Small latin squares, quasigroups, and loops

@article{McKay2007SmallLS,
  title={Small latin squares, quasigroups, and loops},
  author={Brendan D. McKay and Alison M. Meynert and Wendy J. Myrvold},
  journal={Journal of Combinatorial Designs},
  year={2007},
  volume={15}
}
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 The loops of order 8 have been independently found by “QSCGZ” and Guérin (unpublished, 2001 ). 

Computing with small quasigroups and loops

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Classifying partial Latin rectangles

Parity Types, Cycle Structures and Autotopisms of Latin Squares

A fast algorithm for finding the autotopy group of a Latin square, based on the cycle decomposition of its rows, is presented.

Isotopy graphs of Latin tableaux

Bounds on the number of autotopisms and subsquares of a Latin square

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