Minimal Covers of the Klein Quadric

A t-cover of a quadric Q is a set C of t-dimensional subspaces contained in Q such that every point of Q belongs to at least one element of C. We consider t-covers of the Klein quadric Q+(5, q). For t=2, we show that a 2-cover has at least q2+q elements, and we give an exact description of the examples of this cardinality. For t=1, we show that a 1-cover has at least q3+2q+1 elements, and we give examples of covers of that size.