# 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 The loops of order 8 have been independently found by “QSCGZ” and Guérin (unpublished, 2001 ).

## 155 Citations

### Gröbner bases and the number of Latin squares related to autotopisms of order <= 7

- MathematicsJ. Symb. Comput.
- 2007

### Computing with small quasigroups and loops

- Computer Science
- 2015

This paper first outlines the philosophy behind the GAP package LOOPS, and then it focuses on three particular computational problems: construction of loop isomorphisms, classification of small Frattini Moufang loops of order 64, and the search for loops of nilpotency class higher than two with an abelian inner mapping group.

### Parity Types, Cycle Structures and Autotopisms of Latin Squares

- MathematicsElectron. J. Comb.
- 2012

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

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

- MathematicsComb.
- 2013

It is shown that an n×n Latin square has at most nO(log k) subsquares of order k and admits at mostN(log n) autotopisms and the theorem by McKay and Wanless that gave a factorial divisor of Rn is extended.

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