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A class of repeated-root constacyclic codes over $\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ of Type $2$ (1805.05595v2)

Published 15 May 2018 in cs.IT and math.IT

Abstract: Let $\mathbb{F}{pm}$ be a finite field of cardinality $pm$ where $p$ is an odd prime, $n$ be a positive integer satisfying ${\rm gcd}(n,p)=1$, and denote $R=\mathbb{F}{pm}[u]/\langle ue\rangle$ where $e\geq 4$ be an even integer. Let $\delta,\alpha\in \mathbb{F}{pm}{\times}$. Then the class of $(\delta+\alpha u2)$-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 $(\delta+\alpha u2)$-constacyclic codes over $R$ of length $npk$ 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 $(\delta+\alpha u2)$-contacyclic codes over $\mathbb{F}{pm}[u]/\langle u{e}\rangle$ of length $pk$ and their dual codes are presented precisely.

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