PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

Volume 49, Number 3, July–September, 2013
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CONTENTS

Characterization of the Peak Value Behavior of the Hilbert Transform of Bounded Bandlimited Signals
H. Boche and U. J. Mönich
pp. 197–223

Abstract—The peak value of a signal is a characteristic that has to be controlled in many applications. In this paper we analyze the peak value of the Hilbert transform for the space $\mathcal{B}_{\pi}^{\infty}$ of bounded bandlimited signals. It is known that for this space the Hilbert transform cannot be calculated by the common principal value integral, because there are signals for which it diverges everywhere. Although the classical definition fails for $\mathcal{B}_{\pi}^{\infty}$, there is a more general definition of the Hilbert transform, which is based on the abstract $\mathcal{H}^1$-$\text{BMO}(\mathbb{R})$ duality. It was recently shown in [1], that, in addition to this abstract definition, there exists an explicit formula for calculating the Hilbert transform. Based on this formula we study properties of the Hilbert transform for the space $\mathcal{B}_{\pi}^{\infty}$ of bounded bandlimited signals. We analyze its asymptotic growth behavior, and thereby solve the peak value problem of the Hilbert transform for this space. Further, we obtain results for the growth behavior of the Hilbert transform for the subspace $\mathcal{B}_{\pi,0}^{\infty}$ of bounded bandlimited signals that vanish at infinity. By studying the properties of the Hilbert transform, we continue the work [2].

On Estimating the Output Entropy of the Tensor Product of a Phase-Damping Channel and an Arbitrary Channel
G. G. Amosov
pp. 224–231

Abstract—We obtain a lower estimate for the output entropy of a tensor product of the quantum phase-damping channel and an arbitrary channel. We show that this estimate immediately implies that strong superadditivity of the output entropy holds for this channel as well as for the quantum depolarizing channel.

Structure of Steiner Triple Systems $S(2^m-1,3,2)$ of Rank $2^m-m+2$ over $\mathbb{F}_2$
V. A. Zinoviev and D. V. Zinoviev
pp. 232–248

Abstract—The structure of all different Steiner triple systems $S(2^m-1,3,2)$ of rank $2^m-m+2$ over $\mathbb{F}_2$ is described. This induces a natural recurrent method for constructing Steiner triple systems of any rank. In particular, the method gives all different such systems of order $2^m-1$ and rank $\le 2^m-m+2$. The number of such different systems of order $2^m-1$ and rank less than or equal to $2^m-m+2$ which are orthogonal to a given code is found. It is shown that all different triple Steiner systems of order $2^m-1$ and rank $\le 2^m-m+2$ are derivative and Hamming. Furthermore, all such triples are embedded in quadruple systems of the same rank and in perfect binary nonlinear codes of the same rank.

On Statistical Problems in Geolocation
G. K. Golubev and V. G. Potapov
pp. 249–275

Abstract—We consider the problem of estimating the location of an emitter on the Earth surface based on signals received by a ground terminal from two geostationary satellites. Coordinates of the emitter are computed using differential time-of-arrival and differential Doppler shift of received signals. Our main purpose is mathematical analysis of statistical problems that arise in estimating these parameters.

On Expressive Power of Regular Realizability Problems
M. N. Vyalyi
pp. 276–291

Abstract—A regular realizability (RR) problem is a problem of testing nonemptiness of intersection of some fixed language (filter) with a regular language. We show that RR problems are universal in the following sense. For any language $L$ there exists an RR problem equivalent to $L$ under disjunctive reductions in nondeterministic log space. From this result, we derive existence of complete problems under polynomial reductions for many complexity classes, including all classes of the polynomial hierarchy.

A Model of Random Merging of Segments
L. G. Mityushin
pp. 292–297

Abstract—We consider a growing set $U$ of segments with integer endpoints on a line. For every pair of adjacent segments, their union is added to $U$ with probability $q$. At the beginning, $U$ contains all segments of length from 1 to $m$. Let $h_n$ be the probability that the segment $[a,a+n]$ will be created; the critical value $q_c(m)$ is defined as $\sup\{q\mid\lim\limits_{n\to\infty}h_n=0\}$. Lower and upper bounds for $q_c(m)$ are obtained.