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Publisher:

Chapman & Hall/CRC

Publication Date:

2007

Number of Pages:

311

Format:

Hardcover

Edition:

2

Series:

Discrete Mathematics and Its Applications 44

Price:

89.95

ISBN:

9781584888383

Category:

Textbook

[Reviewed by , on ]

Miklós Bóna

12/6/2007

This is an introductory graduate textbook that is very easy to read. There are almost no prerequisites: a little bit of linear algebra, the notion of congruence, and some counting principles up to the notion of binomial coefficients, so nothing that a fourth-year undergraduate majoring in mathematics would not know. Therefore, the book is perfect for a reading course taken by an ambitious undergraduate. It is also a useful reference material for the non-specialist researcher needing basic information on designs.

Besides the classic theory, there are several sections devoted to applications. The first one is on one-factorizations, which includes the scheduling round-robin tournaments. We learn, for instance, that in any round-robin tournament of 2n teams, in which each team plays every other team twice, once home and once away from home, there will be at most two teams that do not have to play home twice in a row, or away from home twice in a row. We also learn that it is possible to schedule such a tournament, though, in a way that each of the remaining 2n–2 teams will only have to play two such consecutive games once. There are other applications, to statistics and cryptography, but in less detail. In the experience of this reviewer, students interested in designs are always very interested in error-correcting codes, and this book devotes less than two pages to them.

There is a sufficient number of exercises, but very few of them come with complete solutions. Students of this reviewer always complain about books lacking them.

Other than that, my only critical remark is that the book does not discuss nearly enough open problems. When an advanced mathematics textbook is all about facts and not about questions, it can leave the false impression that its subject is a finished discipline. That can discourage innovative students from going into the field. That said, instructors looking for a textbook for a basic graduate course on designs should certainly consider this very reader-friendly volume.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.

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

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