Metrical properties of the set of bent functions in view of duality

In the paper, we give a review of metrical properties of the entire set of bent functions and its significant subclasses of self-dual and anti-self-dual bent functions. We present results for iterative construction of bent functions in n + 2 variables based on the concatenation of four bent functions and consider related open problem proposed by one of the authors. Criterion of self-duality of such functions is discussed. It is explored that the pair of sets of bent functions and affine functions as well as a pair of sets of self-dual and anti-self-dual bent functions in n > 4 variables is a pair of mutually maximally distant sets that implies metrical duality. Groups of automorphisms of the sets of bent functions and (anti-)self-dual bent functions are discussed. The solution to the problem of preserving bentness and the Hamming distance between bent function and its dual within automorphisms of the set of all Boolean functions in n variables is considered

The group of automorphisms of the set of self-dual bent functions

A bent function is a Boolean function in even number of variables which is on the maximal Hamming distance from the set of affine Boolean functions. It is called self-dual if it coincides with its dual. It is called anti-self-dual if it is equal to the negation of its dual. A mapping of the set of all Boolean functions in n variables to itself is said to be isometric if it preserves the Hamming distance. In this paper we study isometric mappings which preserve self-duality and anti-self-duality of a Boolean bent function. The complete characterization of these mappings is obtained for n>2. Based on this result, the set of isometric mappings which preserve the Rayleigh quotient of the Sylvester Hadamard matrix, is characterized. The Rayleigh quotient measures the Hamming distnace between bent function and its dual, so as a corollary, all isometric mappings which preserve bentness and the Hamming distance between bent function and its dual are described

О метрических свойствах множества самодуальных бент-функций

Приводится обзор известных метрических свойств множества самодуальных бент- функций. Бент-функция называется самодуальной, если она совпадает со своей дуальной бент-функцией, и анти-самодуальной, если совпадает с отрицанием своей дуальной. Приводится полный спектр расстояний Хэмминга между самодуальными бент-функциями из класса Мэйорана — МакФарланда. Даются результаты, касающиеся характеризации булевых функций, находящихся на максимально возможном удалении от множества самодуальных бент-функций. Описаны группы автоморфизмов множеств самодуальных и анти-самодуальных бент-функций от n переменных, автоморфизмы множества булевых функций от n переменных, которые меняют местами множества самодуальных и анти-самодуальных бент- функций, изометричные отображения, сохраняющие неизменным отношение Рэлея каждой булевой функции от n переменных. Даётся характеризация всех изо- метричных отображений, сохраняющих максимальную нелинейность и расстояние Хэмминга между каждой бент-функций и дуальной к ней

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application