On classification of some finite linear spaces

We classified all finite linear spaces having property: for every pair L, L' of intersecting lines, and every point p not on either of these, there are L lines through p disjoint from both L and L', where the nonnegative number L is constant. This space will be denoted by (2i,0,L)-interlaced. In the present paper, we will show that this property (with L arbitrary) essentially characterizes the semia¢ ne planes with non-constant line size. For linear spaces with constant line size, this property does not seem to be strong enough, but we derive some necessary conditions on the parameters showing that examples will be hard to find.