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

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 \( \frac{\mathbb{F}_{p^m}[u]}{\langle u^3 \rangle} \) exist only when \( p = 2 \). In this work, we classify and enumerate all self-dual cyclic codes of length \( 2^s \) over \( \frac{\mathbb{F}_{2^m}[u]}{\langle u^3 \rangle} \), thereby completing the classification and enumeration of self-dual constacyclic codes of length \( p^s \) over \( \frac{\mathbb{F}_{p^m}[u]}{\langle u^3 \rangle} \). Additionally, we correct and improve results from B. Kim and Y. Lee (2020) in \cite{kim2020classification}.