Classification of Self-Dual Constacyclic Codes of Prime Power Length $p^s$ Over $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^3⟩}$

Let ${\mathbb{F}}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is a prime number and $m$ is a positive integer. Self-dual constacyclic codes of length $p^s$ over $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^3⟩}$ exist only when $p=2$. In this work, we classify and enumerate all self-dual cyclic codes of length $2^s$ over $\frac{{\mathbb{F}}_{2^m}[u]}{⟨u^3⟩}$, thereby completing the classification and enumeration of self-dual constacyclic codes of length $p^s$ over $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^3⟩}$. Additionally, we correct and improve results from B. Kim and Y. Lee (2020) in \cite{kim2020classification}.