Sphere Packings, Lattices and GroupsThe second edition of this timely, definitive, and popular book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to examine related problems such as the kissing number problem, the covering problem, the quantizing problem, and the classification of lattices and quadratic forms. Like the first edition, the second edition describes the applications of these questions to other areas of mathematics and science such as number theory, coding theory, group theory, analog-to-digital conversion and data compression, n-dimensional crystallography, and dual theory and superstring theory in physics. Results as of 1992 have been added to the text, and the extensive bibliography - itself a contribution to the field - is supplemented with approximately 450 new entries. |
Contents
Chapter | 1 |
1 | 12 |
The Kissing Number Problem | 21 |
2 | 31 |
Lattices Quadratic Forms and Number Theory | 41 |
1 | 53 |
Quantizers | 56 |
Chapter 3 | 63 |
12 | 331 |
13 | 337 |
5 | 346 |
Algebraic Constructions for Lattices | 352 |
4 | 364 |
5 | 370 |
1 | 378 |
Classification of Forms of Small Determinant and | 385 |
ErrorCorrecting Codes | 75 |
tDesigns Steiner Systems and Spherical tDesigns | 88 |
Chapter 4 | 94 |
4 | 113 |
3 | 119 |
10 | 129 |
Chapter 5 | 136 |
3 | 142 |
3 | 148 |
Chapter 6 | 157 |
5 | 171 |
Chapter 7 | 181 |
5 | 191 |
Chapter 19 | 196 |
9 | 198 |
Extremal Nonbinary Codes and Complex Lattices | 205 |
Chapter 23 | 220 |
Repeated Differences and Craigs Lattices | 222 |
2 | 233 |
Chapter 9 | 245 |
2 | 252 |
2 | 258 |
Chapter 10 | 267 |
6 | 273 |
5 | 279 |
3 | 287 |
Chapter 8 | 290 |
7 | 294 |
Definitions of the Hexacode | 300 |
7 | 307 |
The Triad Group and the Projective Plane of Order 4 | 314 |
The Classification of Positive Definite Forms | 396 |
Computational Complexity | 402 |
16 | 406 |
The Mass Formulae for Lattices | 408 |
17 | 421 |
18 | 427 |
Construction of the Niemeier Lattices | 434 |
Enumeration of Extremal SelfDual Lattices | 439 |
Decoding Unions of Cosets | 446 |
Chapter 12 | 452 |
B Voronoi Cell for A | 459 |
2 | 461 |
F Voronoi Cell for A | 472 |
The Covering Radius of the Leech Lattice | 476 |
A Characterization of the Leech Lattice | 478 |
Holes Whose Diagram Contains an A Subgraph | 484 |
Holes Whose Diagram Contains a D Subgraph | 495 |
Holes Whose Diagram Contains an E Subgraph | 502 |
5 | 503 |
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 |
9 | 559 |
Constructing Representations for | 566 |
Bibliography | 572 |
4 | 576 |
Supplementary Bibliography | 585 |
| 602 | |
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Common terms and phrases
A₁ algorithm automorphism group binary bound C-set C₁ center density Chap chapter classes code of length codewords column congruent Construction contains coordinates corresponding cosets covering radius Coxeter-Dynkin diagram deep holes defined denote densest determinant diag diagram digits dimensions dodecad dual elements equivalent example finite follows function genus given glue vectors Golay code hexads icosian integral invariant isomorphic kissing number L₁ laminated lattices lattice packing Leech lattice linear log2 Mac6 matrix maximal minimal distance minimal norm minimal vectors modulo n-dimensional Niemeier lattices nonlattice packings nonzero notation obtained octad orthogonal p-adic permutation polynomial polytope prime problem Proof quadratic forms quantizer root lattice root system satisfies self-dual sextet space sphere packing spherical codes spinor square Steiner system subgroup sublattice Table tetrads Theorem theta series transitive unimodular lattices unique Voronoi cell weight enumerator