P℘N functions, complete mappings and quasigroup difference sets
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We investigate pairs of permutations F,G of 𝔽_p^n such that F(x+a)-G(x) is a permutation for every a∈𝔽_p^n. We show that necessarily G(x) = ℘(F(x)) for some complete mapping -℘ of 𝔽_p^n, and call the permutation F a perfect ℘ nonlinear (P℘N) function. If ℘(x) = cx, then F is a PcN function, which have been considered in the literature, lately. With a binary operation on 𝔽_p^n×𝔽_p^n involving ℘, we obtain a quasigroup, and show that the graph of a P℘N function F is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for P℘N functions, respectively, the difference sets in the corresponding quasigroup.
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