The generator coordinate method begins with the variational construction of a set of non-orthogonal mean-field states that span a subspace of the full many-body Hilbert space. These states are then often projected onto states with good quantum numbers to restore symmetries, leading to a set with members that can be similar to one another, and it is sometimes possible to reduce this set without greatly affecting results. Here we propose a greedy algorithm that we call the energy-transition-orthogonality procedure (ENTROP) to select subsets of important states. As applied here, the approach selects on the basis of diagonal energy, orthogonality, and contribution to the matrix element that governs neutrinoless double-$beta$ decay. We present both shell-model and preliminary ab initio calculations of this matrix element for the decay of $^{76}$Ge, with quadrupole deformation parameters and the isoscalar pairing strength as generator coordinates. ENTROP converges quickly, reducing significantly the number of basis states needed for an accurate calculation.
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