Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$
Yonglin Cao, Yuan Cao, Yuan Cao, Yuan Cao, Hai Quang Dinh, Fang‐Wei Fu, Jian Gao, Songsak Sriboonchitta · Advances in Mathematics of Communications · 2018
Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $R = \mathbb{F}_{p^m}[u]/\langle u^2\rangle = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2 = 0)$, where $p$ is a prime and $m$ is a positive integer. For any $λ∈ \mathbb{F}_{p^m}^{×}$, an explicit representation for all distinct $λ$-constacyclic codes over $R$ of length $np^s$ is given by a canonical form decomposition for each code, where $s$ and $n$ are arbitrary positive integers satisfying ${\rm gcd}(p,n) = 1$. For any such code, using its canonical form decomposition the representation for the dual code of the code is provided. Moreover, representations for all distinct cyclic codes, negacyclic codes and their dual codes of length $np^s$ over $R$ are obtained, and self-duality for these codes are determined. Finally, all distinct self-dual negacyclic codes over $\mathbb{F}_5+u\mathbb{F}_5$ of length $2· 3^t· 5^s$ are listed for any positive integer $t$.