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Publisher:

Chapman & Hall/CRC

Publication Date:

2007

Number of Pages:

984

Format:

Hardcover

Edition:

2

Series:

Discrete Mathematics and Its Applications 42

Price:

129.95

ISBN:

1584885068

Category:

Handbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Ezra Brown

02/6/2007

Do you love combinatorial designs? If so, this is a book for you! This volume is an updated and greatly expanded version of the 1996 first edition. With more pages than Euler had publications and more than 2^{11} references, it is just what you expect in a handbook — a comprehensive guide to Everything You Always Wanted To Know About Combinatorial Designs, but didn’t know where to look. The editors have done this by assembling 109 chapters within seven main sections, written individually and collectively by 110 contributors, all experts in the field. The contributors clearly took considerable care in their work: for the most part, this vast compendium is well written. The book has lists, tables, examples, diagrams, theorems, constructions, and even three pictures. (You guess who.)

Section 1 begins, most appropriately with the Fano Plane, also known as the (7,3,1) design. For, this beautiful and deceptively simple combinatorial object illustrates so many different ideas from the world of combinatorial designs: symmetry, balance, sets, arrays, graphs, finite geometries, Latin Squares, matrices, error-correcting codes, number theory, finite fields, groups, and even an optimal strategy for the Seven Hats Problem. They proceed immediately to one of the highlights, a section on the history of designs from ancient times up to 1950. Here is where you learn just how pivotal Thomas Kirkman was to the field, and how R. C. Bose’s 1939 paper found the field a collection of results and left it an important mathematical discipline.

Next, there are four sections devoted to four major areas of study. Section 2 treats block designs, and their generalizations, *t*-designs. A *t-(v,k,λ) design* is a *v*-set *V* together with a set *B* of *k*-subsets of *V* (the blocks) such that every *t*-subset of *V* is contained in exactly *λ* blocks. A *block design* is a 2-(*v,k,λ*) design. Section 2 also treats triple systems, Steiner systems, symmetric designs, and resolvable designs, including the famous Kirkman Schoolgirls Problem. Section 3 deals with the construction, enumeration and existence of Latin squares, mutually orthogonal Latin squares (MOLS) and orthogonal arrays, including an account of the history and eventual disproof of Euler’s Conjecture on MOLS.

Section 4 is concerned with partially balanced designs (PBD) and group divisible designs (GDD). A running example through this section is the famous 3x3 magic square — yes, *that* one, and you know how to construct it. Its twelve rows, columns and extended diagonals form a PBD, a GDD, a 2-(9,3,1) design, a resolvable design, and an affine plane of order three. In Section 5, we meet Hadamard matrices, Hadamard designs, and various generalizations of these. (The incidence matrix of the (7,3,1) design gives rise to a Hadamard matrix of order 8.)

Section 6, the longest section, consists of 65 short chapters on other types of combinatorial designs from Association Schemes to Youden Designs. There are chapters on magic squares, difference sets, Skolem and Langford sequences, Whist tournaments, Room squares, and designs intriguingly called SAMDRR (Spouse-Avoiding Mixed Doubles Round Robin) tournaments. Finally, Section 7 contains a wealth of information about other mathematical areas and their relation to combinatorial designs. These include such areas as number theory, graph theory, linear algebra, codes, finite groups, finite fields, and matroids.

The book contains some fascinating tidbits along the way. For example, the section on scheduling a tournament contains examples from Major League Baseball, the NFL, World Cup Soccer, the Czech National Hockey League, the World Motorcycle Speedway Championships, and jai-alai. Chapters 52 and 58 of Section 6 contain information on authentication codes, secrecy codes and secret-sharing schemes. Finally, the chapter on Latin Squares has a brief section on that most tantalizing and addictive game, Sudoku.

In their introduction, the editors state that the handbook is not designed to be used sequentially, and they are absolutely right. This vast book should not be read like a novel. Researchers in the field will find a chapter on their particular area of interest. If you are new to the area, read the introduction and the historical background first, then dive in. Order a copy for your school’s library, or (if you are a combinatorialist) for yourself. But be careful: combinatorial designs can be addictive!

