PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

Volume 43, Number 1, January–March, 2007
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CONTENTS

One-Mode Quantum Gaussian Channels: Structure and Quantum Capacity
A. S. Holevo
pp. 1–11

Abstract—A complete classification of one-mode Gaussian channels is given up to canonical unitary equivalence. We also comment on the quantum capacity of these channels. A channel complementary to the quantum channel with additive classical Gaussian noise is described, providing an example of a one-mode Gaussian channel which is neither degradable nor antidegradable.

On Inequalities between Mutual Information and Variation
V. V. Prelov
pp. 12–22

Abstract—We continue studying the relationship between mutual information and variational distance started in Pinsker’s paper [1], where an upper bound for the mutual information via variational distance was obtained. We present a simple lower bound, which in some cases is optimal or asymptotically optimal. A uniform upper bound for the mutual information via variational distance is also derived for random variables with a finite number of values. For such random variables, the asymptotic behaviour of the maximum of mutual information is also investigated in the cases where the variational distance tends either to zero or to its maximum value.

Addendum to “Code Spectrum and the Reliability Function: Binary Symmetric Channel”
M. V. Burnashev
pp. 23–25

Abstract—A much simpler proof of Theorem 1 from [1] is presented; we use the notation and enumeration of formulas of [1]. The text below replaces the subsection General Case in [1, Section 4, p. 271].

On $\mathbb{Z}_4$-linear Codes with the Parameters of Reed–Muller Codes
F. I. Solov'eva
pp. 26–32

Abstract—For any pair of integers $r$ and $m$, $0\le r\le m$, we construct a class of quaternary linear codes whose binary images under the Gray map are codes with the parameters of the classical $r$th-order Reed–Muller code $RM(r,m)$.

On Resolvability of Steiner Systems $S(v=2^m,4,3)$ of Rank $r\le v-m+1$ over $\mathbb{F}_2$
V. A. Zinoviev and D. V. Zinoviev
pp. 33–47

Abstract—Two new constructions of Steiner quadruple systems $S(v,4,3)$ are given. Both preserve resolvability of the original Steiner system and make it possible to control the rank of the resulting system. It is proved that any Steiner system $S(v=2^m,4,3)$ of rank $r\le v-m+1$ over $\mathbb{F}_2$ is resolvable and that all systems of this rank can be constructed in this way. Thus, we find the number of all different Steiner systems of rank $r=v-m+1$.

Bound on the Cardinality of a Covering of an Arbitrary Randomness Test by Frequency Tests
K. Yu. Gorbunov
pp. 48–56

Abstract—We improve a well-known asymptotic bound on the number of monotonic selection rules for covering of an arbitrary randomness test by frequency tests. More precisely, we prove that, for any set $S$ (arbitrary test) of binary sequences of sufficiently large length $L$, where $|S|\le 2^{L(1-\delta)}$, for sufficiently small $\delta$ there exists a polynomial (in $1/\delta$) set of monotonic selection rules (frequency tests) which guarantee that, for each sequence $\boldsymbol{t}\in S$, a subsequence can be selected such that the product of its length by the squared deviation of the fraction of zeros in it from $1/2$ is of the order of at least $0.5\ln2\,L[\delta/\ln(1/\delta)](1-2\ln\ln(1/\delta)/\ln(1/\delta))$.

Random Coding Bound for the Second Moment of Multidimensional Lattices
B. D. Kudryashov and K. V. Yurkov
pp. 57–68

Abstract—Efficiency of lattice quantization depends on the parameter of a lattice called the normalized second moment of the Voronoi polyhedron. We apply random-coding methods to study lattices generated by $q$-ary linear codes. We prove that in this class there are lattices with the normalized second moment close to the theoretically attainable limit.