A cross-intersection theorem for vector spaces based on semidefinite programming

Let ℱ and G{script} be families of k- and ℓ-dimensional subspaces, respectively, of a given n-dimensional vector space over a finite field F{double-struck}_q. Suppose that x ∩ y ≠ 0 for all x ∈ ℱ and y ∈ G{script}. By explicitly constructing optimal feasible solutions to a semidefinite programming problem which is akin to Lovász's theta function, we show that (Equation Presented), provided that n ≥ 2k and n ≥ 2 ℓ. The characterization of the extremal families is also established.