A Note on Complete Classification of Repeated-Root \(\sigma\)-Constacyclic Codes of Prime Power Length over \(\mathbb{F}_{p^m}[u]/\langle u^3 \rangle\) and their Hamming Distances

This note presents a counterexample to Propositions 7 and 8 in the paper [1], where the authors determine the values of \( V \) and \( W \). These values are crucial in determining the Hamming distance and MDS codes in the family of certain costacyclic codes over \(\mathbb{F}_{p^m}[u]/\langle u^3 \rangle\), which implies that the results found in [2] are incorrect. Furthermore, we provide corrections to the aforementioned results.