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Small latin squares, quasigroups, and loops
Author(s) -
McKay Brendan D.,
Meynert Alison,
Myrvold Wendy
Publication year - 2007
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.20105
Subject(s) - mathematics , isomorphism (crystallography) , latin square , combinatorics , order (exchange) , quasigroup , orthogonal array , discrete mathematics , statistics , rumen , chemistry , food science , finance , taguchi methods , fermentation , crystal structure , economics , crystallography
We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam, and Thiel, 1990), quasigroups of order 6 (Bower, 2000), and loops of order 7 (Brant and Mullen, 1985). The loops of order 8 have been independently found by “QSCGZ” and Guérin (unpublished, 2001). We also report on the most extensive search so far for a triple of mutually orthogonal Latin squares (MOLS) of order 10. Our computations show that any such triple must have only squares with trivial symmetry groups. © 2006 Wiley Periodicals, Inc. J Combin Designs