Preface ........................................................ ix
Notation ....................................................... xi
1 Fields ....................................................... 1
1.1 Rings and fields ........................................ 1
1.2 Field automorphisms ..................................... 6
1.3 The multiplicative group of a finite field .............. 9
1.4 Exercises .............................................. 10
2 Vector spaces ............................................... 15
2.1 Vector spaces and subspaces ............................ 15
2.2 Linear maps and linear forms ........................... 17
2.3 Determinants ........................................... 19
2.4 Quotient spaces ........................................ 20
2.5 Exercises .............................................. 21
3 Forms ....................................................... 25
3.1 σ-Sesquilinear forms ................................... 25
3.2 Classification of reflexive forms ...................... 27
3.3 Alternating forms ...................................... 30
3.4 Hermitian forms ........................................ 34
3.5 Symmetric forms ........................................ 38
3.6 Quadratic forms ........................................ 40
3.7 Exercises .............................................. 47
4 Geometries .................................................. 51
4.1 Projective spaces ...................................... 51
4.2 Polar spaces ........................................... 54
4.3 Quotient geometries .................................... 60
4.4 Counting subspaces ..................................... 61
4.5 Generalised polygons ................................... 65
4.6 Plьcker coordinates .................................... 71
4.7 Polarities ............................................. 74
4.8 Ovoids ................................................. 76
4.9 Exercises .............................................. 83
5 Combinatorial applications .................................. 93
5.1 Groups ................................................. 93
5.2 Finite analogues of structures in real space ........... 99
5.3 Codes ................................................. 105
5.4 Graphs ................................................ 109
5.5 Designs ............................................... 114
5.6 Permutation polynomials ............................... 117
5.7 Exercises ............................................. 120
6 The forbidden subgraph problem ............................. 124
6.1 The Erdos-Stone theorem ............................... 124
6.2 Even cycles ........................................... 125
6.3 Complete bipartite graphs ............................. 130
6.4 Graphs containing no K2,s ............................. 132
6.5 A probabilistic construction of graphs containing
no KtiS .............................................. 134
6.6 Graphs containing no K3,3 .............................. 135
6.7 The norm graph ........................................ 137
6.8 Graphs containing no K5,5 ............................. 140
6.9 Exercises ............................................. 144
7 MDS codes .................................................. 147
7.1 Singleton bound ....................................... 147
7.2 Linear MDS codes ...................................... 148
7.3 Dual MDS codes ........................................ 151
7.4 The MDS conjecture .................................... 152
7.5 Polynomial interpolation .............................. 154
7.6 The A-functions ....................................... 155
7.7 Lemma of tangents ..................................... 157
7.8 Combining interpolation with the lemma of tangents .... 162
7.9 A proof of the MDS conjecture for k ≤ p ............... 164
7.10 More examples of MDS codes of length q + 1 ............ 165
7.11 Classification of linear MDS codes of length q + 1
for k ≤ p ............................................. 167
7.12 The set of linear forms associated with a linear
MDS code .............................................. 172
7.13 Lemma of tangents in the dual space ................... 174
7.14 The algebraic hypersurface associated with a linear
MDS code .............................................. 177
7.15 Extendability of linear MDS codes ..................... 182
7.16 Classification of linear MDS codes of length q + 1
for k < c√q............................................ 184
7.17 A proof of the MDS conjecture for k < c√q ............. 189
7.18 Exercises ............................................. 189
Appendix A Solutions to the exercises ........................ 191
A.l Fields ................................................ 191
A.2 Vector spaces ......................................... 200
A.3 Forms ................................................. 206
A.4 Geometries ............................................ 213
A.5 Combinatorial applications ............................ 229
A.6 The forbidden subgraph problem ........................ 233
A.7 MDS codes ............................................. 238
Appendix В Additional proofs ................................. 242
B.l Probability ........................................... 242
B.2 Fields ................................................ 243
B.3 Commutative algebra ................................... 247
Appendix С Notes and references .............................. 263
C.l Fields ................................................ 263
C.2 Vector spaces ......................................... 264
C.3 Forms ................................................. 264
C.4 Geometries ............................................ 264
C.5 Combinatorial applications ............................ 266
C.6 The forbidden subgraph problem ........................ 269
C.7 MDS codes ............................................. 270
C.8 Appendices ............................................ 271
References .................................................... 272
Index ......................................................... 282
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