Entropic matroids and their representation

TL;DR

This paper explores entropic matroids, their properties, and their relation to linear matroids, revealing that entropic matroids are linear for certain small support sizes but not universally, with implications for coding and cryptography.

Contribution

It establishes the linearity of entropic matroids for support sizes 2 and 3, and demonstrates non-linearity at support size 9, advancing understanding of their structure.

Findings

Entropic matroids are linear for p=2,3.

Entropic matroids are not linear for p=9.

Connections to secret-sharing and cryptography are discussed.

Abstract

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.