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Classification is a common method to study quantum entanglement,and localunitary equivalence (LU-equivalence) is an effective classification tool.Thepurpose of this work is show how to determine the LU-equivalence of sets ofgeneralized Bell states (GBSs) in a bipartite quantum system$\mathbb{C}^{p^\alpha}\otimes \mathbb{C}^{p^\alpha}$ ($p$ is a prime number and$\alpha$ is a positive integer). The idea is that, for a given GBS set$\mathcal{M}$,try to find all the GBS sets that are LU-equivalent to$\mathcal{M}$, then we can determine whether another GBS set is LU-equivalentto $\mathcal{M}$ by comparison. In order to accomplish this intention,we firstreduce the LU-equivalence of two GBS sets to the unitary conjugate equivalence(UC-equivalence) of the corresponding generalized Pauli matrix (GPM) sets.Thenwe show the necessary and sufficient conditions for a 2-GPM set UC-equivalentto a special 2-GPM set $\{ X^{p^\gamma}, Z^{p^\beta} \}$ ($0\leq \beta, \gamma<\alpha$). The general case, that is, the UC-equivalence of two general GPMsets,follows by the particular case.Moreover, these results are programmable,that is, we provide programs that can give all GPM sets UC-equivalent andunitary equivalent (U-equivalent) to a given GPM set,and then theLU-equivalence of two arbitrary GBS sets can be determined.To illustrate therole of the programs, we show a complete LU-equivalent classification of 4-GBSsets in the system $\mathbb{C}^{4}\otimes \mathbb{C}^{4}$.

How to determine the local unitary equivalence of sets of generalized Bell states in $\mathbb{C}^{p^\alpha}\otimes \mathbb{C}^{p^\alpha}$