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In a recent survey, Schmidt compiled equivalences between generalized bentfunctions, group invariant Butson Hadamard matrices, and abelian splittingrelative difference sets. We establish a broader network of equivalences byconsidering Butson matrices that are cocyclic rather than strictly groupinvariant. This result has several applications; for example, to theconstruction of Boolean functions whose expansions are generalized partiallybent functions, including cases where no bent function can exist.

Generalized partially bent functions, generalized perfect arrays and cocyclic Butson matrices