Combinatorics of symmetric designs

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1.1. Fisher’s Inequality......Page 17
1.2. The First Ray-Chaudhuri–Wilson Inequality......Page 19
1.3. Symmetric designs and Ryser designs......Page 21
1.4. Equidistant families of sets......Page 24
Exercises......Page 27
Notes......Page 28
2.1. Incidence structures......Page 30
2.2. Graphs......Page 35
2.3. Basic properties of (v, b, r, k, λ)-designs......Page 40
2.4. Symmetric designs......Page 44
2.5. The Bruck–Ryser–Chowla Theorem......Page 50
2.6. Automorphisms of symmetric designs......Page 54
2.7. A symmetric (41, 16, 6)-design......Page 58
2.8. A symmetric (79, 13, 2)-design......Page 64
Exercises......Page 69
Notes......Page 72
3.1. Finite fields......Page 75
3.2. Affine planes and nets......Page 77
3.3. The 36 officers problem......Page 82
3.4. Projective planes......Page 88
3.5. Affine geometries over finite fields......Page 92
3.6. Projective geometries over finite fields......Page 95
3.7. Combinatorial characterization of PG…(n, q)......Page 102
3.8. Two infinite families of symmetric designs......Page 111
3.9. Linear codes......Page 113
Exercises......Page 119
Notes......Page 126
4.1. Basic properties of Hadamard matrices......Page 129
4.2. Kronecker product constructions......Page 132
4.3. Conference matrices......Page 134
4.4. Regular Hadamard matrices......Page 142
4.5. From Paley matrices to regular Hadamard matrices......Page 148
4.6. Regular sets of (±1)-matrices......Page 149
4.7. Binary equidistant codes......Page 160
Exercises......Page 166
Notes......Page 168
5.1. Bose’s Inequality......Page 170
5.2. Affine α-resolvable designs......Page 177
5.3. Resolvable 2-designs......Page 179
5.4. Embedding of resolvable designs in symmetric designs......Page 188
5.5. Resolvable 2-designs and equidistant codes......Page 198
Notes......Page 200
6.1. Basic properties of t-designs......Page 202
6.2. The Second Ray-Chaudhuri–Wilson Inequality......Page 207
6.3. Hadamard 3-designs......Page 209
6.4. Cameron’s Theorem......Page 211
6.5. Golay codes and Witt designs......Page 214
6.6. Symmetric designs with parameters (56, 11, 2) and (176, 50, 14)......Page 219
Exercises......Page 223
Notes......Page 226
7.1. Strongly regular graphs......Page 228
7.2. Eigenvalues of strongly regular graphs......Page 235
7.3. Switching in strongly regular graphs......Page 239
7.4. Symmetric designs with polarities......Page 249
7.5. Symmetric designs and digraphs......Page 255
Exercises......Page 259
Notes......Page 261
8.1. Association schemes......Page 263
8.2. Quasi-symmetric designs......Page 266
8.3. Multiples of symmetric designs......Page 275
8.4. Quasi-3 symmetric designs......Page 279
8.5. Block schematic designs with three intersection numbers......Page 286
8.6. Designs with a nearly affine decomposition......Page 292
8.7. A symmetric (71, 15, 3)-design......Page 296
Notes......Page 302
9.1. Group invariant matrices and group rings......Page 305
9.2. Singer and Paley–Hadamard difference sets......Page 315
9.3. Symmetries in a group ring......Page 317
9.4. Building blocks and building sets......Page 323
9.5. McFarland, Spence, and Davis–Jedwab difference sets......Page 326
9.6. Relative difference sets......Page 329
Exercises......Page 335
Notes......Page 337
10.1. Basic properties of BGW-matrices......Page 339
10.2. BGW-matrices with classical parameters......Page 347
10.3. BGW-matrices and relative difference sets......Page 352
10.4. Kronecker product constructions......Page 357
10.5. BGW-matrices and projective geometries......Page 370
Exercises......Page 381
Notes......Page 382
11.1. A symmetric (66, 26, 10)-design......Page 384
11.2. Global decomposition of symmetric designs......Page 385
11.3. Six infinite families of globally decomposable symmetric designs......Page 390
11.4. Productive Hadamard matrices......Page 392
11.5. Symmetric designs with irregular global decomposition......Page 399
11.6. Decomposable symmetric designs and regular graphs......Page 402
11.7. Local decomposition of symmetric designs......Page 407
11.8. Infinite families of locally decomposable symmetric designs......Page 413
11.9. An infinite family of designs with a nearly affine decomposition......Page 418
Notes......Page 422
12.1. Tight subdesigns......Page 423
12.2. Examples of tight subdesigns......Page 428
12.3. Normal subdesigns......Page 437
12.4. Symmetric designs with M-arcs......Page 440
Notes......Page 443
13.1. Quasi-residuals of non-existing symmetric designs......Page 445
13.2. Linear non-embeddability conditions......Page 447
13.3. BGW-matrices and non-embeddability......Page 452
13.4. Non-embeddable quasi-derived designs......Page 459
Exercises......Page 461
Notes......Page 462
14.1. Basic properties of Ryser designs......Page 463
14.2. Type-1 Ryser designs......Page 472
14.3. Ryser designs of prime index......Page 480
14.4. Ryser designs of small index......Page 483
14.5. Ryser designs of small gcd......Page 491
Notes......Page 502
Appendix......Page 504
References......Page 511
Index......Page 530