论文标题
p $ \ wp $ n函数，完整的映射和准群差异集
P$\wp$N functions, complete mappings and quasigroup difference sets
论文作者
论文摘要
我们调查成对的排列$ f，g $ of $ \ mathbb {f} _ {p^n} $，这样$ f（x+a）-g（x）$都是每个$ a \ in \ mathbb {f} _ {f} _ {p^n} $的置换。我们表明，对于某些完整的映射$ - \ wp $的$ g（x）= \ wp（f（x））$，$ \ mathbb {f} _ {p^n} $，并将排列$ f $称为完美的$ \ wp $ nlinear（p $ \ wp $ n）函数。如果$ \ wp（x）= cx $，则$ f $是一种PCN功能，最近在文献中考虑过。使用$ \ mathbb {f} _ {p^n} \ times \ mathbb {f} _ {p^n} $涉及$ \ wp $的二进制操作，我们获得了一个Quasigroup，并显示P $ \ wp $ n函数的图表是在各种Quasigroup中的差异。我们进一步指出，从此类准群差异集获得的对称设计的变体。最后，我们分别为p $ \ wp $ n函数分别分析了相应的差异集（通过相应的准群的自然定义），在相应的准群中差异集。
We investigate pairs of permutations $F,G$ of $\mathbb{F}_{p^n}$ such that $F(x+a)-G(x)$ is a permutation for every $a\in\mathbb{F}_{p^n}$. We show that necessarily $G(x) = \wp(F(x))$ for some complete mapping $-\wp$ of $\mathbb{F}_{p^n}$, and call the permutation $F$ a perfect $\wp$ nonlinear (P$\wp$N) function. If $\wp(x) = cx$, then $F$ is a PcN function, which have been considered in the literature, lately. With a binary operation on $\mathbb{F}_{p^n}\times\mathbb{F}_{p^n}$ involving $\wp$, we obtain a quasigroup, and show that the graph of a P$\wp$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$\wp$N functions, respectively, the difference sets in the corresponding quasigroup.
