A translation of Problemy Peredachi Informatsii

Volume 35, Number 3, July–September, 1999
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Generalized Spectra of Binary BCH Codes
Zhang Ichun
pp. 191–204

Abstract—Generalized weight spectra of binary linear codes are studied. An analog of the MacWilliams identities relating generalized spectra of a linear binary code and its dual code is established. Asymptotic (as $n\rightarrow\infty$) expressions for elements of the $r$-spectrum of a $t$-error-correcting binary BCH code of length $n=2^m-1$ are obtained under some restrictions on $t$. This spectrum is computed in the particular case where $r=2$, $t=2$, and $m$ is odd.


Weighted Modules and Representations of Codes
Th. Honold and A. A. Nechaev
pp. 205–223

Abstract—A weight on a finite module is called egalitarian if the average weights of elements of any two its nonzero submodules are equal and it is called homogeneous if in addition the weights of any two associated elements are equal. The criteria of the existence of the egalitarian and homogeneous weights on an arbitrary finite module and the description of possible homogeneous weights are given. These results generalize the analogous results of Constantinescu and Heise for the ring $\mathbb{Z}_m$. Those finite modules which admit a homogeneous weight are called weighted and characterized in terms of the composition factors of their socle. A homogeneous weight in terms of Möbius and Euler functions for finite modules is described and effectively calculated. As an application, besides the known presentation of the generalized Kerdock code, also isometric representations of the Golay codes and the generalized Reed–Muller codes as short linear codes over modules are given.


Perfect Codes for Metrics Generated by Primitive 2-Error-Correcting Binary BCH Codes
E. M. Gabidulin and J. Simonis
pp. 224–230

Abstract—For any positive integer $m$, a metric on $\Bbb{F}_{2}^{2m}$ is considered which is induced by the quasi-perfect $[2^{m}-1,2^{m}-2m-1,5]$ binary BCH code. The isometry group is determined. Constructions of codes are given which are perfect with respect to this metric. In addition, easy decoding methods for these codes are proposed.


New Modification of a Fourth-Order Cumulant Periodogram
V. G. Alekseev
pp. 231–235

Abstract—We investigate the correlation properties of a new modification of a fourth-order cumulant periodogram, which is intended to serve as a half-finished product for designing a nonparametric estimate of the trispectral density of a stationary stochastic process $\{X(k),\: k\in\mathbb{Z}\}$.


On the Lower Bound of the Exact Asymptotics for the Large-Deviation Probabilities of Statistical Estimates
M. S. Ermakov
pp. 236–247

Abstract—The Hajek–Le Cam theorem of locally asymptotic minimax bound of the risk of statistical estimates is extended to the problem of lower bound of the exact asymptotics for the large-deviation probability of statistical estimates. The result obtained can be considered as the exact lower bound of efficiency for confidence estimation of a multivariate parameter, inasmuch as the estimated confidence level usually assumes small values.


Two-Sided Bounds of Random Sums with Subexponential Summands
V. Kalashnikov and G. Tsitsiashvili
pp. 248–258

Abstract—Two-sided bounds of the distribution function of a sum of independent identically distributed positive random variables are derived in the case where the number of summands is random, independent of the values of the summands, and has a discrete distribution of a general form. Such random sums model important characteristics of communication networks, queueing systems, reliability models, and risk models. The basic goal of the work is deriving asymptotically correct upper and lower bounds in the case of subexponentially distributed summands. The bounds proposed are new.


Axiomatic Description of Categories of Information Transformers
P. V. Golubtsov
pp. 259–274

Abstract—It is proposed to consider any uniform class of information transformers (ITs) as a family of morphisms of a category that satisfy a certain set of axioms. The paper defines basic concepts for information transformers and studies their main properties in terms of categories of ITs. In particular it generalizes the Bayesian approach to decision-making problems. It also introduces two different approaches to comparison of informativeness of ITs and investigates their interrelations. It is shown that classes of ITs of equivalent informativeness form a partially ordered bounded Abelian monoid. Several examples of concrete categories of information transformers are examined.


Aim Functions and Sequential Estimation of the Source Model for Universal Coding
Yu. M. Shtarkov
pp. 275–285

Abstract—The aim functions proposed earlier are used to analyze the properties of multimodel universal coding with the sequential estimation of the source model and to develop the algorithms of fuzzy estimation and weighted fuzzy estimation. For the set of context-tree models and estimation or fuzzy estimation algorithms, the same upper bound of the maximal individual redundancy of coding of any message is derived.