Finite Geometry and Combinatorial ApplicationsCambridge University Press, 2 jul 2015 - 285 páginas The projective and polar geometries that arise from a vector space over a finite field are particularly useful in the construction of combinatorial objects, such as latin squares, designs, codes and graphs. This book provides an introduction to these geometries and their many applications to other areas of combinatorics. Coverage includes a detailed treatment of the forbidden subgraph problem from a geometrical point of view, and a chapter on maximum distance separable codes, which includes a proof that such codes over prime fields are short. The author also provides more than 100 exercises (complete with detailed solutions), which show the diversity of applications of finite fields and their geometries. Finite Geometry and Combinatorial Applications is ideal for anyone, from a third-year undergraduate to a researcher, who wishes to familiarise themselves with and gain an appreciation of finite geometry. |
Índice
Vector spaces | 15 |
Forms | 25 |
Geometries | 51 |
Combinatorial applications | 93 |
The forbidden subgraph problem | 124 |
MDS codes | 147 |
Appendix A Solutions to the exercises | 191 |
Appendix B Additional proofs | 242 |
Notes and references | 263 |
| 272 | |
| 282 | |
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Finite Geometry and Combinatorial Applications Simeon Ball No hay ninguna vista previa disponible - 2015 |
Términos y frases comunes
affine plane algebraic automorphism bijection bilinear code of length collinear columns of G common neighbours construct coordinates cosets denote dimension dual elements of F entry ex(n Example 7.2 Exercise finite field generalised quadrangle graph G Hence hyperoval hyperplane implies incidence structure intersect isomorphism Kakeya set Lemma length q Let f Let G linear code linear forms linear MDS code linearly equivalent linearly independent lines incident M(id matrix mutually orthogonal latin non-degenerate non-square non-zero vector number of points one-dimensional subspaces orthogonal latin squares ovoid plane of order polar space projective plane projective space Proof Let Proof Suppose Prove q is odd quadratic form Reed–Solomon code respect set of points singular vectors space of rank squares of order strongly regular graph subgroup subset subspace of Vk tangent totally singular subspace unique vector space vertex Vk F zero