On the structure and classification of SOMAs: Generalizations of mutually orthogonal Latin squares
type: Item
instance of: scholarly article
title: On the structure and classification of SOMAs: Generalizations of mutually orthogonal Latin squares
zbMATH Open document ID: 0922.05010
author: Leonard H. Soicher
published in: The Electronic Journal of Combinatorics
publication date: 1999-10-03T00:00:00Z
full work available at URL: https://eudml.org/doc/119940
full work available at URL: v6i1r32.html
review text: Summary: Let \(k\geq 0\) and \(n\geq 2\) be integers. A SOMA, or more specifically a \(\text{SOMA} (k,n)\), is an \(n\times n\) array \(A\), whose entries are \(k\)-subsets of a \(kn\)-set \(\Omega\), such that each element of \(\Omega\) occurs exactly once in each row and exactly once in each column of \(A\), and no 2-subset of \(\Omega\) is contained in more than one entry of \(A\). A \(\text{SOMA} (k,n)\) can be constructed by superposing \(k\) mutually orthogonal Latin squares of order \(n\) with pairwise disjoint symbol-sets, and so a \(\text{SOMA} (k,n)\) can be seen as a generalization of \(k\) mutually orthogonal Latin squares of order \(n\). We first study the structure of SOMAs, concentrating on how SOMAs can decompose. We then report on the use of computational group theory and graph theory in the discovery and classification of SOMAs. In particular, we discover and classify \(\text{SOMA} (3,10)\)s with certain properties, and discover two \(\text{SOMA} (4,14)\)s (SOMAs with these parameters were previously unknown to exist). Some of the newly discovered \(\text{SOMA} (3,10)\)s come from superposing a Latin square of order 10 on a \(\text{SOMA} (2,10)\).
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 03B30
zbMATH DE Number: 1345962
zbMATH Keywords: mutually orthogonal Latin squares
describes a project that uses: nauty
describes a project that uses: GRAPE
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1345962