PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii


Volume 52, Number 3, July–September, 2016
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Degradable Channels, Less Noisy Channels, and Quantum Statistical Morphisms: An Equivalence Relation
F. Buscemi
pp. 201–213

Abstract—Two partial orderings among communication channels, namely “being degradable into” and “being less noisy than,” are reconsidered in the light of recent results about statistical comparisons of quantum channels. Though our analysis covers at once both classical and quantum channels, we also provide a separate treatment of classical noisy channels and show how in this case an alternative self-contained proof can be constructed, with its own particular merits with respect to the general result.

 

Classical Capacities of Quantum Channels with Environment Assistance
S. Karumanchi, S. Mancini, A. Winter, and D. Yang
pp. 214–238

Abstract—A quantum channel physically is a unitary interaction between an information carrying system and an environment, which is initialized in a pure state before the interaction. Conventionally, this state, as also the parameters of the interaction, is assumed to be fixed and known to the sender and receiver. Here, following the model introduced by us earlier [1], we consider a benevolent third party, i.e., a helper, controlling the environment state, and show how the helper's presence changes the communication game. In particular, we define and study the classical capacity of a unitary interaction with helper, in two variants: one where the helper can only prepare separable states across many channel uses, and one without this restriction. Furthermore, two even more powerful scenarios of pre-shared entanglement between helper and receiver, and of classical communication between sender and helper (making them conferencing encoders) are considered.

 

Estimates for Discontinuity Jumps of Information Characteristics of Quantum Systems and Channels
M. E. Shirokov
pp. 239–264

Abstract—Quantitative analysis of discontinuity of information characteristics of quantum states and channels is presented. Estimates for discontinuity jump (loss) of the von Neumann entropy for a given converging sequence of states are obtained. It is shown, in particular, that for any sequence the loss of entropy is upper bounded by the loss of mean energy (with the coefficient characterizing the Hamiltonian of a system). Then we prove that discontinuity jumps of basic measures of classical and quantum correlations in composite quantum systems are upper bounded by the loss of one of the marginal entropies (with a corresponding coefficient). Quantitative discontinuity analysis of the output entropy of a quantum operation and of basic information characteristics of a quantum channel considered as functions of a pair (channel, input state) is presented.

 

On the Symmetry Group of the Mollard Code
I. Yu. Mogilnykh and F. I. Solov'eva
pp. 265–275

Abstract—We study the symmetry group of a binary perfect Mollard code $M(C,D)$ of length $tm+t+m$ containing as its subcodes the codes $C^1$ and $D^2$ formed from perfect codes $C$ and $D$ of lengths $t$ and $m$, respectively, by adding an appropriate number of zeros. For the Mollard codes, we generalize the result obtained in [1], for the symmetry group of Vasil'ev codes; namely, we describe the stabilizer $\operatorname{Stab}_{D^2}\operatorname{Sym}(M(C,D))$ of the subcode $D^2$ in the symmetry group of the code $M(C,D)$ (with the trivial function). Thus we obtain a new lower bound on the order of the symmetry group of the Mollard code. A similar result is established for the automorphism group of Steiner triple systems obtained by the Mollard construction but not necessarily associated with perfect codes. To obtain this result, we essentially use the notions of “linearity” of coordinate positions (points) of a nonlinear perfect code and a nonprojective Steiner triple system.

 

Multicomponent Codes with Maximum Code Distance
E. M. Gabidulin and N. I. Pilipchuk
pp. 276–283

Abstract—We consider subspace codes, called multicomponent codes with zero prefix (MZP codes), whose subspace code distance is twice their dimension. We find values of parameters for which the codes are of the maximum cardinality. We construct combined codes where the last component of the multicomponent code is the code from [1] found by exhaustive search for particular parameter values. As a result, we obtain a family of subspace codes with maximum cardinality for a number of parameters. We show that the family of maximum-cardinality codes can be extended by using dual codes.

 

On Perfect Codes That Do Not Contain Preparata-like Codes
D. S. Krotov and A. Yu. Vasil'eva
pp. 284–288

Abstract—We show that for every length of the form $4^k-1$ there exists a binary $1$-perfect code that does not contain any Preparata-like code.

 

On Extending Propelinear Structures of the Nordstrom–Robinson Code to the Hamming Code
I. Yu. Mogil'nykh
pp. 289–298

Abstract—A code is said to be propelinear if its automorphism group contains a subgroup which acts on the codewords regularly. Such a subgroup is called a propelinear structure on the code. With the aid of computer, we enumerate all propelinear structures on the Nordstrom–Robinson code and analyze the problem of extending them to propelinear structures on the extended Hamming code of length 16. The latter result is based on the description of partitions of the Hamming code of length 16 into Nordstrom–Robinson codes via Fano planes, presented in the paper. As a result, we obtain a record-breaking number of propelinear structures in the class of extended perfect codes of length 16.

 

The Problem of Fair Division for a Hybrid Resource
M. L. Blank
pp. 299–307

Abstract—We propose an elementary solution to the apartment rent division problem. This problem belongs to the class of problems of “fair division,” but differs from its standard setting by “heterogeneity,” i.e., the presence of both a conventional continuous component and a discrete one, a fixed set of rooms. A combinatorial-topological approach to solving this problem in a finite number of steps (each of which requires a survey of all participants), actively used in the literature, allows to obtain an approximate decision only. We propose a fundamentally different setting, based on a priori estimates of each room by the participants and allowing, in principle, to consider various optimization tasks as well. Our approach is particularly relevant in the case of a large number of participants. We also note that the proposed approach allows to find a solution in a number of cases where the combinatorial-topological approach does not work.

 

The Newton and Coulomb Laws as Information Transfer by Virtual Particles
V. A. Malyshev
pp. 308–318

Abstract—In elementary particle physics the philosophy of virtual particles is widely used. We use this philosophy to obtain the famous inverse square law of classical physics. We define a formal model without fields or forces but with a virtual (auxiliary) particle, information carrier. This formal model admits a very simple (school level) interpretation with two classical particles and one virtual. Then we prove (in a mathematically rigorous way) that the trajectories in our model converge to standard Newtonian trajectories of classical physics.