2022/12/25 by Nurdagül Anbar, Anbar, Nurdagul, Tekgul Kalyci +7 · 1 citation
Engineering · #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2212.12943
openalex publication_date 2022/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate pairs of permutations F,G of \mathbbFpn such that F(x+a)-G(x) is a permutation for every a∈\mathbbFpn. We show that necessarily G(x) = \wp(F(x)) for some complete mapping -\wp of \mathbbFpn, and call the permutation F a perfect \wp nonlinear (P\wpN) function. If \wp(x) = cx, then F is a PcN function, which have been considered in the literature, lately. With a binary operation on \mathbbFpn×\mathbbFpn involving \wp, we obtain a quasigroup, and show that the graph of a P\wpN 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\wpN functions, respectively, the difference sets in the corresponding quasigroup.