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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.

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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