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Entropic Matroids and Their Representation

1
Department of Mathematics, École polytechnique fédérale de Lausanne, 1015 Lausanne, Switzerland
2
Department of Mathematics, Princeton University, Princeton, NJ 08540, USA
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(10), 948; https://doi.org/10.3390/e21100948
Received: 30 July 2019 / Revised: 16 September 2019 / Accepted: 24 September 2019 / Published: 27 September 2019
(This article belongs to the Special Issue Information Measures with Applications)
This paper investigates entropic matroids, that is, matroids whose rank function is given as the Shannon entropy of random variables. In particular, we consider p-entropic matroids, for which the random variables each have support of cardinality p. We draw connections between such entropic matroids and secret-sharing matroids and show that entropic matroids are linear matroids when p = 2 , 3 but not when p = 9 . Our results leave open the possibility for p-entropic matroids to be linear whenever p is prime, with particular cases proved here. Applications of entropic matroids to coding theory and cryptography are also discussed. View Full-Text
Keywords: matroids; entropy function; extremal dependencies; combinatorics; coding matroids; entropy function; extremal dependencies; combinatorics; coding
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Abbe, E.; Spirkl, S. Entropic Matroids and Their Representation. Entropy 2019, 21, 948.

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