 | Moore E.H. Difference sets: connecting algebra, combinatorics, and geometry / E.H.Moore, H.S.Pollatsek. - Providence: AMS, 2013. - xiii, 298 p. - (Student mathematical library; vol.67). - Bibliogr.: p.287-291. - Ind.: p.293-295. - ISBN 978-0-8218-9176-6
|
Preface ........................................................ xi
Chapter 1. Introduction ......................................... 1
Chapter 2. Designs ............................................ 11
2.1 Incidence structures ...................................... 11
2.2 t-Designs ................................................. 14
2.3 Affine planes ............................................. 20
2.4 Symmetric designs ......................................... 26
2.5 Projective geometry ....................................... 30
Chapter 3. Automorphisms of Designs ............................ 37
3.1 Group actions ............................................. 37
3.2 Automorphisms of symmetric designs ........................ 40
Chapter 4. Introducing Difference Sets ......................... 45
4.1 Definition and examples ................................... 46
4.2 Difference sets and designs ............................... 54
4.3 Integral group ring ....................................... 59
4.4 Equivalence ............................................... 65
Chapter 5. Bruck-Ryser-Chowla Theorem .......................... 71
5.1 The BRC Theorem ........................................... 72
5.2 Proof of BRC for v odd .................................... 76
5.3 Partial converse and extension of BRC ..................... 84
Chapter 6. Multipliers ......................................... 87
6.1 Definition and examples ................................... 87
6.2 Existence of numerical multipliers ........................ 91
6.3 Multipliers fix sD ........................................ 94
6.4 Using multipliers ......................................... 96
6.5 Multipliers in non-cyclic groups .......................... 99
Chapter 7. Necessary Group Conditions ......................... 103
7.1 Intersection numbers ..................................... 103
7.2 Turyn's exponent bound ................................... 112
7.3 Dillon's dihedral trick .................................. 116
Chapter 8. Difference Sets from Geometry ...................... 121
8.1 Singer difference sets ................................... 121
8.2 Turyn's construction ..................................... 125
8.3 McFarland difference sets ................................ 129
Chapter 9. Families from Hadamard Matrices .................... 135
9.1 Hadamard matrices ........................................ 135
9.2 Paley-Hadamard family: ν = 4n - 1 ........................ 141
9.3 Hadamard family: ν = 4n .................................. 155
Chapter 10. Representation Theory ............................. 167
10.1 Definitions and examples ................................. 167
10.2 Equivalent representations ............................... 177
10.3 Maschke's Theorem ........................................ 179
10.4 Representations and difference sets ...................... 191
Chapter 11. Group Characters .................................. 197
11.1 Definitions and examples ................................. 198
11.2 The Fundamental Theorem .................................. 201
11.3 Proof of the Fundamental Theorem ......................... 209
11.4 Characters and difference sets ........................... 220
11.5 Character tables ......................................... 228
Chapter 12. Using Algebraic Number Theory ..................... 233
12.1 Why algebraic number theory? ............................. 233
12.2 Definitions and basic facts .............................. 235
12.3 Seeking difference sets .................................. 240
12.4 Proving Turyn's exponent bound ........................... 247
Chapter 13. Applications ...................................... 253
13.1 Binary sequences ......................................... 253
13.2 Imaging with coded masks ................................. 257
13.3 Error correcting codes ................................... 261
13.4 Quantum information and MUBs ............................. 263
Appendix A. Background ........................................ 267
Appendix B. Notation .......................................... 273
Appendix C. Hints and Solutions to Selected Exercises ......... 277
Bibliography .................................................. 287
Index ......................................................... 293
Index of Parameters ........................................... 297
|
|
