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Helsinki University of Technology, 
     Laboratory for Theoretical Computer Science

The Steiner triple systems of order 19


Petteri Kaski and Patric R. J. Östergård. The Steiner triple systems of order 19. Mathematics of Computation, 73:2075–2092, 2004.


Using an orderly algorithm, the Steiner triple systems of order are classified; there are pairwise nonisomorphic such designs. For each design, the order of its automorphism group and the number of Pasch configurations it contains are recorded; of the designs are anti-Pasch. There are three main parts of the classification: 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 final triple systems. Isomorph rejection is based on the graph canonical labeling software emphnauty supplemented with a vertex invariant based on Pasch configurations. 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 .


automorphism group, orderly algorithm, Pasch configuration, Steiner triple system

Suggested BibTeX entry:

    author = {Petteri Kaski and Patric R. J. {\"O}sterg{\aa}rd},
    journal = {Mathematics of Computation},
    pages = {2075--2092},
    title = {The {S}teiner triple systems of order 19},
    volume = {73},
    year = {2004},

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Latest update: 19 January 2010.