A note on “H. Q. Dinh et al., Hamming distance of repeated-root constacyclic codes of length $2p^{s}$ over ${\mathbb{F}}_{p^{m}}+ u{\mathbb{F}}_{p^{m}}$”

Abstract

Let ${\mathcal{R}}$ be the finite chain ring ${\mathcal{R}}={\mathbb{F}}_{p^{m}}+ u{\mathbb{F}}_{p^{m}}(u^{2} = 0)$, where p is an odd prime number and m is a positive integer. For $\eta \in {\mathbb{F}}_{p^{m}}^{*}$, the Hamming distances of all $\eta$-constacyclic codes of length $2p^{s}$ over ${\mathcal{R}}$ had already been studied in Dinh et al. (in AAECC, 2020. https://doi.org/10.1007/s00200-020-00432-0). However, such a study is incomplete. In this paper, we provide corrections to some results that appeared in Dinh et al. (2020) and we completely solve the problem of determination of the Hamming distance of $\eta$-constacyclic codes of length $2p^{s}$ over ${\mathcal{R}}$.