PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

CONTENTS                  

Symmetric Rank Codes
E. M. Gabidulin and N. I. Pilipchuk
pp. 103–117

Abstract—As is well known, a finite field $\mathbb{K}_n=\operatorname{\it GF}(q^{n})$ can be described in terms of $n\times n$ matrices $A$ over the field $\mathbb{K}=\operatorname{\it GF}(q)$ such that their powers $A^{i}$, $i=1,2,\ldots,q^{n}-1$, correspond to all nonzero elements of the field. It is proved that, for fields $\mathbb{K}_n$ of characteristic $2$, such a matrix $A$ can be chosen to be symmetric. Several constructions of field-representing symmetric matrices are given. These matrices $A^{i}$ together with the all-zero matrix can be considered as a $\mathbb{K}_n$-linear matrix code in the rank metric with maximum rank distance $d=n$ and maximum possible cardinality $q^{n}$. These codes are called symmetric rank codes. In the vector representation, such codes are maximum rank distance (MRD) linear $[n,1,n]$ codes, which allows one to use known rank-error-correcting algorithms. For symmetric codes, an algorithm of erasure symmetrization is proposed, which considerably reduces the decoding complexity as compared with standard algorithms.

It is also shown that a linear $[n,k,d=n-k+1]$ MRD code $\mathcal{V}_k$ containing the above-mentioned one-dimensional symmetric code as a subcode has the following property: the corresponding transposed code is also $\mathbb{K}_n$-linear. Such codes have an extended capability of correcting symmetric errors and erasures.

On Weight Distributions of Shifts of Goethals-like Codes
V. A. Zinoviev and T. Helleseth
pp. 118–134

Abstract—We study weight distributions of shifts of codes from a well-known family: the $3$-error-correcting binary nonlinear Goethals-like codes of length $n=2^m$, where $m\ge 6$ is even. These codes have covering radius $\rho=6$. We know the weight distribution of any shift of weight $i= 1$, $2$, $3$, $5$, or $6$. For a shift of weight $4$, the weight distribution is uniquely defined by the number of leaders in this shift, i.e., the number of vectors of weight $4$. We also consider the weight distribution of shifts of codes with minimum distance $7$ obtained by deleting any one position of a Goethals-like code of length $n$.

The Nonexistence of Ternary $[284,6,188]$ Codes
R. Daskalov and E. Metodieva
pp. 135–146

Abstract—Let $[n,k,d]_q$ codes be linear codes of length $n$, dimension $k$, and minimum Hamming distance $d$ over $GF(q)$. Let $n_q(k,d)$ be the smallest value of $n$ for which there exists an $[n,k,d]_q$ code. It is known from [Hamada, N. and Helleseth, T., Math. Japon., 2000, vol. 52, no. 1, pp. 31–43; Maruta, T., Table on $n_3(6,d)$] that $284\le n_3(6,188)\le 285$ and $285\le n_3(6,189)\le 286$. In this paper, the nonexistence of $[284,6,188]_3$ codes is proved, whence we get $n_3(6,188)=285$ and $n_3(6,189)=286$.

Quaternary Codes and Biphase Sequences from $\mathbb{Z}_8$-Codes
D. V. Zinoviev and P. Solé
pp. 147–158

Abstract—Composing the Carlet map with the inverse Gray map, a new family of cyclic quaternary codes is constructed from $5$-cyclic $\mathbb{Z}_8$-codes. This new family of codes is inspired by the quaternary Shanbag–Kumar–Helleseth family, a recent improvement on the Delsarte–Goethals family. We conjecture that these $\mathbb{Z}_4$-codes are not linear. As applications, we construct families of low-correlation quadriphase and biphase sequences.

On Components of Preparata Codes
N. N. Tokareva
pp. 159–164

Abstract—The paper considers the interrelation between $i$-components of an arbitrary Preparata-like code $P$ and $i$-components of a perfect code $C$ containing $P$. It is shown that each $i$-component of $P$ can uniquely be completed to an $i$-component of $C$ by adding a certain number of special codewords of $C$. It is shown that the set of vertices of $P$ in a characteristic graph of an arbitrary $i$-component of $C$ forms a perfect code with distance $3$.

Criterion of Infinite Topological Entropy for Multidimensional Cellular Automata
E. L. Lakshtanov and E. S. Langvagen
pp. 165–167

Abstract—The paper proves the infinity of the topological entropy for a wide class of multi-dimensional cellular automata, namely, automata with spaceships.

Probability-Theoretic and Statistical Analysis of Dictionary Texts
S. A. Nekrasov
pp. 168–174

Abstract—We consider probability-theoretic and statistical models and methods for computing the characteristics of dictionary structures. Results of a statistical analysis of several commonly used dictionaries are presented in order to test the adequacy of the computing methods proposed.

On the Asymptotic Power of the Likelihood Ratio Criterion for Testing the Hypothesis of Nonstationarity of an Autoregressive Series with Cauchy Innovations
O. V. Gaas
pp. 175–185

Abstract—The hypothesis of stationarity of an autoregressive time series is tested, where the innovation noise has infinite variance, namely, is subject to the Cauchy distribution. The main result obtained is the limit distribution of the likelihood ratio.