Steiner triple systems

A Steiner triple system is a non empty Steiner system, i.e. a non empty, finite linear space with exactly three points on each line. The usual counting formulas for Steiner systems imply that there are exactly

lines, and each line pencil has exactly elements. In particular, the number v is congruent to either 1 or 3 modulo 6. In fact, the following classical theorem holds.

Theorem. There exists a Steiner triple system on v points if and only if .

The number of isomorphism types of the small Steiner triple systems is well known.

v isomorphism types
1 1
3 1
7 1
9 1
13 2
15 80
19 >2000000

It's commonly assumed that the number of isomorphism types of Steiner triple systems with v=19 points is about 109. But unfortunately, this is out of the range of todays computers.

Contributed by Hauke Klein
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