Repeated-Root Constacyclic Codes of Length \(3p^s\) over the Finite Non-Chain Ring \(\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}\) and their Duals

This study aims to determine the algebraic structures of \(\alpha\)-constacyclic codes of length \(3p^s\) over the finite commutative non-chain ring \(\mathcal{R}=\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}\), for a prime \(p \neq 3.\) For the unit \(\alpha\), we consider two different instances: when \(\alpha\) is a cube in \(\mathcal{R}\) and when it is not. Analyzing the first scenario is relatively easy. When \(\alpha\) is not a unit in \(\mathcal{R}\), we consider several subcases and determine the algebraic structures of constacyclic codes in those cases. Further, we also provide the number of codewords and the duals of \(\alpha\)-constacyclic codes.