A class of constacyclic codes over \(\frac{\mathbb{F}_{p^m}[u, v]}{\left\langle u^k, v^2, u v-v u\right\rangle}\)

Let \( p \) be a prime number, and let \( k \) and \( m \) be positive integers with \( k \geq 2 \). This paper studies the algebraic structure of \(\lambda\)-constacyclic codes of arbitrary length over the finite commutative ring \( R = \frac{\mathbb{F}_{p^m}[u, v]}{ \langle u^k, v^2, uv - vu \rangle } \), where \(\lambda\) is a unit in \( R \) given by \( \lambda = \sum\limits_{i=0}^{k-1} \lambda_i u^i + v\sum\limits_{i=0}^{k-1} \lambda_i' u^i \), with \(\lambda_i, \lambda_i' \in \mathbb{F}_{p^m}\) and \(\lambda_0, \lambda_1 \neq 0\). We provide a complete classification of these constacyclic codes, determine their dual structures, and compute their Hamming distances when the code length is \( p^s \).