On regular planar spaces of type (k,n)

In this paper, we study finite regular planar spaces (S,L,P) of type (k,n) whose planes pairwise have non-empty intersection. We show that |L|⩾|P| and if |L|=|P|, then (S,L,P) is PG(4,n); if |L|>|P| and n⩽30000, then (S,L,P) is PG(3,n).