Violating the Ingleton inequality with finite groups

Mao, Wei; Hassibi, Babak

doi:10.1109/allerton.2009.5394878

Cited by 13 publications

(17 citation statements)

References 14 publications

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“…Before we proceed to present the details of our results, we would like to mention some recent developments after our first paper [1] on this subject. In [23], Boston and Nan mainly study symmetric groups and discover many new Ingleton violations in the related groups.…”

Section: Discussionmentioning

confidence: 99%

On Ingleton-Violating Finite Groups

Mao

Thill

Hassibi

2017

IEEE Trans. Inform. Theory

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Given n discrete random variables, its entropy vector is the 2 n − 1 dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a one-toone correspondence between such an entropy vector and a certain group-characterizable vector obtained from a finite group and n of its subgroups [3]. This correspondence may be useful for characterizing the space of entropic vectors and for designing network codes. If one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al[4] that linear network codes cannot achieve capacity in general network coding problems (since linear network codes form an abelian group). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. It is therefore of interest to identify groups that violate the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group S 5 to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family P GL(2, q) with a prime power q ≥ 5, i.e., the projective group of 2 × 2 nonsingular matrices with entries in F q . We further interpret this family of groups, and their subgroups, using the theory of group actions and identify the subgroups as certain stabilizers. We also extend the construction to more general groups such as P GL(n, q) and GL(n, q). The families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes, and we discuss some considerations for constructing such group network codes. Index TermsPortions of this work were presented at the

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Section: Discussionmentioning

confidence: 99%

On Ingleton-Violating Finite Groups

Mao

Thill

Hassibi

2017

IEEE Trans. Inform. Theory

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“…It is also far better than the maximum violation index value of Ingleton violating example P GL (2, p) of [7] whose maximum value 0.0082 occurs for p = 13. See Fig.…”

Section: Ingleton Violation Via Mcmcmentioning

confidence: 81%

MCMC methods for entropy optimization and nonlinear network coding

Shadbakht

Hassibi

2010

2010 IEEE International Symposium on Information Theory

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Abstract-Although determining the space of entropic vectors for n random variables, denoted by Γ * n , is crucial for solving a large class of network information theory problems, there has been scant progress in explicitly characterizing Γ * n for n ≥ 4. In this paper, we present a certain characterization of quasi-uniform distributions that allows one to numerically stake out the entropic region via a random walk to any desired accuracy. When coupled with Monte Carlo Markov Chain (MCMC) methods, one may "bias" the random walk so as to maximize certain functions of the entropy vector. As an example, we look at maximizing the violation of the Ingleton inequality for four random variables and report a violation well in excess of what has been previously available in the literature. Inspired by the MCMC method, we also propose a framework for designing optimal nonlinear network codes via performing a random walk over certain truth tables. We show that the method can be decentralized and demonstrate its efficacy by applying it to the Vamos network and a certain storage problem from [1].

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Section: Introductionmentioning

confidence: 97%

“…For 4 random variables, examples of violators of the Ingleton inequality have been found using group theory [1]. This paper extends the approach of [1] to the case of 5 random variables, and exploits group theory in order to find examples of random variables violating linear rank inequalities. First, we prove in Section II that random variables from finite groups violate the Ingleton inequality for 4 random variables if and only they violate the Ingleton inequalities for 5 random variables.…”

Section: Introductionmentioning

confidence: 97%

Groups and information inequalities in 5 variables

Markin

Thomas

Oggier

2013

2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Linear rank inequalities in 4 subspaces are characterized by Shannon-type inequalities and the Ingleton inequality in 4 random variables. Examples of random variables violating these inequalities have been found using finite groups, and are of interest for their applications in nonlinear network coding [1]. In particular, it is known that the symmetric group S5 provides the first instance of a group, which gives rise to random variables that violate the Ingleton inequality. In the present paper, we use group theoretic methods to construct random variables which violate linear rank inequalities in 5 random variables. In this case, linear rank inequalities are fully characterized [8] using Shannon-type inequalities together with 4 Ingleton inequalities and 24 additional new inequalities. We show that finite groups which do not produce violators of the Ingleton inequality in 4 random variables will also not violate the Ingleton inequalities for 5 random variables. We then focus on 2 of the 24 additional inequalities in 5 random variables and formulate conditions for finite groups which help us eliminate those groups that obey the 2 inequalities. In particular, we show that groups of order pq, where p, q are prime, always satisfy them, and exhibit the first violator, which is the symmetric group S4.

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Violating the Ingleton inequality with finite groups

Cited by 13 publications

References 14 publications

On Ingleton-Violating Finite Groups

On Ingleton-Violating Finite Groups

MCMC methods for entropy optimization and nonlinear network coding

Groups and information inequalities in 5 variables

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