Related papers: Asymptotic enumeration of correlation-immune boole…

The number of $n$-ary balanced correlation immune (resilient) Boolean functions of order $\frac{n}{2}$ is not less than $n^{2^{(n/2)-2}(1+o(1))}$ as $n\rightarrow\infty$. Keywords: resilient function, correlation immune function, orthogonal…

Algebraic immunity of Boolean function $f$ is defined as the minimal degree of a nonzero $g$ such that $fg=0$ or $(f+1)g=0$. Given a positive even integer $n$, it is found that the weight distribution of any $n$-variable symmetric Boolean…

In this paper, we explicitly construct a large class of symmetric Boolean functions on $2k$ variables with algebraic immunity not less than $d$, where integer $k$ is given arbitrarily and $d$ is a given suffix of $k$ in binary…

It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover, it is $2n/3-1$ if and only if the function corresponds to an equitable $2$-partition of the…

Algebraic immunity has been proposed as an important property of Boolean functions. To resist algebraic attack, a Boolean function should possess high algebraic immunity. It is well known now that the algebraic immunity of an $n$-variable…

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic…

A Boolean function is symmetric if it is invariant under all permutations of its arguments; it is quasi-symmetric if it is symmetric with respect to the arguments on which it actually depends. We present a test that accepts every…

To resist algebraic attack, a Boolean function should possess good algebraic immunity (AI). Several papers constructed symmetric functions with the maximum algebraic immunity $\lceil \frac{n}{2}\rceil $. In this correspondence we prove that…

Correlation-immune (CI) multi-output Boolean functions have the property of keeping the same output distribution when some input variables are fixed. Recently, a new application of CI functions has appeared in the system of resisting…

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number…

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$…

We study the number of queries needed to identify a monotone Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$. A query consists of a 0-1-sequence, and the answer is the value of $f$ on that sequence. It is well-known that the number of…

We consider asymptotic behavior of the correlation functions of the characteristic polynomials of the hermitian sample covariance matrices $H_n=n^{-1}A_{m,n}^*A_{m,n}$, where $A_{m,n}$ is a $m\times n$ complex matrix with independent and…

With $G_k(n)$ the number of functions of $n$ boolean variables definable by $k$-SAT formulae, we prove that $G_3(n)$ is asymptotic to $2^{n+\binom{n}{3}}$. This is a strong form of the case $k=3$ of a conjecture of Bollob\'as, Brightwell…

Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{n_1\cdots n_k\le x} F(n_1,\ldots,n_k)$, taken over the hyperbolic region $\{(n_1,\ldots,n_k)\in {\Bbb N}^k: n_1\cdots n_k\le x\}$, where $F:{\Bbb N}^k\to {\Bbb C}$ is a given…

We use the convolution method for arithmetic functions of several variables to deduce an asymptotic formula for the number of $k$-tuples of positive integers with components which are pairwise non-coprime and $\le x$. More generally, we…

We deduce the non-asymptotical bilateral estimates for moment inequalities for sums of non-negative independent random variables, based on the correspondent estimates for the so-called Bell functions and the Poisson distribution.

This paper concerns the estimation of sums of functions of observable and unobservable variables. Lower bounds for the asymptotic variance and a convolution theorem are derived in general finite- and infinite-dimensional models. An explicit…

A $\{00,01,10,11\}$-valued function on the vertices of the $n$-cube is called a $t$-resilient $(n,2)$-function if it has the same number of $00$s, $01$s, $10$s and $11$s among the vertices of every subcube of dimension $t$. The Friedman and…

In longitudinal and spatial studies, observations often demonstrate strong correlations that are stationary in time or distance lags, and the times or locations of these data being sampled may not be homogeneous. We propose a nonparametric…