Design Theory  book cover
2nd Edition

Design Theory

ISBN 9781420082968
Published October 23, 2008 by Chapman and Hall/CRC
272 Pages 63 B/W Illustrations

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Book Description

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 students understand and enjoy this branch of mathematics.

This edition now offers a thorough development of the embedding of Latin squares and combinatorial designs. It also presents some pure mathematical ideas, including connections between universal algebra and graph designs.

The authors focus on several basic designs, including Steiner triple systems, Latin squares, and finite projective and affine planes. They produce these designs using flexible constructions and then add interesting properties that may be required, such as resolvability, embeddings, and orthogonality. The authors also construct more complicated structures, such as Steiner quadruple systems.

By providing both classical and state-of-the-art construction techniques, this book enables students to produce many other types of designs.

Table of Contents

Steiner Triple Systems

The Existence Problem

≡ 3 (mod 6): The Bose Construction

≡ 1 (mod 6): The Skolem Construction

≡ 5 (mod 6): The 6n + 5 Construction

Quasigroups with Holes and Steiner Triple Systems

The Wilson Construction

Cyclic Steiner Triple Systems

The 2n + 1 and 2n + 7 Constructions

λ-Fold Triple Systems

Triple Systems of Index λ > 1

The Existence of Indempotent Latin Squares

2-fold Triple Systems

λ= 3 and 6

λ-Fold Triple Systems in General

Quasigroup Identities and Graph Decompositions

Quasigroup Identities

Mendelsohn Triple Systems Revisited

Steiner Triple Systems Revisited

Maximum Packings and Minimum Coverings

The General Problem

Maximum Packings

Minimum Coverings

Kirkman Triple Systems

A Recursive Construction

Constructing Pairwise Balanced Designs

Mutually Orthogonal Latin Squares


The Euler and MacNeish Conjectures

Disproof of the MacNeish Conjecture

Disproof of the Euler Conjecture

Orthogonal Latin Squares of Order ≡ 2 (mod 4)

Affine and Projective Planes

Affine Planes

Projective Planes

Connections between Affine and Projective Planes

Connection between Affine Planes and Complete Sets of MOLS

Coordinating the Affine Plane

Intersections of Steiner Triple Systems

Teirlinck’s Algorithm

The General Intersection Problem


Embedding Latin Rectangles—Necessary Conditions

Edge-Coloring Bipartite Graphs

Embedding Latin Rectangles: Ryser’s Sufficient Conditions

Embedding Idempotent Commutative Latin Squares: Cruse’s Theorem

Embedding Partial Steiner Triple Systems

Steiner Quadruple Systems


Constructions of Steiner Quadruple Systems

The Stern and Lenz Lemma

The (3v – 2u)-Construction

Appendix A: Cyclic Steiner Triple Systems

Appendix B: Answers to Selected Exercises



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…it is remarkable how quickly the book propels the reader from the basics to the frontiers of design theory … Combined, these features make the book an excellent candidate for a design theory text. At the same time, even the seasoned researcher of triple systems will find this a useful resource.

—Peter James Dukes (3-VCTR-MS; Victoria, BC), Mathematical Reviews, 2010