Determine the order of magnitude of n_q(5,2,2;2)

Determine the correct order of magnitude, as a function of the field size q, of the maximum number n_q(5,2,2;2) of lines in PG(4,q) such that every plane contains at most two lines.

Background

The paper studies n_q(r,h,f;s), the maximum number of (h-1)-spaces in PG(r-1,q) such that every subspace of codimension f contains at most s selected elements. For (r,h,f,s)=(5,2,2,2), this is the maximum number of lines in PG(4,q) for which every plane contains at most two lines.

The authors obtain lower bounds from constructions and an upper bound of order q^4, while their known constructions provide only order q^3. Consequently, the correct asymptotic order in q is unresolved.