Design Theory, Second Edition

by Lindner; Charles C.

• ISBN13: 9781420082968
• ISBN10: 1420082965
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-10-23
• Publisher: Chapman & Hall/

Summary

Design Theory, Second Edition presents some of the most important techniques used for constructing combinatorial designs. It augments the descriptions of the constructions with many figures to help readers understand and enjoy this branch of mathematics.

Table of Contents

Steiner Triple Systems	p. 1
The existence problem	p. 1
v [identical with] 3 (mod 6): The Bose Construction	p. 4
v [identical with] 1 (mod 6): The Skolem Construction	p. 9
v [identical with] 5 (mod 6): The 6n + 5 Construction	p. 14
Quasigroups with holes and Steiner triple systems	p. 17
Constructing quasigroups with holes	p. 17
Constructing Steiner triple systems using quasigroups with holes	p. 22
The Wilson Construction	p. 27
Cyclic Steiner triple systems	p. 31
The 2n + 1 and 2n + 7 Constructions	p. 35
[lambda]-Fold Triple Systems	p. 45
Triple systems of index [lambda] > 1	p. 45
The existence of idempotent latin squares	p. 47
2-Fold triple systems	p. 50
Constructing 2-fold triple systems	p. 50
Mendelsohn triple systems	p. 55
[lambda] = 3 and 6	p. 59
[lambda]-Fold triple systems in general	p. 62
Quasigroup Identities and Graph Decompositions	p. 65
Quasigroup identities	p. 65
Mendelsohn triple systems revisited	p. 70
Steiner triple systems revisited	p. 72
Maximum Packings and Minimum Coverings	p. 77
The general problem	p. 77
Maximum packings	p. 82
Minimum coverings	p. 87
Kirkman Triple Systems	p. 95
A recursive construction	p. 95
Constructing pairwise balanced designs	p. 103
Mutually Orthogonal Latin Squares	p. 119
Introduction	p. 119
The Euler and MacNeish Conjectures	p. 123
Disproof of the MacNeish Conjecture	p. 135
Disproof of the Euler Conjecture	p. 138
Orthogonal latin squares of order n [identical with] 2 (mod 4)	p. 141
Affine and Projective Planes	p. 155
Affine planes	p. 155
Projective planes	p. 157
Connections between affine and projective planes	p. 159
Connection between affine planes and complete sets of MOLS	p. 161
Coordinatizing the affine plane	p. 165
Intersections of Steiner Triple Systems	p. 169
Teirlinck's Algorithm	p. 169
The general intersection problem	p. 175
Embeddings	p. 185
Embedding latin rectangles - necessary conditions	p. 185
Edge-coloring bipartite graphs	p. 187
Embedding latin rectangles: Pyser's Sufficient Conditions	p. 191
Embedding idempotent communtative latin squares: Cruse's Theorem	p. 194
Embedding partial Steiner triple systems	p. 198
Steiner Quadruple Systems	p. 207
Introduction	p. 207
Constructions of Steiner Quadruple Systems	p. 214
The Stern and Lenz Lemma	p. 220
The (3v - 2u)-Construction	p. 229
Appendices	p. 247
Cyclic Steiner Triple Systems	p. 249
Answers to Selected Exercises	p. 251
References	p. 259
Index	p. 263
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