Chapter 1 Basic Theory of Association Schemes 1
1.1 Definition of Association Scheme 1
1.2 Examples 4
1.3 The Eigenvalues of Association Schemes 7
1.4 The Krein Parameters 12
1.5 S-Rings and Duality 16
1.6 Primitivity and Imprimitivity 20
1.7 Subschemes and Quotient Schemes 25
1.8 The Polynomial Property 29
1.9 The Automorphisms 34
Chapter 2 Association Schemes of Rectangular Matrices 36
2.1 Definition and Primitivity 36
2.2 The Polynomial Property of Association Schemes of Rectangular Matrices 39
2.3 Recurrence Formulas for Intersection Numbers 42
2.4 The Duality of Association Schemes of Rectangular Matrices 48
2.5 The Automorphisms of Mat(m×n,q) 50
Chapter 3 Association Schemes of Alternate Matrices 52
3.1 Primitivity and P-polynomial Property 52
3.2 The Parameters of Γ（1） 54
3.3 Recurrences for Intersection Numbers 58
3.4 Recurrences for Intersection Numbers:Continued 62
3.5 The Self-duality of Alt(n,q) 66
3.6 The Automorphisms of Alt(n,q) 67
Chapter 4 Association Schemes of Hermitian Matrices 69
4.1 Primitivity and P-polynomial Property 69
4.2 The Parameters of Graph Г（1） 71
4.3 Recurrences for Intersection Numbers 74
4.4 Recurrences for Intersection Numbers:Continued 77
4.5 The Self-duality of Her(n,q2) 80
4.6 The Automorphisms of Her(n,q2) 81
Chapter 5 Association Schemes of Symmetric Matrices in Odd Characteristic 82
5.1 The Normal Forms of Symmetric Matrices 82
5.2 The Association Schemes of Symmetric Matrices and Their Primitivity 83
5.3 Sym(n,q)for Small n 87
5.4 A Few Enumeration Formulas from Orthogonal Geometry 92
5.5 Calculation of Intersection Numbers 96
5.6 Calculation of Intersection Numbers:Continued 101
5.7 The Association Scheme Quad(n,q) 109
5.8 The Self-duality of Sym(n,q) 121
5.9 The Automorphisms of Sym(n,q) 122
Chapter 6 Association Schemes of Symmetric Matrices in Even Characteristic 128
6.1 The Normal Forms of Symmetric Matrices 128
6.2 The Imprimitivity of Sym(n,q) 129
6.3 The Association Scheme Sym(2,q) 130
6.4 Some Results of Pseudo-symplectic Geometry 135
6.5 Calculation of Intersection Numbers 138
6.6 Calculation of Intersection Numbers:Continued 145
6.7 A Fusion Scheme of Sym(n,q) 150
6.8 The Automorphisms of Sym(n,q) 154
Chapter 7 Association Schemes of Quadratic Forms in Even Characteristic 156
7.1 The Normal Forms of Quadratic Forms 156
7.2 Qua(2,q)and Qua(3,q) 160
7.3 Some Enumeration Formulas from Orthogonal Geometry 166
7.4 Calculation of Intersection Numbers 172
7.5 The Duality of Association Schemes of Quadratic Forms 186
7.6 The Imprimitivity of Association Schemes of Quadratic Forms 190
7.7 Two Fusion Schemes of Qua(n,q) 192
7.8 The Automorphisms of Association Schemes of Quadratic Forms 199
Chapter 8 The Eigenvalues of Association Schemes of Quadratic Forms 207
8.1 The Eigenvalues of Association Scheme Qua(2,q) 207
8.2 Some Lemmas on Additive Characters 209
8.3 The 1-extensions and f(n)r 211
8.4 Values of f(n)r on the Union Classes C(n)2i 215
8.5 The 2-extensions and f(n)2k 218
8.6 Values of f(n)2k on Classes C(n)2i and C(n)2i ∪C(n)2i-1 227
8.7 Dual Schemes of Two Fusion Schemes of Qua(n,q) 229
8.8 Eigenvalues of Small Association Schemes of Quadratic Forms 230
References 233
Index 235