A note on complete classification of $(\delta +\alpha u^2)$-constacyclic codes of length $p^k$ over ${\text{F}}_{p^m}+u{\text{F}}_{p^m}+u^2{\text{F}}_{p^m}$

For units $\delta$ and $\alpha$ in ${\text{F}}_{p^m}$, the structure of $(\delta +\alpha u^2)$-constacyclic codes of length $p^k$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+u^2{\mathbb{F}}_{p^m}$ is studied and self-dual $(\delta +\alpha u^2)$-constacyclic codes are analyzed.