PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

CONTENTS                  

On Approximation of Infinite-Dimensional Quantum Channels
M. E. Shirokov and A. S. Holevo
pp. 73–90

Abstract—We develop an approximation approach to infinite-dimensional quantum channels based on a detailed investigation of continuity properties of entropic characteristics of quantum channels and operations (trace-nonincreasing completely positive maps) as functions of a pair “channel, input state.” Obtained results are then applied to the problems of continuity of the $\chi$-capacity as a function of a channel, strong additivity of the $\chi$-capacity for infinite-dimensional channels, and approximating representation for the convex closure of the output entropy of an arbitrary quantum channel.

On the Optimum Choice of a Threshold in a Frequency Hopping OFDMA System
V. V. Zyablov and D. S. Osipov
pp. 91–98

Abstract—We consider a frequency hopping orthogonal frequency-division multiple access (FH OFDMA) system different from those previously described. To find the probability density function of the variable used by the detector, we propose an approach based on the characteristic function method and Bessel function technique. Based on this approach, we obtain an expression for the desired density function (in the most probable case of a collision of multiplicity two) and an expression for the optimum threshold value, which enables us to compute it to any desired degree of accuracy. Efficiency of our approach is justified by simulation results.

Self-checking Circuits and Decoding Algorithms for Binary Hamming and BCH Codes and Reed–Solomon Codes over $\operatorname{\it GF}(2^m)$
I. M. Boyarinov
pp. 99–111

Abstract—We consider problems of detecting errors in combinational circuits and algorithms for the decoding of linear codes. We show that a totally self-checking combinatorial circuit for the decoding of a binary Hamming $[n,k]$ code can be constructed if and only if $n=2^r-1$, $r=n-k$. We introduce the notion of a totally self-checking combinational circuit detecting error clusters of size at most $\mu$; for shortened Hamming $[n,k]$ codes, we construct totally self-checking decoding combinational circuits detecting error clusters of size at most $\mu$, $2\le\mu\lt n-k$. We describe single-error protected and self-checking algorithms: the extended Euclidean algorithm and decoding algorithms for binary BCH codes and Reed--Solomon codes over $\operatorname{\it GF}(2^m)$.

Asymptotic Bounds for the Rate of Colored Superimposed Codes
V. S. Lebedev
pp. 112–118

Abstract—We introduce the notion of a $q$-ary $(r_0,r_1,\ldots,r_{q-1})$ superimposed code. We obtain upper and lower asymptotic bounds for the rate of these (colored) codes.

Nonparametric Hypothesis Testing with Small Type I or Type II Error Probabilities
M. S. Ermakov
pp. 119–137

Abstract—For the problem of signal detection in Gaussian white noise, we obtain lower bounds for the asymptotics of moderate deviation probabilities of type I and type II errors. These asymptotics are attained on tests of the $\chi^2$ type. Using these lower bounds, we find lower bounds for nonparametric confidence estimation in the moderate deviation zone.

Exact Asymptotics of Small Deviations for a Stationary Ornstein–Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L^p$-Norm, $2\le p\le\infty$
V. R. Fatalov
pp. 138–155

Abstract—We prove results on exact asymptotics of the probabilities $$ \operatorname{\bf P}\Biggl\{\int\limits_0^1 |\eta(t)|^p\,dt\le\varepsilon^p\Biggr\},\quad \varepsilon \to 0, $$ where $2\le p\le\infty$, for two types of Gaussian processes $\eta(t)$, namely, a stationary Ornstein–Uhlenbeck process and a Gaussian diffusion process satisfying the stochastic differential equation $$ \begin{cases} dZ(t)=dw(t)+g(t) Z(t)\,dt,\quad t\in[0,1],\\ Z(0)=0. \end{cases} $$ Derivation of the results is based on the principle of comparison with a Wiener process and Girsanov's absolute continuity theorem.

Activity Maxima in Random Networks in the Heavy Tail Case
A. V. Lebedev
pp. 156–160

Abstract—We consider a model of information network described by an undirected random graph, where each node has a random information activity whose distribution possesses a heavy tail (with regular variation). We investigate the cases of networks described by classical and power-law random graphs. We derive sufficient conditions under which the maximum of aggregate activities (over a node and its nearest neighbors) asymptotically grows in the same way as the maximum of individual activities and the Fréchet limit law holds for them.

Occurrence Indices of Elements in Linear Recurrence Sequences over Primary Residue Rings
D. N. Bylkov and O. V. Kamlovskii
pp. 161–168

Abstract—We study distances to the first occurrence (occurrence indices) of a given element in a linear recurrence sequence over a primary residue ring $\mathbb{Z}_{p^n}$. We give conditions on the characteristic polynomial $F(x)$ of a linear recurrence sequence $u$ which guarantee that all elements of the ring occur in $u$. For the case where $F(x)$ is a reversible Galois polynomial over $\mathbb{Z}_{p^n}$, we give upper bounds for occurrence indices of elements in a linear recurrence sequence $u$. A situation where the characteristic polynomial $F(x)$ of a linear recurrence sequence $u$ is a trinomial of a special form over $\mathbb{Z}_4$ is considered separately. In this case we give tight upper bounds for occurrence indices of elements of $u$.

Viktor Dmitrievich Kolesnik: In Memoriam
pp. 169–170