Remarks on shapes of decision diagrams and classes of multiple-valued functions

Abstract

The paper studies binary and ternary functions that have decision diagrams of identical shape in the original and spectral (Fourier) domain. These functions are called Fourier-sweet functions. This class of functions involves certain classes of bent functions and quadratic forms in both binary and ternary cases. Bent functions and quadratic forms have applications in cryptography and error-correcting codes. Not all bent functions are Fourier-sweet functions. It follows, that Fourier-sweet functions are capable of capturing the differences among the classes of bent functions, and at the same time link them to quadratic forms. Representation by shape invariant decision diagrams in the original and spectral domain might provide some better insight into features of bent functions and quadratic forms. The functions represented by the disjoint quadratic forms in the binary case and diagonal forms in the ternary case are elementary Fourier-sweet functions. In both binary and ternary cases, the application of affine transformations, under certain precisely specified restrictions, to the elementary Fourier-sweet functions produces other Fourier-sweet functions.