On metrical properties of the set of self-dual bent functions

For every bent function $f$ its dual bent function $\widetilde{f}$ is uniquely defined. If $\tilde{f}=f$ then $f$ is called self-dual bent and it is called anti-self-dual bent if $\tilde{f}=f\oplus 1$. In this work we give a review of metrical properties of the set of self-dual bent functions. We give a complete Hamming distance spectrum between self-dual Maiorana — McFarland bent functions. The set of Boolean functions which are maximally distant from the set of self-dual bent functions is discussed. We give a characterization of automorphim groups of the sets of self-dual and anti-self-dual bent functions in $n$ variables as well as the description of isometric mappings that define bijections between the sets of self-dual and anti-self dual bent functions. The set of isometric mappings which preserve the Rayleigh quotient of a Boolean function is given. As a corollary all isometric mappings which preserve bentness and the Hamming distance between bent function and its dual are given.