A Note on an Analytic Approach to the Problem of Matroid Representability, The Cardinality of Sets of k-Independent Vectors over Finite Fields and the Maximum Distance Separable Conjecture

TL;DR

This paper introduces new quantities related to matroid theory to identify non-representability over finite fields, providing bounds and connections to the MDS conjecture and properties of random matrices.

Contribution

It defines novel measures for matroids that help determine non-representability over finite fields and links these to the independence properties of random matrices.

Findings

Bounds on the proportion of dependent subsets imply non-representability.

Connections established between matroid quantities and independence in random matrices.

Results relate to the MDS conjecture and vector independence over finite fields.

Abstract

We introduce various quantities that can be defined for an arbitrary matroid, and show that certain conditions on these quantities imply that a matroid is not representable over $\mathbb{F}_q$ where $q$ is a prime power. Mostly, for a matroid of rank $r$, we examine the proportion of size-$(r-k)$ subsets that are dependent, and give bounds, in terms of the cardinality of the matroid and $q$, for this proportion, below which the matroid is not representable over $\mathbb{F}_q$. We also explore connections between the defined quantities and demonstrate that they can be used to prove that random matrices have high proportions of subsets of columns independent. Our study relates to the results of our papers [4,5,11] dealing with the cardinality of sets of $k$-independent vectors over $\mathbb{F}_q$ and the Maximal Distance Separation Conjecture over $\mathbb{F}_q$.