Ezra Brown (brown@math.vt.edu) is Alumni Distinguished Professor of Mathematics at Virginia Tech, with degrees from Rice and LSU. He is a number theorist by trade, but his first publication was about tournaments and Hadamard matrices. He is a fairly regular contributor to the MAA journals. He sings (everything from blues to opera), plays a tolerable jazz piano, and his wife Jo is teaching him to be a gardener. He occasionally bakes biscuits for his students.

PREFACE

INTRODUCTION

NEW! Opening the Door

NEW! Design Theory: Antiquity to 1950

BLOCK DESIGNS

2-(v, k, ?) Designs of Small Order

NEW! Triple Systems

BIBDs with Small Block Size

t-Designs with t = 3

Steiner Systems

Symmetric Designs

Resolvable and Near-Resolvable Designs

LATIN SQUARES

Latin Squares

Quasigroups

Mutually Orthogonal Latin Squares (MOLS)

Incomplete MOLS

Self-Orthogonal Latin Squares (SOLS)

Orthogonal Arrays of Index More Than One

Orthogonal Arrays of Strength More Than Two

PAIRWISE BALANCED DESIGNS

PBDs and GDDs: The Basics

PBDs: Recursive Constructions

PBD-Closure

NEW! Group Divisible Designs

PBDs, Frames, and Resolvability

Pairwise Balanced Designs as Linear Spaces

HADAMARD MATRICES AND RELATED DESIGNS

Hadamard Matrices and Hadamard Designs

Orthogonal Designs

D-Optimal Matrices

Bhaskar Rao Designs

Generalized Hadamard Matrices

Balanced Generalized Weighing Matrices and Conference Matrices

Sequence Correlation

Complementary, Base and Turyn Sequences

NEW! Optical Orthogonal Codes

OTHER COMBINATORIAL DESIGNS

Association Schemes

Balanced Ternary Designs

Balanced Tournament Designs

NEW! Bent Functions

NEW! Block-Transitive Designs

Complete Mappings and Sequencings of Finite Groups

Configurations

Correlation-Immune and Resilient Functions

Costas Arrays

NEW! Covering Arrays

Coverings

Cycle Decompositions

Defining Sets

NEW! Deletion-Correcting Codes

Derandomization

Difference Families

Difference Matrices

Difference Sets

Difference Triangle Sets

Directed Designs

Factorial Designs

Frequency Squares and Hypercubes

Generalized Quadrangles

Graph Decompositions

NEW! Graph Embeddings and Designs

Graphical Designs

NEW! Grooming

Hall Triple Systems

Howell Designs

NEW! Infinite Designs

Linear Spaces: Geometric Aspects

Lotto Designs

NEW! Low Density Parity Check Codes

NEW! Magic Squares

Mendelsohn Designs

NEW! Nested Designs

Optimality and Efficiency: Comparing Block Designs

Ordered Designs, Perpendicular Arrays and Permutation Sets

Orthogonal Main Effect Plans

Packings

Partial Geometries

Partially Balanced Incomplete Block Designs

NEW! Perfect Hash Families

NEW! Permutation Codes and Arrays

NEW! Permutation Polynomials

NEW! Pooling Designs

NEW! Quasi-3 Designs

Quasi-Symmetric Designs

(r, ?)-designs

Room Squares

Scheduling a Tournament

Secrecy and Authentication Codes

Skolem and Langford Sequences

Spherical Designs

Starters

Superimposed Codes and Combinatorial Group Testing

NEW! Supersimple Designs

Threshold and Ramp Schemes

(t,m,s)-Nets

Trades

NEW! Turán Systems

Tuscan Squares

t-Wise Balanced Designs

Whist Tournaments

Youden Squares and Generalized Youden Designs

RELATED MATHEMATICS

Codes

Finite Geometry

NEW! Divisible Semiplanes

Graphs and Multigraphs

Factorizations of Graphs

Computational Methods in Design Theory

NEW! Linear Algebra and Designs

Number Theory and Finite Fields

Finite Groups and Designs

NEW! Designs and Matroids

Strongly Regular Graphs

NEW! Directed Strongly Regular Graphs

Two-Graphs

BIBLIOGRAPHY

INDEX

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