There are 398 nonisomorphic nontrivial linear spaces on 16 points having no lines of size two. In addition, there are 157,151 proper linear spaces on 17 points having one line of size five and 42 lines of size three. 0 1993 John Wiley & Sons, Inc.
1. BACKGROUND
A linear space is a pair ( P , 2); P is a set of points, and 2 is a set of subsets of points called lines, For every pair of distinct points in P , exactly one line of 2 contains both.
The number of points is the order of the linear space.
Linear spaces have been widely studied as pairwise balanced designs, taking the points to be elements, and the lines to be blocks. Of most interest are those linear spaces having all lines of size at least three; we call such linear spaces proper. We further call a linear space nontrivial if it contains more than one line. Kelly and Nwankpa [6] determined all isomorphism types of proper linear spaces of order at most 14. Brouwer [l] determined the 120 isomorphism classes of proper linear spaces of order 15, of which 80 are the well-known linear spaces in which all lines have size three, the Steiner triple systems of order 15 (see [7]). Beyond this, little is known. Franek, Mathon, and Rosa [4] determined that there are 23 isomorphism classes of linear spaces on 16 points having four lines of size four, and 32 lines of size 3. However, the classification of proper linear spaces on 16 points remained far from complete. We complete the classification here.
In a linear space ( P , Y ) of order 16, let the type of a point x be the multiset (111 -1: n E 1 E 9}. The type is thus a partition of 15 into parts each at least two