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 Fq​. 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 a prime power, for
this proportion, below which the matroid is not representable over
Fq​. 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