Small latin squares, quasigroups, and loops

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 The loops of order 8 have been independently found by “QSCGZ” and Guérin (unpublished, 2001 ).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

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.

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

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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.



On the number of 8×8 latin squares

On the Number of Latin Squares

Abstract.We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order 11, (2) answer some questions of Alter by showing that

Mutually orthogonal latin squares: a brief survey of constructions

Nonexistence of a Triple of Orthogonal Latin Squares of Order 10 with Group of Order 25 - A Search Made Short

Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture

If is the prime power decomposition of an integer v, and we define the arithmetic function n(v) by then it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually

Counting Two-graphs Related to Trees

This paper gives formulae for the numbers of labelled objects in each of these classes of pentagon-free two-graphs and of pentagons and hexagons, as well as the number of labelled reduced two- graphs in each class, using various enumeration results for trees.

The Non-Existence of Finite Projective Planes of Order 10

A finite projective plane of order n, with n > 0, is a collection of n 2+ n + 1 lines and n2 + n + 1 points such that 1. every line contains n + 1 points, 2. every point is on n + 1 lines, 3. any two

The 6 × 6 Latin squares

  • R. FisherF. Yates
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1934
The problem of the enumeration of the different arrangements of n letters in an n × n Latin square, that is, in a square in which each letter appears once in every row and once in every column, was

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It is confirmed as a result of an exhaustive computer search that there is no Latin power set of the kind sought and the non-existence of a 2-fold perfect (10, 9, 1)-Mendelsohn design which was conjectured to exist by Denes.

Atomic Latin squares of order eleven

A Latin square is pan‐Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i ≠ j. A Latin square is atomic if all of its conjugates are