Representability of uniform matroids over finite fields

Characterize the finite fields over which the uniform matroid U_{s,t} is representable, equivalently determining for which fields a matrix representation exists in which every s-element subset of the t columns is linearly independent.

Background

The problem of determining the largest t for which a truncation of a projective geometry has a U_{s,t}-minor contains, as the special case m=0, the problem of determining when U_{s,t} is representable over a field.

The paper identifies this representability question as an open problem in projective geometry and coding theory. Even coarse bounds for the truncation problem are relevant to lower bounds for vertex Turán densities of the hypercube.