Chapter 11 - Distances Between Bent Functions

Abstract

Hamming distances between bent functions are studied. In general, we follow the PhD thesis of N.A. Kolomeec published in 2014 and totally devoted to the topic of this chapter. It is shown that the minimal possible distance between two distinct bent functions in n variables is equal to 2n/2. Moreover, bent functions are at this distance if and only if they differ in all elements of some affine subspace of dimension n/2 and both functions are affine on it. It was proven by Kolomeec that if f is a bent function in n variables, then the number of bent functions at distance 2n/2 from it is not more than 2n/2∏i=1n/2(2i+1); this bound is achieved if and only if f is a quadratic bent function. Complete classification of all bent functions at distance 2n/2 from a quadratic bent function is given. Some new themes such as locally metrically nonequivalent bent functions are discussed. Finally, we consider the graph of minimal distances of bent functions and discuss a problem regarding its connectivity.