Volume 32, Issue 5

For Reed–Muller codes we consider subcodes of codimension 1. A classification of Hadamard products of such subcodes is obtained. With the use of this classification it has been shown that in most cases the problem of recovery of the secret key of a code-based cryptosystem employing such subcodes is equivalent to the problem of recovery of the secret key of the same cryptosystem based on Reed–Muller codes, which is known to be tractable.

We consider local probabilities of lower deviations for branching process Zn=Xn,1+⋯+Xn,Zn−1 ${{Z}_{n}}={{X}_{n,1}}+\cdots +{{X}_{n,{{Z}_{n-1}}}}$in random environment η . We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables X i,j are geometric ones.We suppose that the associated random walk Sn=ξ1+⋯+ξn ${{S}_{n}}={{\xi }_{1}}+\cdots +{{\xi }_{n}}$has positive mean μ and satisfies left-hand Cramer’s condition Eexp(hξi)<∞ if h−<h<0 $\mathbf{E}\exp \left( h{{\xi }_{i}} \right)<\infty \text{ if }{{h}^{-}}<h<0$for some h−<−1. ${{h}^{-}}<-1.$Under these assumptions, we find the asymptotic representation of local probabilities P(Zn=⌊ exp(θn) ⌋) for θ∈[ θ1,θ2 ]⊂(μ−;μ) $\mathbf{P}\left( {{Z}_{n}}=\left\lfloor \exp (\theta n) \right\rfloor \right)\text{ for }\theta \in \left[ {{\theta }_{1}},{{\theta }_{2}} \right]\subset \left( {{\mu }^{-}};\mu \right)$for some non-negative μ − .

With each positive integer one can naturally associate a graph in the form of a tree. This paper is concerned with the average values of the number of edges, the number of leaves and the height of trees corresponding to positive integers not greater than a given boundary.

In 1988, Friese et al. put forward lower estimates for the lengths of shortest nonzero vectors for “almost all” lattices of some families in the dimension 3. In 2004, the author of the present paper obtained a similar result for the dimension 4. Here by means of results obtained in part of the paper we show that these estimates also hold in the dimension 5.

The Boolean majority function and the generalized Boolean majority function of an even number n of variables are considered. For these functions exact values of the Walsh coefficients and the curvature are calculated.

We show that triangular families of Boolean functions comprise an exponentially small fraction of proper families of a given order. We prove that if F is a proper family of Boolean functions, then the number of solutions of an equation F ( x ) = A is even. Finally, we describe a new class of proper families of Boolean functions.