Table of Contents
Preface v
Preface to the Japanese version ix
1 Classical design theory and classical coding theory 1
1.1 Introduction to graph theory 1
1.2 Strongly regular graphs and Moore graphs 5
1.3 Classical t-designs: definitions and basic properties 14
1.4 Examples of designs 19
1.5 Introduction to classical coding theory 34
1.6 Examples and existence problems of codes 39
2 Association schemes 49
2.1 The definition of association schemes 49
2.2 Bose-Mesner algebras 53
2.3 Commutative association schemes 55
2.4 Character tables of association schemes 57
2.5 Intersection matrices and Bose-Mesner algebras 64
2.6 Dual Bose-Mesner algebras and Terwilliger algebras 65
2.7 Various concepts on association schemes 70
2.7.1 Duality of association schemes 70
2.7.2 Fusion schemes of association schemes 71
2.7.3 Primitive association schemes, distribution graphs, representation graphs 73
2.7.4 Subschemes and quotient schemes 78
2.8 Distance-regular graphs and P-polynomial association schemes 84
2.9 Q-polynomial association schemes 87
2.10 Character tables of various association schemes 89
2.10.1 Association schemes on finite abelian groups 89
2.10.2 The character table of the Hamming scheme H(d,q) 90
2.10.3 The character table of the Johnson scheme J(v,d) 93
2.11 Embeddings into spheres 100
3 Codes and designs in association schemes (Delsarte theory on association schemes) 109
3.1 Introducing linear programming 109
3.2 Subsets of association schemes 113
3.2.1 Subsets of association schemes 113
3.2.2 Codes in P-polynomial schemes 118
3.2.3 Designs in Q-polynomial schemes 120
3.2.4 On the strength and the degree of a Q-polynomial scheme 124
3.3 Combinatorial designs and designs in Johnson schemes 128
3.4 Codes in Hamming schemes 130
3.5 Tight designs in Johnson schemes 134
3.5.1 Existence and non-existence of tight designs 134
3.5.2 Classification of tight 4-designs in Johnson schemes 136
3.6 Tight t-designs for odd t in Johnson schemes and Hamming schemes 143
4 Codes and designs in association schemes (continued) 145
4.1 The Assmus-Mattson theorem and its extensions (Relative designs in Delsarte theory) 145
4.2 t-Designs in regular semilattices 157
5 Algebraic combinatorics on spheres and general remarks on algebraic combinatorics 175
5.1 Finite subsets on spheres 175
5.1.1 Study of finite sets on the sphere from the viewpoint of coding theory 175
5.1.2 Design theoretical study of finite subsets on the sphere 178
5.1.3 Connections of spherical designs with group theory, number theory, modular forms 196
5.2 Study of finite subsets on other spaces 215
5.2.1 Finite subsets on projective spaces (compact symmetric spaces of rank 1) 215
5.2.2 Finite subsets on compact symmetric spaces of general ranks 216
5.2.3 Finite subsets in Euclidean spaces 218
5.2.4 Connections with analysis (in particular, numerical analysis, approximation theory, orthogonal polynomials, and cubature formulas) 228
5.2.5 Analogy between Euclidean or hyperbolic t-designs and relative t-designs in association schemes 230
6 P- and Q-polynomial schemes 241
6.1 P-polynomial/Q-polynomial schemes revisited 241
6.1.1 Distance-regular graphs revisited 241
6.1.2 Q-polynomial schemes revisited 251
6.1.3 P-polynomial schemes and Q-polynomial schemes 256
6.1.4 Orthogonal polynomials 265
6.2 Tridiagonal pairs (TD-pairs) 274
6.2.1 Weight space decompositions 277
6.2.2 TD-relations 282
6.3 Leonard pairs (L-pairs) 291
6.3.1 Standard bases, dual systems of orthogonal polynomials 295
6.3.2 Pre-L-pairs 297
6.3.3 Terwilliger's lemma 304
6.3.4 AW-relations 317
6.3.5 Classification 327
6.3.6 Dual systems of AW-polynomials 335
6.4 Known P- and Q-polynomial schemes 346
6.4.1 The core of P- and Q-polynomial schemes 349
6.4.2 P- and Q-polynomial schemes derived from the core part 367
6.4.3 Towards the classification of P- and Q-polynomial schemes 387
Bibliography 399
Index 421
