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Nonlinear functions and difference sets on group actions. (English) Zbl 1371.05036
Summary: There are many generalizations of the classical Boolean bent functions. Let $$G$$, $$H$$ be finite groups and let $$X$$ be a finite $$G$$-set. $$G$$-perfect nonlinear functions from $$X$$ to $$H$$ have been studied in several papers. They are generalizations of perfect nonlinear functions from $$G$$ itself to $$H$$. By introducing the concept of a $$(G, H)$$-related difference family of $$X$$, we obtain a characterization of $$G$$-perfect nonlinear functions on $$X$$ in terms of a $$(G, H)$$-related difference family. When $$G$$ is abelian, we prove that there is a normalized $$G$$-dual set $$\widehat{X}$$ of $$X$$, and characterize a $$G$$-difference set of $$X$$ by the Fourier transform on a normalized $$G$$-dual set $${{\widehat{X}}}$$. We will also investigate the existence and constructions of $$G$$-perfect nonlinear functions and $$G$$-bent functions. Several known results are direct consequences of our results.

##### MSC:
 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 05E18 Group actions on combinatorial structures 65T50 Numerical methods for discrete and fast Fourier transforms
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