# P℘N functions, complete mappings and quasigroup difference sets

Anbar Meidl, Nurdagül and Kalaycı, Tekgül and Meidl, Wilfried and Riera, Constanza and Stănică, Pantelimon (2023) P℘N functions, complete mappings and quasigroup difference sets. Journal of Combinatorial Designs, 31 (12). pp. 667-690. ISSN 1063-8539 (Print) 1520-6610 (Online)

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We investigate pairs of permutations F,G $F,G$ of Fpn ${{\mathbb{F}}}_{{p}<^>{n}}$ such that F(x+a)-G(x) $F(x+a)-G(x)$ is a permutation for every a & ISIN;Fpn $a\in {{\mathbb{F}}}_{{p}<^>{n}}$. We show that, in that case, necessarily G(x)=P(F(x)) $G(x)=\wp (F(x))$ for some complete mapping -P $-\wp$ of Fpn ${{\mathbb{F}}}_{{p}<^>{n}}$, and call the permutation F $F$ a perfect P $\wp$ nonlinear (PP $\wp$N) function. If P(x)=cx $\wp (x)=cx$, then F $F$ is a PcN function, which have been considered in the literature, lately. With a binary operation on FpnxFpn ${{\mathbb{F}}}_{{p}<^>{n}}\times {{\mathbb{F}}}_{{p}<^>{n}}$ involving P $\wp$, we obtain a quasigroup, and show that the graph of a PP $\wp$N function F $F$ 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 PP $\wp$N functions, respectively, for the difference sets in the corresponding quasigroup.