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We introduce various quantities that can be defined for an arbitrary matroid,and show that certain conditions on these quantities imply that a matroid isnot representable over $\mathbb{F}_q$ where $q$ is a prime power. Mostly, for amatroid of rank $r$, we examine the proportion of size-$(r-k)$ subsets that aredependent, 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 anddemonstrate that they can be used to prove that random matrices have highproportions of subsets of columns independent. Our study relates to the resultsof our papers [4,5,11] dealing with the cardinality of sets of $k$-independentvectors over $\mathbb{F}_q$ and the Maximal Distance Separation Conjecture over$\mathbb{F}_q$.

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