PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

CONTENTS                  

Remarks on Reverse Pinsker Inequalities
X. Y. Gui and Y. C. Huang
pp. 297–299

Abstract—In this note we propose a simplified approach to recent reverse Pinsker inequalities due to O. Binette. More precisely, we give direct proofs of optimal variational bounds on $f$-divergence with possible constraints on relative information extrema. Our arguments are closer in spirit to those of Sason and Verdú.

Coupling of Several Random Variables
V. V. Prelov
pp. 300–305

Abstract—We consider the problem of finding conditions under which an $\alpha$-coupling is possible for several random variables $X_1,X_2,\ldots,X_k$ with a finite or countably infinite range of values and with given probability distributions, i.e., the possibility of constructing a joint distribution of these random variables such that $\Pr\{X_1=X_2=\ldots=X_k\}=\alpha$.

On One Construction Method for Hadamard Matrices
M. Villanueva, V. A. Zinoviev, and D. V. Zinoviev
pp. 306–328

Abstract—Using a concatenated construction for $q$-ary codes, we construct codes over $\mathbb{Z}_q$ in the Lee metrics which after a proper mapping to the binary alphabet (which in the case of $\mathbb{Z}_4$ is the well-known Gray map) become binary Hadamard codes (in particular, Hadamard matrices). Our construction allows to increase the rank and the kernel dimension of the resulting Hadamard code. Using computer search, we construct new nonequivalent Hadamard matrices of orders $32$, $48$, and $64$ with various fixed values of the rank and the kernel dimension in the range of possible values. It was found that in a special case, our construction coincides with the Kronecker (or Sylvester) construction and can be regarded as a version of a presently known [1] modified Sylvester construction which uses one Hadamard matrix of order $m$ and $m$ (not necessarily distinct) Hadamard matrices of order $k$. We generalize this modified construction by proposing a more general Sylvester-type construction based on two families of (not necessarily distinct) Hadamard matrices, namely, on $k$ matrices of order $m$ and $m$ matrices of order $k$. The resulting matrix is of order $mk$, as in the construction from [1].

Correcting a Single Error in Feedback Channels
I. V. Vorobyev, C. Deppe, A. V. Lebedev, and V. S. Lebedev
pp. 329–340

Abstract—We address the problem of correcting a single error in an arbitrary discrete memoryless channel with error-free instantaneous feedback. For the case of a one-time feedback, we propose a method for constructing optimal transmission strategies. The obtained result allows us to prove that for a binary channel, two feedbacks are sufficient to transmit the same number of messages as in the case of complete feedback. We also apply the developed techniques to a binary asymmetric channel to construct transmission strategies for small lengths.

Nonoverlapping Convex Polytopes with Vertices in a Boolean Cube and Other Problems in Coding Theory
A. Janabekova, G. A. Kabatiansky, I. Kamel, and T. F. Rabie
pp. 341–351

Abstract—We establish relations between several problems that are quite far from each other at first glance and formulate a number of open problems.

On Codes with Distances $d$ and $n$
P. Boyvalenkov, K. Delchev, V. A. Zinoviev, and D. V. Zinoviev
pp. 352–371

Abstract—We enumerate all $q$-ary additive (in particular, linear) block codes of length $n$ and cardinality $N\ge q^2$ with exactly two distances: $d$ and $n$. For arbitrary codes of length $n$ with distances $d$ and $n$, we obtain upper bounds on the cardinality via linear programming and using relationships to 2-distance sets on a Euclidean sphere.

Predictors for High Frequency Signals Based on Rational Polynomial Approximation of Periodic Exponentials
N. G. Dokuchaev
pp. 372–381

Abstract—We present linear integral predictors for continuous-time high-frequency signals with a finite spectrum gap. The predictors are based on approximation of a complex-valued periodic exponential (complex sinusoid) by rational polynomials.

Lower Bound on the Minimum Number of Edges in Subgraphs of Johnson Graphs
Ya. K. Shubin
pp. 382–388

Abstract—We prove a new lower bound on the minimum number of edges in subgraphs of Johnson graphs in the general case.

Some Classes of Balanced Functions over Finite Fields with a Small Value of the Linear Characteristic
O. V. Kamlovskii and K. N. Pankov
pp. 389–402

Abstract—We present balanced functions over finite fields with a small value of the linear characteristic. Previously, linear characteristics of similar classes of functions were studied for the two-element field only.