Doubly Exponentially Many Ingleton Matroids

Nelson, Peter; Pol, J.G. van der

doi:10.1137/17m1160094

Cited by 9 publications

(12 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1.1. First, we prove in Theorem 3.14 an interesting consequence of the results by Nelson and van der Pol [45]. Namely, every almost entropic sparse paving matroid must satisfy Ingleton inequality.…”

Section: Our Resultsmentioning

confidence: 74%

“…In addition, by using the improved linear programming technique, we find new lower bounds on the information ratio of secret sharing schemes for several matroid ports. By combining our bounds for matroids on eight points with the results in [45], we present in Theorem 5.1 lower bounds that apply to every sparse paving matroid that does not satisfy Ingleton inequality. We found a lower bound on the information ratio of linear secret sharing schemes for the ports of the tic-tac-toe matroid and some of the aforementioned 171 related matroids.…”

Section: Our Resultsmentioning

confidence: 99%

“…As a consequence, the class of Ingleton-compliant sparse paving matroids has a finite number of forbidden minors [45,Theorem 1.3]. In contrast, the set of excluded minors for the class of Ingleton-compliant matroids is infinite [43].…”

Section: Corollary 313 If a Sparse Paving Matroid M Is Not Ingleton-compliant Then There Is A Minor Of M On Eight Points That Is Not Inglmentioning

confidence: 99%

“…By using the result in Proposition 3.12, Nelson and van der Pol [45] proved that the number of Ingleton-compliant matroids is doubly exponential on the size of the ground set. This indicates that the power of Ingleton inequality in the classification of matroids is quite limited.…”

Section: Theorem 314 If a Sparse Paving Matroid Is Not Ingleton-compliant Then It Is Not Zycompliant And Hence It Is Neither Almost Entromentioning

confidence: 99%

See 3 more Smart Citations

Common information, matroid representation, and secret sharing for matroid ports

Bamiloshin

Ben-Efraim

Farràs

et al. 2020

Des. Codes Cryptogr.

View full text Add to dashboard Cite

Linear information and rank inequalities as, for instance, Ingleton inequality, are useful tools in information theory and matroid theory. Even though many such inequalities have been found, it seems that most of them remain undiscovered. Improved results have been obtained in recent works by using the properties from which they are derived instead of the inequalities themselves. We apply here this strategy to the classification of matroids according to their representations and to the search for bounds on secret sharing for matroid ports.

show abstract

Section: Our Resultsmentioning

confidence: 74%

Section: Our Resultsmentioning

confidence: 99%

Section: Corollary 313 If a Sparse Paving Matroid M Is Not Ingleton-compliant Then There Is A Minor Of M On Eight Points That Is Not Inglmentioning

confidence: 99%

Section: Theorem 314 If a Sparse Paving Matroid Is Not Ingleton-compliant Then It Is Not Zycompliant And Hence It Is Neither Almost Entromentioning

confidence: 99%

See 2 more Smart Citations

Common information, matroid representation, and secret sharing for matroid ports

Bamiloshin

Ben-Efraim

Farràs

et al. 2020

Des. Codes Cryptogr.

View full text Add to dashboard Cite

show abstract

“…This motivates the following definition. In [23] it was shown that Vámos-like matroids violate the Ingleton inequalities. Mayhew and Royle show that among the matroids on eight elements [21], there are 44 non-representable matroids and 39 of them are Vámos-like.…”

Section: Number Of Pairs (E V) For Whichmentioning

confidence: 99%