Repeated-Root Constacyclic Codes of Length $3p^s$ over the Finite Non-Chain Ring $\frac{{\mathbb{F}}_{p^m}[u,v]}{⟨u^2,v^2,uv-vu⟩}$ 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 $\mathcal{R}=\frac{{\mathbb{F}}_{p^m}[u,v]}{⟨u^2,v^2,uv-vu⟩}$, for a prime $p\mathrm{\ne }3.$ 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.