Maximal complete permutations over $ \mathbb{F}_2^n $
Xiaofang Xu, Lisha Li, Bing Chen, Xiangyong Zeng · Advances in Mathematics of Communications · 2022
We study maximal complete permutations over \begin{document}$ \mathbb{F}_2^n $\end{document} , i.e., complete permutations have a single fixed point and a cycle of length \begin{document}$ 2^n-1 $\end{document} . We characterize several classes of Boolean functions with some linear structures. By using some \begin{document}$ \gamma $\end{document} -cycle permutations, triple-cycle permutations and involutions which are related to Boolean functions having some linear structures, three general constructions of maximal complete permutations are presented from a given maximal linear complete permutation of \begin{document}$ \mathbb{F}_2^n $\end{document} . Besides, maximal complete permutations are also constructed through composing certain maximal complete permutations with themselves.