О продолжении до бент-функций и оценке сверху их числа

Булева бент-функция f от n переменных является продолжением булевой функции g от k < n переменных, если g является сужением f на фиксированную аффинную плоскость размерности k . Доказывается, что продолжение всегда существует, если k С n/2. Получена оценка сверху для числа продолжений. Оценка усиливается для случая k = n - 1, когда g является почти-бент-функцией. В результате мы улучшаем известные оценки сверху для числа бент-функций

Upper bounds on the numbers of binary plateaued and bent functions

The logarithm of the number of binary n-variable bent functions is asymptotically less than $(2^n)/3$ as n tends to infinity. Keywords: boolean function, Walsh--Hadamard transform, plateaued function, bent function, upper boun

A STUDY OF BINARY DECISION DIAGRAM CHARACTERISTICS OF BENT BOOLEAN FUNCTIONS

Bent Boolean functions exist only for an even number of variables, moreover, they are unbalanced. Therefore, they are used in coding theory and in many areas of computer science. General form of bent functions is still unknown. One way of representing Boolean functions is with a reduced ordered binary decision diagram (ROBDD). The strength of ROBDDs is that they can represent Boolean functions data with a high level of redundancy in a compact form, as long as the data is encoded in such a way that the redundancy is exposed. This paper investigates characteristics of bent functions with focus on their ROBDD parameters. Decision diagram experimental framework has been used for implementation of a program for calculation of the ROBDD parameters. The results presented in this paper are intended to be used to create methods for the construction of bent functions using a ROBDD as a data structure from which the bent functions can be discovered

On biunimodular vectors for unitary matrices

A biunimodular vector of a unitary matrix $A \in U(n)$ is a vector v \in \mathbb{T}^n\subset\bc^n such that $Av \in \mathbb{T}^n$ as well. Over the last 30 years, the sets of biunimodular vectors for Fourier matrices have been the object of extensive research in various areas of mathematics and applied sciences. Here, we broaden this basic harmonic analysis perspective and extend the search for biunimodular vectors to arbitrary unitary matrices. This search can be motivated in various ways. The main motivation is provided by the fact, that the existence of biunimodular vectors for an arbitrary unitary matrix allows for a natural understanding of the structure of all unitary matrices

Counting all bent functions in dimension eight 99270589265934370305785861242880

International audienceno abstrac