Group divisible designs in MOLS of order ten

The maximum number of mutually orthogonal latin squares (MOLS) of order 10 is known to be between 2 and 6. A hypothetical set of four MOLS must contain at least one of the types of group divisible designs (GDDs) classified here. The proof is based on a dimension argument modified from work by Doughe...

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Bibliographic Details
Published in:Designs, codes, and cryptography Vol. 71; no. 2; pp. 283 - 291
Main Authors: Dukes, Peter, Howard, Leah
Format: Journal Article
Language:English
Published: Boston Springer US 01.05.2014
Springer
Subjects:
Circuits
Coding and Information Theory
Combinatorics
Combinatorics. Ordered structures
Computer Science
Cryptology
Data Structures and Information Theory
Designs and configurations
Discrete Mathematics in Computer Science
Exact sciences and technology
Geometry
Group theory
Group theory and generalizations
Information and Communication
Mathematics
Sciences and techniques of general use
GDD
Dimension
Net
Primary 05B15
Code
MOLS
Secondary 51E14
Orthogonal array
Incidence geometry
Latin square
Group divisible design
Symmetry group
Block design
Classification
ISSN:0925-1022, 1573-7586
Online Access:Get full text
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Summary:The maximum number of mutually orthogonal latin squares (MOLS) of order 10 is known to be between 2 and 6. A hypothetical set of four MOLS must contain at least one of the types of group divisible designs (GDDs) classified here. The proof is based on a dimension argument modified from work by Dougherty. The argument has recently led to the discovery of a counterexample to Moorhouse’s conjecture on the rank of nets, found by Howard and Myrvold. Although it is known that even three MOLS can admit no nontrivial symmetry group, we are hopeful this classification via GDDs and dimension can offer some structure to aid the eventual goal of exhausting the search for four MOLS of order 10.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-012-9729-8