Linear network coding and the model theory of linear rank inequalities

Gomez, Arley; Mejía, Carolina; Montoya, J. Andres

doi:10.1109/netcod.2014.6892128

Cited by 3 publications

(2 citation statements)

References 5 publications

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“…Because of space bounds we cannot investigate, in more detail, the algorithmic consequences of SH, we invite our readers to study the companion paper [21].…”

Section: The Importance Of Being Semialgebraicmentioning

confidence: 99%

Network coding and the model theory of linear information inequalities

Gomez

Mejía

Montoya

2014

2014 International Symposium on Network Coding (NetCod)

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Let n ≥ 4, can the entropic region of order n be defined by a finite list of polynomial inequalities? This question was first asked by Chan and Grant. We show that if it were the case one could solve many algorithmic problems coming from Network Coding, Index Coding and Secret Sharing. Unfortunately, it seems that the entropic regions of order larger than four are not semialgebraic. Actually, we guess that it is the case and we provide strong evidence supporting our conjecture.

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“…Because of space bounds we cannot investigate, in more detail, the algorithmic consequences of SH, we invite our readers to study the companion paper [21].…”

Section: The Importance Of Being Semialgebraicmentioning

confidence: 99%

Network coding and the model theory of linear information inequalities

Gomez

Mejía

Montoya

2014

2014 International Symposium on Network Coding (NetCod)

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“…Actually, it seems that it is the only way of obtaining nontrivial lower bounds for the optimal information ratio of general access structures. It is worth to remark that linear rank inequalities are important tools in the study of other communication problems, like for example, lower bounds on the linear capacity of information networks (see [8]).…”

Section: Introductionmentioning

confidence: 99%

Linear secret sharing and the automatic search of linear rank inequalities

Mejía¹

2015

ams

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We study the problem of computing linear rank inequalities, and the related problem of computing lower bounds on the linear share complexity of access structures. We prove that if one knows a generating set for the cone of linear rank inequalities on n + 1 variables, then he can use this generating set and linear programming (or semi-infinite programming) to compute the exact linear optimal information ratio of any access structure on n parties. This theorem shows that it is useful and important to compute generating sets for the cones of linear rank inequalities on a given number of variables. Then, we study the only method we know to cope with the later task, the so called common information method. We investigate the completeness of this method and we prove some preliminary results suggesting that the method is not complete (i.e. there are linear rank inequalities which cannot be obtained via the studied method).

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Defining the almost-entropic regions by algebraic inequalities

Gomez

Mejía

Montoya

2017

IJICOT

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We study the definability of the almost-entropic regions by finite lists of algebraic inequalities. First, we study linear information inequalities and polyhedrality, we present a proof of a theorem of Matus, which claims that the almost-entropic regions are not polyhedral. Then, we study polynomial inequalities and semilagebraicity, we show that the semialgebracity of the almost-entropic regions is something that depends on the essentially conditionality of a certain class of conditional information inequalities. Those results suggest that the almost-entropic regions are not semialgebraic. We conjecture that those regions are not decidable.

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Linear network coding and the model theory of linear rank inequalities

Cited by 3 publications

References 5 publications

Network coding and the model theory of linear information inequalities

Network coding and the model theory of linear information inequalities

Linear secret sharing and the automatic search of linear rank inequalities

Defining the almost-entropic regions by algebraic inequalities

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