Combinatorics of symmetric designs

Yury J. Ionin, Mohan S. Shrikhande0521818338, 9780521818339, 9780511161681

Providing a unified exposition of the theory of symmetric designs with emphasis on recent developments, this volume covers the combinatorial aspects of the theory, giving particular attention to the construction of symmetric designs and related objects. The last five chapters are devoted to balanced generalized weighing matrices, decomposable symmetric designs, subdesigns of symmetric designs, non-embeddable quasi-residual designs, and Ryser designs. The book concludes with a comprehensive bibliography of over 400 entries. Detailed proofs and a large number of exercises make it suitable as a text for an advanced course in combinatorial designs.

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Table of contents :
Cover……Page 1
Annotation……Page 2
Series……Page 3
Title page……Page 4
Date-line……Page 5
Contents……Page 6
Preface……Page 10
1.1 Fisher’s Inequality……Page 13
1.2 The First Ray-Chaudhuri-Wilson Inequality……Page 15
1.3 Symmetric designs and Ryser designs……Page 17
1.4 Equidistant families of sets……Page 20
Exercises……Page 23
Notes……Page 24
2.1 Incidence structures……Page 26
2.2 Graphs……Page 31
2.3 Basic properties of $(v,b,r,k,lambda)$-designs……Page 36
2.4 Symmetric designs……Page 40
2.5 The Bruck-Ryser-Chowla Theorem……Page 46
2.6 Automorphisms of symmetric designs……Page 50
2.7 A symmetric (41, 16, 6)-design……Page 54
2.8 A symmetric (79, 13, 2)-design……Page 60
Exercises……Page 65
Notes……Page 68
3.1 Finite fields……Page 71
3.2 Affine planes and nets……Page 73
3.3 The 36 officers problem……Page 78
3.4 Projective planes……Page 84
3.5 Affine geometries over finite fields……Page 88
3.6 Projective geometries over finite fields……Page 91
3.7 Combinatorial characterization of $PG_{n-1}(n,q)$……Page 98
3.8 Two infinite families of symmetric designs……Page 107
3.9 Linear codes……Page 109
Exercises……Page 115
Notes……Page 122
4.1 Basic properties of Hadamard matrices……Page 125
4.2 Kronecker product constructions……Page 128
4.3 Conference matrices……Page 130
4.4 Regular Hadamard matrices……Page 138
4.5 From Paley matrices to regular Hadamard matrices……Page 144
4.6 Regular sets of ($pm$1)-matrices……Page 145
4.7 Binary equidistant codes……Page 156
Exercises……Page 162
Notes……Page 164
5.1 Bose’s Inequality……Page 166
5.2 Affine $alpha$-resolvable designs……Page 173
5.3 Resolvable 2-designs……Page 175
5.4 Embedding of resolvable designs in symmetric designs……Page 184
5.5 Resolvable 2-designs and equidistant codes……Page 194
Notes……Page 196
6.1 Basic properties of $t$-designs……Page 198
6.2 The Second Ray-Chaudhuri-Wilson Inequality……Page 203
6.3 Hadamard 3-designs……Page 205
6.4 Cameron’s Theorem……Page 207
6.5 Golay codes and Witt designs……Page 210
6.6 Symmetric designs with parameters (56,11,2) and (176,50,14)……Page 215
Exercises……Page 219
Notes……Page 222
7.1 S trongly regular graphs……Page 224
7.2 Eigenvalues of strongly regular graphs……Page 231
7.3 Switching in strongly regular graphs……Page 235
7.4 Symmetric designs with polarities……Page 245
7.5 Symmetric designs and digraphs……Page 251
Exercises……Page 255
Notes……Page 257
8.1 Association schemes……Page 259
8.2 Quasi-symmetric designs……Page 262
8.3 Multiples of symmetric designs……Page 271
8.4 Quasi-3 symmetric designs……Page 275
8.5 Block schematic designs with three intersection numbers……Page 282
8.6 Designs with a nearly affine decomposition……Page 288
8.7 A symmetric (71,15,3)-design……Page 292
Notes……Page 298
9.1 Group invariant matrices and group rings……Page 301
9.2 Singer and Paley-Hadamard difference sets……Page 311
9.3 Symmetries in a group ring……Page 313
9.4 Building blocks and building sets……Page 319
9.5 McFarland, Spence, and Davis-Jedwab difference sets……Page 322
9.6 Relative difference sets……Page 325
Exercises……Page 331
Notes……Page 333
10.1 Basic properties of BGW-matrices……Page 335
10.2 BGW-matrices with classical parameters……Page 343
10.3 BGW-matrices and relative difference sets……Page 348
10.4 Kronecker product constructions……Page 353
10.5 BGW-matrices and projective geometries……Page 366
Exercises……Page 377
Notes……Page 378
11.1 A symmetric (66,26,10)-design……Page 380
11.2 Global decomposition of symmetric designs……Page 381
11.3 Six infinite families of globally decomposable symmetric designs……Page 386
11.4 Productive Hadamard matrices……Page 388
11.5 Symmetric designs with irregular global decomposition……Page 395
11.6 Decomposable symmetric designs and regular graphs……Page 398
11.7 Local decomposition of symmetric designs……Page 403
11.8 Infinite families of locally decomposable symmetric designs……Page 409
11.9 An infinite family of designs with a nearly affine decomposition……Page 414
Notes……Page 418
12.1 Tight subdesigns……Page 419
12.2 Examples of tight subdesigns……Page 424
12.3 Normal subdesigns……Page 433
12.4 Symmetric designs with $M$-arcs……Page 436
Notes……Page 439
13.1 Quasi-residuals of non-existing symmetric designs……Page 441
13.2 Linear non-embeddability conditions……Page 443
13.3 BGW-matrices and non-embeddability……Page 448
13.4 Non-embeddable quasi-derived designs……Page 455
Exercises……Page 457
Notes……Page 458
14.1 Basic properties of Ryser designs……Page 459
14.2 Type-1 Ryser designs……Page 468
14.3 Ryser designs of prime index……Page 476
14.4 Ryser designs of small index……Page 479
14.5 Ryser designs of small gcd……Page 487
Notes……Page 498
Appendix……Page 500
References……Page 507
Index……Page 526