Introduction to Combinatorial Designs Book

List of Figures     xiii
Preface     xv
Basic Concepts     1
Combinatorial Designs     1
Some Examples of Designs     4
Block Designs     6
Systems of Distinct Representatives     11
Balanced Designs     15
Pairwise Balanced Designs     15
Balanced Incomplete Block Designs     20
Another Proof of Fisher's Inequality     29
t-Designs     31
Finite Geometries     35
Finite Affine Planes     35
Finite Fields     38
Construction of Finite Affine Geometries     45
Finite Projective Geometries     49
Some Properties of Finite Geometries     55
Ovals in Projective Planes     55
The Desargues Configuration     58
Difference Sets and Difference Methods     63
Difference Sets     63
Construction of Difference Sets     65
Properties of Difference Sets     71
General Difference Methods     73
Singer Difference Sets     79
More about Block Designs     85
Residual and Derived Designs     85
Resolvability     89
The Main Existence Theorem     97
Sums of Squares     97
The Bruck-Ryser-Chowla Theorem     105
Another Proof     112
Latin Squares     117
Latin Squares and Subsquares     117
Orthogonality     121
Idempotent Latin Squares     126
Transversal Designs     130
More about Orthogonality     137
Spouse-Avoiding Mixed Doubles Tournaments     137
Three Orthogonal Latin Squares     140
Bachelor Squares     146
One-Factorizations     153
Basic Ideas     153
The Variability of One-Factorizations     158
Starters     163
Applications of One-Factorizations     167
An Application to Finite Projective Planes     167
Tournament Applications of One-Factorizations     169
Tournaments Balanced for Carryover     173
Steiner Triple Systems     179
Construction of Triple Systems     179
Subsystems     184
Simple Triple Systems     189
Cyclic Triple Systems     191
Large Sets and Related Designs     195
Kirkman Triple Systems and Generalizations      201
Kirkman Triple Systems     201
Kirkman Packings and Coverings     209
Hadamard Matrices     217
Basic Ideas     217
Hadamard Matrices and Block Designs     222
Further Hadamard Matrix Constructions     226
Regular Hadamard Matrices     230
Equivalence     236
Room Squares     243
Definitions     243
Starter Constructions     246
Subsquare Constructions     251
The Existence Theorem     255
Howell Rotations     260
Further Applications of Design Theory     265
Statistical Applications     265
Information and Cryptography     273
Golf Designs     276
References     281
Answers and Solutions     293
Index     307