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Sparse Steiner triple systems of order 21
Author(s) -
Kokkala Janne I.,
Östergård Patric R. J.
Publication year - 2021
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21757
Subject(s) - steiner system , isomorphism (crystallography) , combinatorics , mathematics , order (exchange) , set (abstract data type) , discrete mathematics , computer science , chemistry , finance , crystal structure , economics , crystallography , programming language
A ( k , l ) ‐configuration is a set of l blocks on k points. For Steiner triple systems, ( l + 2 , l ) ‐configurations are of particular interest. The smallest nontrivial such configuration is the Pasch configuration, which is a ( 6 , 4 ) ‐configuration. A Steiner triple system of order v , an STS ( v ) , is r ‐sparse if it does not contain any ( l + 2 , l ) ‐configuration for 4 ≤ l ≤ r . The existence problem for anti‐Pasch Steiner triple systems has been solved, but these have been classified only up to order 19. In the current work, a computer‐aided classification shows that there are 83,003,869 isomorphism classes of anti‐Pasch STS(21)s. Exploration of the classified systems reveals that there are three 5‐sparse STS(21)s but no 6‐sparse STS(21)s. The anti‐Pasch STS(21)s lead to 14 Kirkman triple systems, none of which is doubly resolvable.