Linear spaces with at most 12 points

Abstract

The 28,872,973 linear spaces on 12 points are constructed. The parameters of the geometries play an important role. In order to make generation easy, w e construct possible parameter sets for geometries rst (purely algebraically). Afterwards, the corresponding geometries are tried to construct. We de ne line types, point t ypes, point cases and also re ned line types. These are the rst three steps of a general decomposition according to the parameters which we call TDO. The depth of parameter precalculation can be varied, thereby obtaining a handy tool to react in a exible way t o d i e r e n t grades of di culty of the problem.

Matroid Enumeration for Incidence Geometry

“…If a field F is GF (2) and GF (3), M is called a binary matroid and a ternary matroid, respectively. F 7 is known as the unique non-orientable one among binary rank 3 matroids.…”

“…Compared to 77.36% of rank 4 matroids on nine elements [36], our result also gives additional evidence that paving matroids do indeed predominate. Some previous studies on enumeration of matroids focus on simple matroids, including the result of Blackburn, Crapo and Higgs on matroids on n ≤ 8 elements [5] and one of Betten and Betten on rank 3 matroids on n ≤ 12 elements [3]. We also show the number of simple matroids in Table 4 and our results are in complete agreement with these previous results.…”

“…Since Blackburn, Crapo and Higgs [5] enumerated all isomorph-free simple matroids on n ≤ 8 elements in 1973, many studies have been done to extend the classification of matroids. For rank 3 simple matroids, Betten and Betten [3] reached n ≤ 12 elements by exploiting special properties of them. Recently Mayhew and Royle [28] achieved the enumeration of all isomorph-free matroids of any rank on n ≤ 9 elements.…”

On linear spaces and matroids of arbitrary cardinality

“…These properties are, in case of finite E, well known (cf. for instance [13,Chapters 2,3]); however, for the sake of completeness, we carry over the rather short proofs to arbitrary E. Lemma 2.6. Assume M = (E, F ) is some matroid of rank m < ∞ with F as its closed sets.…”

“…More recently, linear spaces defined on some finite set E are studied and constructed and in case of #E ≤ 12 even classified (see [3]). Some special classes of linear spaces are also examined in [4] and [5].…”