m$ where $p$ is an odd prime, $n$ be a positive integer satisfying $\mathrm{g}\mathrm{c}\mathrm{d}(n,p)=1$, and denote $R={\mathbb{F}}_{p^m}[u]\mathrm{/}⟨u^e⟩$ where $e\ge 4$ be an even integer. Let $\delta ,\alpha \in {\mathbb{F}}_{p^m}^{\times }$. Then the class of $(\delta +\alpha u^2)$-constacyclic codes over $R$ is a significant subclass of constacyclic codes over $R$ of Type 2. For any integer $k\ge 1$, an explicit representation and a complete description for all distinct $(\delta+\alpha u^2)$-constacycliccodesover$R$oflength$np^k$ and their dual codes are given. Moreover, formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct $(\delta +\alpha u^2)$-contacyclic codes over $$\mathbb{F}$_{p^m}[u]/\langle u^{e}\rangle$oflength$p^k$ and their dual codes are presented precisely.