Twisted partial difference sets, twisted LP-packings and bent partitions

Recently, the first constructions of bent partitions of elementary abelian groups that do not induce partial difference sets or Latin square type partial difference set packings (LP-packings) have been presented (Anbar et al.; Wang et al., 2025). Motivated by observations on the differential properties of examples of these bent partitions, the notions of twisted partial difference sets and twisted LP-packings in abelian groups \(\mathcal {G}\) $\mathcal{G}$ are introduced in this article. Basic properties of twisted partial difference sets and twisted LP-packings, as well as properties of the corresponding character values, are investigated. It is shown that the sets arising from all recently introduced bent partitions are twisted partial difference sets, and that these bent partitions induce twisted LP-packings. As a consequence, all nontrivial bent partitions of elementary abelian groups known so far are shown to correspond either to LP-packings or to twisted LP-packings.