The Steiner triple systems of order 19

Using an orderly algorithm, the Steiner triple systems of order 19 are classied; there are 11;084;874;829 pairwise nonisomorphic such designs. For each design, the order of its automorphism group and the number of Pasch congurations it contains are recorded; 2;591 of the designs are anti-Pasch. There are three main parts of the classication: constructing an initial set of blocks, the seeds; completing the seeds to triple systems with an algorithm for exact cover; and carrying out isomorph rejection of the nal triple systems. Isomorph rejection is based on the graph canonical labeling software nauty supplemented with a vertex invariant based on Pasch congurations. The possibility of using the (strongly regular) block graphs of these designs in the isomorphism tests is utilized. The aforementioned value is in fact a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters (57; 24; 11; 9).

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