Largest t-wise linearly independent subsets over finite fields

Determine the maximum size of a subset of F_q^r that contains no t+1 linearly dependent vectors.

Background

The construction of small Gaussian and spherical designs in the paper uses subsets of F_q^r in which every t elements are linearly independent. The paper proves a lower bound of (1/4)q^{r/(t-1)-1} for the size of such a set, and explains that improving this bound could improve the resulting design-size upper bounds.

The authors therefore ask for the extremal size of a subset avoiding linearly dependent subsets of size t+1. This is a concrete finite-field extremal problem directly connected to improving constructions of fixed-strength spherical designs.