PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

Volume 8, Number 1, January–March, 1972
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CONTENTS

Scalar-Matrix Differentiation and Its Applications to Constructive Problems of Communication Theory
A. A. Amosov and V. V. Kolpakov
pp. 1–9

Abstract—The authors have developed s method, called scalar-matrix differentiation, which is designed to ascertain the extrema of scalar functions of a matrix argument in a space of matrix operators. It is shown that this method enables one to formalize the solution of a number of variational problems in information, communication, and control theory. Rules are given for the most useful scalar-matrix derivatives and differentials. Some applications of the method are described.

Capacity of Multipath Channels
D. D. Klovskii and V. A. Soifer
pp. 10–16

Abstract—The capacity of multipath channels under slow-fading conditions is determined. The distribution for each of $N$ paths is described by a general Gaussian model. The capacity is determined both under separable reception conditions and under conditions such that separable reception is impossible. In the former case two-path combination techniques are analyzed: incoherent summation and cophasing.

On Perfect Codes
V. A. Zinoviev and V. K. Leont'ev
pp. 17–24

Abstract—It is shown that the length of a nontrivial perfect code over the Galois field $GF(q)$ correcting $t\ge 8$ errors is strictly bounded on both sides. This result implies that for values of $q=2,3,4,5,7,$ and $8$ nontrivial perfect codes other than the already known Hamming and Golay codes are nonexistent.

Redundancy of Linear Codes with Orthogonal and $\lambda$-Connected Checks
V. S. Kugurakov
pp. 25–32

Abstract—Lower and upper bounds are derived for the minimum possible number of redundant symbols of linear (cyclic in particular) codes of length $n$ correcting one additive error by orthogonal and $\lambda$-connected checks. Some asymptotic results are also given.

The Information Content of Observations, and Asymptotically Sufficient Statistics
M. S. Pinsker
pp. 33–46

Abstract—Expressions and estimates are given for the quantity of information contained in observations with respect to an estimated parameter for a fixed and a random number of observations. Conditions are indicated under which the maximum-likelihood, Pitman, and other estimates are asymptotically sufficient statistics (i.e., estimates in which the quantity of information about the estimates parameter coincides with the quantity of information in the observations.

On the Mathematical Theory of Quantum Communication Channels
A. S. Holevo
pp. 47–54

Abstract—The author presents the fundamental tenets of an algebraic approach to the description of communication channels whereby it is possible to analyze classical and quantum channels from a unified point of view. The concepts of encoding and decoding are discussed, and special consideration is given to boson channels. A decoding yielding “almost-Shannon” information $\ln\biggl(1+\dfrac{s}{N+1}\biggr)$ is indicated for a Gaussian boson channel.

Communication Reliability in Complex Retranslation Systems
pp. 55–61

Abstract—An ordered system of retranslation nods is analyzed in which each node is linked by nondirectional communication channels with the two preceding and two succeeding nodes. On the basis that the faultless-state probability of the nodes and channels is given and that their failures are independent, the characteristics of the system are determined.

Large-Time Behavior of Stochastic Approximation Processes
R. Z. Khas'minskii
pp. 62–70

Abstract—Asymptotic normality is proved for certain continuous stochastic approximation (SA) procedures under reasonable assumptions, along with the following more general result: under suitable coordinate and time transformations a SA process will asymptotically approach a Gaussian Markov process. As in [R.Z. Khas'minskii, Stability of Systems of Differential Equations under Random Disturbances of Their Parameters, Fizmatgiz, Moscow, 1969; M.B. Nevel'son, R.Z. Khas'minskii, Probl. Peredachi Inf., 1971, vol. 7, no. 2, pp. 58–69], the limit theorems and properties of SA procedures are deducted from theorems on the solutions of systems, stable in a certain sense, of stochastic differential equations.

Autocorrelative Properties of Certain Types of Binary Sequences
M. I. Pelekhatyi and E. A. Golubev
pp. 71–76

Abstract—The authors investigate the autocorrelative properties of sequences $\alpha=\alpha_0\alpha_1\dots\alpha_{n-1}$, $\alpha_i\in\{-1;1\}$, constructed by means of the two-valued character $\psi(\cdot)$ of the multiplicative group of the field $GF(p)$ (prime $p\ge 3$) according to the following rules: a) for $n=p$, set $\alpha_0=\delta$ and $\alpha_i=\psi(i)$; b) for $n=p-1$, set $\alpha_i=\psi(\omega^i_j+1)$ when $\omega^i_j+1\not\equiv0$ and $\alpha_i=\delta$ when $\omega^i_j+1=0$ [$\omega_j$ is a primitive root, fixed for a given distance, of the field $GF(p)$, $j=1,2,\dots,\varphi(p-1)$, where $\varphi(\cdot)$ is the Euler function], with $\delta=\pm1$. The aperiodic autocorrelation of the given sequences were analyzed on a digital computer. It is established by comparison with previous results that in the majority of cases the investigated sequences have the best aperiodic correlation functions.

On the Compression of Literal Messages
O. L. Smirnov and F. I. Yunakov
pp. 77–78

Abstract—The following problem is formulated in the paper: find an ordering of words of a glossary $A=\{a_i\}$ over a $q$-alphabet such as will minimize the average code length in the literal encoding of the first $r$ words and arbitrary encoding of the remaining words. A technique based on direct scanning of all possible ordering variants is proposed for the solution of the problem.

Optimal Energy-Sensitive Detection of a Partially Polarized Optical Signal
A. S. Drikker and R. R. Krasovskii
pp. 79–81

Abstract—Optimal energy-sensitive detection in the optical region is analyzed for a signal having an arbitrary degree of polarization. An expression is derived for the photocurrent distribution under the action of a signal plus noise at the photodetector, and an optimal detector is synthesized.