For any finite abelian G and any target sum x, the k-subset family summing to x is a nontrivial 2-design only if G is an elementary abelian p-group.
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When Do Subset Sums in Finite Abelian Groups Support $2$-Designs?
For any finite abelian G and any target sum x, the k-subset family summing to x is a nontrivial 2-design only if G is an elementary abelian p-group.