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

Abstract: Let $\mathbb{F}<em>{p<sup>m}$ be a finite field of cardinality $p<sup>m$ where $p$ is an odd prime, $n$ be a positive integer satisfying ${\rm gcd}(n,p)=1$, and denote $$R=\mathbb{F}</em>{p^{m}[u]/\langle} u<sup>e\rangle$$ where $e\geq 4$ be an even integer. Let $$\delta,\alpha\in \mathbb{F}<em>{p<sup>m}<sup>{\times}$$. Then the class of $(\delta+\alpha u<sup>2)$-constacyclic codes over $R$ is a significant subclass of constacyclic codes over $R$ of Type 2. For any integer $k\geq 1$, an explicit representation and a complete description for all distinct $(\delta+\alpha u<sup>2)$-constacyclic codes over $R$ of length $np<sup>k$ 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 $(\delta+\alpha u<sup>2)$-contacyclic codes over $$\mathbb{F}</em>{p^{m}[u]/\langle} u<sup>{e}\rangle$$ of length $p<sup>k$ and their dual codes are presented precisely.