PROBLEMS OF INFORMATION TRANSMISSION

A translation of *Problemy Peredachi Informatsii*

Volume 12, Number 3, July–September, 1976

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** Class of Correcting Codes for Errors With a Lattice Configuration
**

V. R. Sidorenko

pp. 165–171

**Abstract**—The article describes a quasioptimal class of codes for
correcting errors of lattice configuration over parallel channels; codes consisting
of rectangular matrices, in a special rank metric, are considered. Asymptotically
precise limits for code rates are obtained. For square matrices of order $n$, codes
with codes distance $2,\,3,\,4,\,n$ are constructed, as well as codes which will
detect bursts of lattice errors.

** Synthesis of Single-Error-Correcting $AN$-Codes
**

V. M. Gritsenko

pp. 172–176

**Abstract**—The theory of power residues is employed for synthesizing
arithmetic $AN$-codes to the base $r>2$ that can correct an arbitrary error in one
position. A procedure is given for synthesizing simple and composite generators
$A$ with specified properties.

** Statistical Description of an Adaptive Quantizer
**

V. S. Molodtsov, N. I. Pilipchuk, and V. P. Yakovlev

pp. 177–183

**Abstract**—Integral equations are derived for the joint
distribution of signal values and quantization range which is the controlled
parameter of the quantizer. Solutions are obtained for a Gaussian signal of
Markov type for both large and small correlation coefficients of adjacent
readings, and also for an arbitrarily distributed signal with limited
variations.

** Two-Sample Problem of Hypothesis Testing With Large Samples
**

A. I. Pinskii

pp. 183–188

**Abstract**—Regularity conditions are given such that, for a
two-sample problem of hypothesis testing, it is possible to set up rules that
are asymptotically optimal in the class of differentially asymptotically
similar rules.

** On Approximation of Binary Random
Vectors in Discriminant-Analysis Problems
**

L. G. Malinovskii

pp. 188–192

**Abstract**—With reference to the construction of a discriminant
function, this paper makes a comparison of the approximation quality of the
probability distributions of binary random vectors for which the number of
independent parameters is $2^l-1$. A comparison is made of the Bahadur function and
the probability density function of the normal distribution law, with number of
parameters equal to $l(l+1)/2$. It is shown that these approximations are equally
effective for setting up a classification rule; this makes it easier to investigate
binary random vectors and random vectors containing binary and continuous
features.

** On Asymptotic Behavior of Certain
Estimates of Shift and Scale Parameters
**

L. B. Klebanov and I. A. Melamed

pp. 193–204

**Abstract**—Assume that we have $n$ observations of the form
$x_i=\theta+\sigma\varepsilon_i$, where the errors $\varepsilon_i$ are independent
and identically distributed with distribution function $F(x)$. “Polynomial”
Pitman estimates are derived for the parameters $\theta$ and $\varepsilon$, and their
asymptotic behavior is investigated. The results are applied to studying the Pitman
estimates of $\theta$ and $\varepsilon$.

** Homodyne Reception of Quantized Electromagnetic Signals
**

A. S. Drikker

pp. 205–214

**Abstract**—The author investigates indirect measurement of
coherent electromagnetic signals, a process termed homodyne reception by
analogy with the classical process. It is shown that homodyne reception makes
it possible to achieve asymptotically, for large filling numbers, the
capacity of an ideal channel. Unlike the classical case, maximum capacity is
achieved at a homodyne power that is roughly equal to the signal power.

** Asymmetry Principle of Measurement Logic
**

A. P. Stakhov

pp. 214–220

**Abstract**—A property of measurement asymmetry is found which
leads to the elaboration of a nontrivial theorem of optimum measurement
algorithms that is of applied interest for the theory of enumerative coding
and the arithmetic of digital machines (“Fibonacci reckoning
systems”).

** Unstable Multicomponent Systems
**

A. L. Toom

pp. 220–225

**Abstract**—Systems are considered that consist of an infinite
number of interconnected finite probabilistic automata operating in discrete
time. Systems are called ergodic if they “forget” their initial
state in the limit with respect to time. A sufficient condition for
ergodicity is given. Two examples are cited of families of systems that are
ergodic for positive parameter values and possess the opposite property of
zero values.

** Slow-Down in Universal Simulation
**

A. V. Koganov

pp. 225–231

**Abstract**—The article considers computing media (iterative
structures) on lattices, and derives estimates for the slow-down that occurs
in simulating a large class of media by one such medium. It is shown that the
requirement of a guaranteed slow-down on the entire class of media with fixed
input and output alphabets is not compatible with the requirement of
element-by-element recoding of the initial state of the simulated medium to
the initial state of the model (regular simulation). Examples of universal
models with minimum possible guaranteed slow-down and examples of regular
universal models are given.

** On Minimal Matroid Coverings
**

A. K. Kel'mans, M. V. Lomonosov, and V. P. Polesskii

pp. 231–241

**Abstract**—Matroid $M$ on finite set $E$ is considered; packings and
coverings of $E$ and of its subsets by sets from $M$ are examined.

** Minimax Bound on the Risk of Nonparametric Density Estimates
**

A. M. Samarov

pp. 242–244

**Abstract**—The author obtains a minimax lower bound, exact in order of
magnitude as $n\to\infty$, for moments $\|\gamma_n-f\|$, where $f(\cdot)$ is an
unknown density from some class of sufficiently smooth functions, $\gamma_n(\cdot)$
is the density estimate, and $\|\cdot\|$ is the norm in $L_2(-\infty,\infty)$. The
results are allied to those of Chentsov and Farrell [N. N. Chentsov,
*Statistical Decision Rules and Optimal Conclusions* (in Russian), Nauka,
Moscow, 1972; R. Farrell, *Ann. Math. Statist.*, 1972, vol. 43,
no. 1, pp. 170–180].