Sphere Packings, Lattices and GroupsThe main themes. This book is mainly concerned with the problem of packing spheres in Euclidean space of dimensions 1,2,3,4,5, . . . . Given a large number of equal spheres, what is the most efficient (or densest) way to pack them together? We also study several closely related problems: the kissing number problem, which asks how many spheres can be arranged so that they all touch one central sphere of the same size; the covering problem, which asks for the least dense way to cover n-dimensional space with equal overlapping spheres; and the quantizing problem, important for applications to analog-to-digital conversion (or data compression), which asks how to place points in space so that the average second moment of their Voronoi cells is as small as possible. Attacks on these problems usually arrange the spheres so their centers form a lattice. Lattices are described by quadratic forms, and we study the classification of quadratic forms. Most of the book is devoted to these five problems. The miraculous enters: the E 8 and Leech lattices. When we investigate those problems, some fantastic things happen! There are two sphere packings, one in eight dimensions, the E 8 lattice, and one in twenty-four dimensions, the Leech lattice A , which are unexpectedly good and very 24 symmetrical packings, and have a number of remarkable and mysterious properties, not all of which are completely understood even today. |
Contents
7 | 307 |
12 | 316 |
18 | 327 |
Bounds on Kissing Numbers | 337 |
Chapter 15 | 352 |
Rational Invariants of Quadratic Forms | 370 |
10 | 396 |
Chapter 16 | 406 |
5 | 81 |
11 | 87 |
Chapter 4 | 94 |
2 | 110 |
Chapter 23 | 116 |
2 | 118 |
2 | 124 |
Chapter 5 | 136 |
3 | 142 |
Chapter 6 | 157 |
The Main Results | 163 |
Dimensions 9 to 16 | 170 |
Dimensions 17 to 24 | 176 |
Construction A | 182 |
Extremal Type I Codes and Lattices | 189 |
Chapter 19 | 196 |
Constructions A and B for Complex Lattices | 197 |
Extremal Nonbinary Codes and Complex Lattices | 205 |
7 | 223 |
2 | 233 |
Chapter 9 | 245 |
2 | 252 |
2 | 258 |
Chapter 10 | 267 |
5 | 292 |
Chapter 11 | 299 |
Chapter 17 | 421 |
Enumeration of Extremal SelfDual Lattices | 439 |
Decoding Unions of Cosets | 446 |
B Generalized Octahedron or Crosspolytope | 452 |
B Voronoi Cell for A | 459 |
F Voronoi Cell for A | 472 |
The Covering Radius of the Leech Lattice | 476 |
Holes Whose Diagram Contains an A Subgraph | 484 |
Chapter 18 | 494 |
Holes Whose Diagram Contains a D Subgraph | 495 |
Holes Whose Diagram Contains an E Subgraph | 502 |
The Environs of a Deep Hole | 510 |
The Enumeration of the Small Holes | 519 |
Chapter 27 | 527 |
Enumeration of the Leech Roots | 541 |
The Lattices I for n 19 | 547 |
The Monster Group and its 196884Dimensional Space | 554 |
The Dictionary | 560 |
Chapter 30 | 568 |
269 | 573 |
Even Unimodular 24Dimensional Lattices | 590 |
B B Venkov | 632 |
641 | |
654 | |
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Common terms and phrases
A₁ algorithm automorphism group binary bound C-set center density Chap chapter classes codewords column components congruent construction contains coordinates corresponding covering radius Coxeter Coxeter-Dynkin diagram deep holes defined denote determinant diag digits dimension dodecad dual elements equivalent example Figure follows genus given glue vectors Golay code hexacodeword hexads icosian inner product integral invariant isomorphic kissing number laminated lattices lattice packing lattice points Leech lattice Leech roots linear log2 Mac6 Mathieu group matrix minimal norm minimal vectors modulo n-dimensional Niemeier lattices nodes nonzero notation obtained octad orthogonal p-adic permutation polynomial polytope problem Proof quadratic forms root lattices root system satisfies self-dual sextet shown in Fig space sphere packings spherical codes spinor square Steiner system subgroup Table tetrads Theorem theta series unimodular lattice unique vectors of norm vertices Voronoi cell weight enumerator