PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

CONTENTS                  

The Probability of Error in Block Transmission in a Memoryless Gaussian Channel with Feedback
M. S. Pinsker
pp. 1–14

Abstract—Block methods of transmission in a stationary, memoryless Gaussian channel with complete feedback are described and investigated. It is shown that, for any transmission rate $R$, $0\lt R\lt C$, where $C$ is the channel capacity, the optimum probability of error coincides, to within a factor, with the lower boundaries for the probability of error obtained by Shannon [Bell Syst. Tech. J., 1959, vol. 38, p. 611] for discrete time and Fano [R.M. Fano, Transmission of Information, New York, M.I.T. Press, 1961] for continuous time. For $R\to 0$ as $\tau\to\infty$ a method of transmission is constructed such that the index of the exponent of the probability of error coincides with the index of the exponent of the probability of error for the transmission of two signals.

Cyclic Reed–Muller Codes and Their Decoding
V. D. Kolesnik and E. T. Mironchikov
pp. 15–19

Abstract—A large number of cyclic codes with a majority decoding scheme are constructed on the basis of the so-called quasi-separated checks. A general method of decoding for these codes (orthogonalization in $L$ steps) was described by Massey. In this paper another approach to the decoding of these codes which leads to considerably simpler decoding schemes is considered. This approach is based on the analogy of Reed–Muller and $M(n,k)$-codes, constructed by means of finite projective geometries. Cyclic analogs of Reed–Muller codes are described, and a decoding method similar to Reed’s method is deduced.

The Random Design of Experiments
I. Vajda
pp. 20–28

Abstract—The mean error probability for a discrete statistical model of the sequential design of experiments is studied. The asymptotic behavior of this probability for an unlimited increase of the number of parameters or of the sample size is considered.

Bounds for the Exponent of the Probability of Error for a Semicontinuous Memoryless Channel
E. A. Haroutunian
pp. 29–39

Abstract—Coincident lower and upper bounds for the exponent of the probability of incorrect decoding for optimum transmission of information along a semicontinuous memoryless channel at rates close to the channel capacity are constructed. The bounds coincide in the case of a discrete channel with the Fano [R.M. Fano, Transmission of Information, New York, M.I.T. Press, 1961] bounds, but have a new analytic form.

On the Synchronization of Two-Way Networks of Automata
V. I. Levenshtein
pp. 40–51

Abstract—The problem of synchronizing two-way networks of identical automata is considered. For any $n$ ($n\ge 2$) an automaton $\mathfrak A$ is constructed such that any two-way network of an arbitrary number of such automata is synchronized within a certain time.

Approximation by Binary Functions
G. G. Men'shikov
pp. 52–59

Abstract—It is known that the approximate representation of information by functions assuming only the values $\pm1$ is extremely economical. Problems connected with such an approximation are considered in this paper.

Transmission Capacity and Rate in Channels with a Random Parameter
I. A. Ovseevich
pp. 60–63

Abstract—The transmission rate and capacity of channels with a continuous random variable as parameter are calculated.

Solution of Equations of the Third Degree in a Field of Characteristic 3
M. V. Matveeva
pp. 64–66

Abstract—Polynomials of the third degree on $GF(3^k)$ are considered. A condition is deduced for which $GF(3^k)$ will be the field of the expansion of a given polynomial. It is shown that to find the roots of such polynomials it is sufficient to solve a linear system with $k-1$ unknowns. In the cases $k=3,4,5$ explicit expressions are also given for the roots in terms of the coefficients. The results explained can be used for decoding triple Bose–Chaudhuri codes correcting three errors [W.W. Peterson, Error-Correcting Codes, Cambridge, M.I.T. Press, 1961; M.V. Matveeva, Probl. Peredachi Inf., 1968, vol. 4, no. 1, pp. 20–27].

A Method of Investigating Streams of Errors in Communication Channels
B. E. Aksenov and A. M. Aleksandrov
pp. 67–71

Abstract—A method of investigating streams of errors in communication channels is considered. The distribution of the number of errors in a time segment of fixed length is found, and the connection of this distribution with the distribution function of the intervals between the errors is established. An example is given illustrating the application of the proposed method.

Control of the Numbers of a Cell Population Consisting of Cells of Two Types
A. V. Vasil'ev
pp. 72–73

Abstract—In many tissues of living organisms we encounter examples of cell populations consisting of two or more types of cells with limited lifetimes. Here cells of one type may divide into two cells of the same type or be transformed into cells of other types, the total number of cells of each type being maintained at a certain level. In this paper, by computer study of the simplest mathematical model of such a population, we learned how long the total number of cells can be maintained within defined limits. Some simple control methods by which each cell makes the decision “to divide into two,” “to transform into a cell of a different type,” or “to vanish” independently of the others, as a result of information about the total number of cells of a given type, are considered.