Introduction to Combinatorial Designs: 2nd Edition (Hardback) book cover

Introduction to Combinatorial Designs

2nd Edition

By W.D. Wallis

Chapman and Hall/CRC

328 pages | 42 B/W Illus.

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pub: 2007-05-17
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Description

Combinatorial theory is one of the fastest growing areas of modern mathematics. Focusing on a major part of this subject, Introduction to Combinatorial Designs, Second Edition provides a solid foundation in the classical areas of design theory as well as in more contemporary designs based on applications in a variety of fields.

After an overview of basic concepts, the text introduces balanced designs and finite geometries. The author then delves into balanced incomplete block designs, covering difference methods, residual and derived designs, and resolvability. Following a chapter on the existence theorem of Bruck, Ryser, and Chowla, the book discusses Latin squares, one-factorizations, triple systems, Hadamard matrices, and Room squares. It concludes with a number of statistical applications of designs.

Reflecting recent results in design theory and outlining several applications, this new edition of a standard text presents a comprehensive look at the combinatorial theory of experimental design. Suitable for a one-semester course or for self-study, it will prepare readers for further exploration in the field.

To access supplemental materials for this volume, visit the author’s website at http://www.math.siu.edu/Wallis/designs

Table of Contents

Basic Concepts

Combinatorial Designs

Some Examples of Designs

Block Designs

Systems of Distinct Representatives

Balanced Designs

Pairwise Balanced Designs

Balanced Incomplete Block Designs

Another Proof of Fisher’s Inequality

t-Designs

Finite Geometries

Finite Affine Planes

Finite Fields

Construction of Finite Affine Geometries

Finite Projective Geometries

Some Properties of Finite Geometries

Ovals in Projective Planes

The Desargues Configuration

Difference Sets and Difference Methods

Difference Sets

Construction of Difference Sets

Properties of Difference Sets

General Difference Methods

Singer Difference Sets

More about Block Designs

Residual and Derived Designs

Resolvability

The Main Existence Theorem

Sums of Squares

The Bruck–Ryser–Chowla Theorem

Another Proof

Latin Squares

Latin Squares and Subsquares

Orthogonality

Idempotent Latin Squares

Transversal Designs

More about Orthogonality

Spouse-Avoiding Mixed Doubles Tournaments

Three Orthogonal Latin Squares

Bachelor Squares

One-Factorizations

Basic Ideas

The Variability of One-Factorizations

Starters

Applications of One-Factorizations

An Application to Finite Projective Planes

Tournament Applications of One-Factorizations

Tournaments Balanced for Carryover

Steiner Triple Systems

Construction of Triple Systems

Subsystems

Simple Triple Systems

Cyclic Triple Systems

Large Sets and Related Designs

Kirkman Triple Systems and Generalizations

Kirkman Triple Systems

Kirkman Packings and Coverings

Hadamard Matrices

Basic Ideas

Hadamard Matrices and Block Designs

Further Hadamard Matrix Constructions

Regular Hadamard Matrices

Equivalence

Room Squares

Definitions

Starter Constructions

Subsquare Constructions

The Existence Theorem

Howell Rotations

Further Applications of Design Theory

Statistical Applications

Information and Cryptography

Golf Designs

References

ANSWERS AND SOLUTIONS

INDEX

About the Series

Discrete Mathematics and Its Applications

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
COM046000
COMPUTERS / Operating Systems / General
MAT036000
MATHEMATICS / Combinatorics