  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
$\mathbb{F}_q$.
