Design Theory, Second Edition

Design Theory, Second Edition

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Features

  • Develops a deep understanding of the methods through both formal descriptions of the constructions and the accompanying illustrations
  • Focuses on construction methods to provide students with the expertise to produce nonstandard experimental designs when needed
  • Includes important results in combinatorial designs, such as the existence of orthogonal Latin squares, balanced incomplete and pairwise balanced designs, affine and projective planes, and quadruple systems
  • Contains extensive new material on embeddings, directed designs, universal algebraic representations of designs, and intersection properties of designs

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 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

Introduction

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

Embeddings

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

Introduction

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

References

Index


Editorial Reviews

…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

 
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