More on Regular Linear Spaces

A linear space is a point line incidence geometry with the property that every pair of points is on exactly one line and every line has at least two points. We consider finite linear spaces, i.e. linear spaces with a finite number of points. In [2], the current authors enumerated all linear spaces of order at most 12. More specific types of linear spaces have also been studied. We denote the set of points (resp. lines) by P (resp. B). We denote the incidence structure as a pair (P ,B), and we identify a line with the set of points on it. We also call a line block, using the language of Design Theory, which is a related area. A linear space is regular (in the sense of [1]) if for every integer i, the number of lines of size i through a point is independent of the point. Let ai be the number of lines of size i through a point, for 2 ≤ i ≤ v, where v is the total number of