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Том 55, № 3 (2019)
- Год: 2019
- Статей: 9
- URL: https://journal-vniispk.ru/0032-9460/issue/view/10158
Information Theory
Upper Bounds for the Holevo Information Quantity and Their Use
Аннотация
We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve the estimates existing in the literature. We also present an upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on the metric divergence of an ensemble. In the case of a multi-mode quantum oscillator, this upper bound is tight for large energy. Upper bounds for the Holevo capacity of finite-dimensional quantum channels depending on metric characteristics of the channel output are obtained.
201-217
Optimal Upper Bounds for the Divergence of Finite-Dimensional Distributions under a Given Variational Distance
Аннотация
We consider the problem of finding the maximum values of divergences D(P‖Q) and D(Q‖P) for probability distributions P and Q ranging in the finite set \(\mathcal{N}=\left\{1,\;2,...,n\right\}\) provided that both the variation distance V (P,Q) between them and either the probability distribution Q or (in the case of D(P‖Q)) only the value of the minimal component qmin of the probability distribution Q are given. Precise expressions for the maximum values of these divergences are obtained. In several cases these expressions allow us to write out some explicit formulas and simple upper and lower bounds for them. Moreover, explicit formulas for the maximum of D(P‖Q) for given V (P,Q) and qmin and also for the maximum of D(Q‖P) for given Q and V (P,Q) are obtained for all possible values of these parameters.
218-225
Coding Theory
Divisible Arcs, Divisible Codes, and the Extension Problem for Arcs and Codes
Аннотация
In an earlier paper we developed a unified approach to the extendability problem for arcs in PG(k - 1, q) and, equivalently, for linear codes over finite fields. We defined a special class of arcs called (t mod q)-arcs and proved that the extendabilty of a given arc depends on the structure of a special dual arc, which turns out to be a (t mod q)-arc. In this paper, we investigate the general structure of (t mod q)-arcs. We prove that every such arc is a sum of complements of hyperplanes. Furthermore, we characterize such arcs for small values of t, which in the case t = 2 gives us an alternative proof of the theorem by Maruta on the extendability of codes. This result is geometrically equivalent to the statement that every 2-quasidivisible arc in PG(k - 1, q), q ≥ 5, q odd, is extendable. Finally, we present an application of our approach to the extendability problem for caps in PG(3, q).
226-240
Generalization of IPP Codes and IPP Set Systems
Аннотация
A quarter century ago Chor, Fiat, and Naor proposed mathematical models for revealing a source of illegal redistribution of digital content (tracing traitors) in the broadcast encryption framework, including the following two combinatorial models: nonbinary IPP codes, based on an (n, n)-threshold secret sharing scheme, and IPP set systems, based on the general (w, n)-threshold secret sharing scheme. We propose a new scheme combining the main ideas of nonbinary IPP codes and IPP set systems, which can also be considered as a generalization of nonbinary IPP codes to the case of constant-weight codes. In the simplest case of a coalition of size two, we compare the new scheme with previously known ones.
241-253
Some q-ary Cyclic Codes from Explicit Monomials over \(\mathbb{F}_{q}m\)
Аннотация
Cyclic codes as a subclass of linear codes have practical applications in communication systems, consumer electronics, and data storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to construct some cyclic codes by the sequence approach. More precisely, we determine the dimension and the generator polynomials of three classes of q-ary cyclic codes defined by some sequences with explicit polynomials over \(\mathbb{F}_{q}m\). The minimum distance of such cyclic codes is also discussed. Some of these codes are optimal according to code tables. Moreover, the third class of cyclic codes provides some answers for Open Problem 3 proposed by Ding and Zhou in [1].
254-274
On Alphabetic Coding for Superwords
Аннотация
We consider alphabetic coding of superwords. We establish an unambiguity coding criterion for the cases of finite and infinite codes. We prove that in the case of an infinite code the ambiguity detection problem is m-complete in the ∃1∀0 class of Kleene’s analytical hierarchy.
275-282
Traceability Codes and Their Generalizations
Аннотация
Codes with the identifiable “parent” property appeared as one of solutions for the broadcast encryption problem. We propose a new, more general model of such codes, give an overview of known results, and formulate some unsolved problems.
283-294
Large Systems
295-297
Erratum
298-298