Small Latin squares, quasigroups, and loops

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 The loops of order 8 have been independently found by "QSCGZ" and Guerin (unpublished, 2001).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. (c) 2006 Wiley Periodicals, Inc.