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Introduction to Combinatorial Designs

Edition: 
2
Publisher: 
Chapman & Hall/CRC
Number of Pages: 
311
Price: 
89.95
ISBN: 
9781584888383

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.

Date Received: 
Thursday, November 1, 2007
Reviewable: 
Yes
Include In BLL Rating: 
Yes
W. D. Wallis
Series: 
Discrete Mathematics and Its Applications 44
Publication Date: 
2007
Format: 
Hardcover
Category: 
Textbook
Miklós Bóna
12/6/2007

 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

Publish Book: 
Modify Date: 
Thursday, December 6, 2007

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