✦ LIBER ✦

Small latin squares, quasigroups, and loops

✍ Scribed by Brendan D. McKay; Alison Meynert; Wendy Myrvold

Publisher: John Wiley and Sons
Year: 2007
Tongue: English
Weight: 169 KB
Volume: 15
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.20105

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

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 The loops of order 8 have been independently found by “QSCGZ” and Guérin (unpublished, 2001).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

📜 SIMILAR VOLUMES

Cycle structure of autotopisms of quasig

Cycle structure of autotopisms of quasigroups and latin squares

✍ Douglas S. Stones; Petr Vojtěchovský; Ian M. Wanless 📂 Article 📅 2012 🏛 John Wiley and Sons 🌐 English ⚖ 542 KB

## Abstract An autotopism of a Latin square is a triple (α, β, γ) of permutations such that the Latin square is mapped to itself by permuting its rows by α, columns by β, and symbols by γ. Let Atp(__n__) be the set of all autotopisms of Latin squares of order __n__. Whether a triple (α, β, γ) of pe

Quasigroups, Loops, and Associative Laws

✍ Kenneth Kunen 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 140 KB

We investigate the question of which weakenings of the associative law imply that a quasigroup is a loop. In particular, we completely settle the question for all Ž laws of ''Bol᎐Moufang type'' those written with four variables, three of which are . distinct .

Finite embedding theorems for partial La

Finite embedding theorems for partial Latin squares, quasi-groups, and loops

✍ Charles C Lindner 📂 Article 📅 1972 🏛 Elsevier Science 🌐 English ⚖ 409 KB

Latin squares with no small odd plexes

✍ Judith Egan; Ian M. Wanless 📂 Article 📅 2008 🏛 John Wiley and Sons 🌐 English ⚖ 242 KB

## Abstract A __k__‐plex in a Latin square of order __n__ is a selection of __kn__ entries in which each row, column, and symbol is represented precisely __k__ times. A transversal of a Latin square corresponds to the case __k__ = 1. We show that for all even __n__ > 2 there exists a Latin square o

On even and odd latin squares

✍ Jeannette C.M Janssen 📂 Article 📅 1995 🏛 Elsevier Science 🌐 English ⚖ 378 KB

Latin squares with one subsquare

✍ I. M. Wanless 📂 Article 📅 2001 🏛 John Wiley and Sons 🌐 English ⚖ 207 KB