## 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 | 8 |

3 | 24 |

Chapter 2 | 31 |

3 | 52 |

2 | 59 |

Chapter 4 | 94 |

6 | 114 |

12 | 316 |

18 | 327 |

Bounds on Kissing Numbers | 337 |

Algebraic Constructions for Lattices | 352 |

Rational Invariants of Quadratic Forms | 370 |

9 | 387 |

7 | 393 |

11 | 400 |

2 | 124 |

The 24Dimensional Leech Lattice | 131 |

6 | 139 |

4 | 145 |

Chapter 19 | 146 |

3 | 148 |

8 | 155 |

Chapter 13 | 157 |

The Main Results | 162 |

Chapter 7 | 181 |

5 | 191 |

9 | 202 |

Chapter 23 | 220 |

3 | 227 |

2 | 233 |

Chapter 9 | 245 |

2 | 252 |

2 | 258 |

Chapter 10 | 267 |

5 | 279 |

3 | 287 |

Chapter 8 | 290 |

7 | 294 |

2 | 300 |

7 | 307 |

Chapter 16 | 406 |

Chapter 17 | 421 |

Chapter 20 | 443 |

Chapter 21 | 451 |

3 | 460 |

3 | 471 |

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 Cellular Structure of the Leech Lattice | 513 |

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 |

11 | 560 |

Chapter 30 | 568 |

Even Unimodular 24Dimensional Lattices | 590 |

B B Venkov | 632 |

657 | |

666 | |

672 | |

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### Common terms and phrases

algebraic algorithm automorphism group binary C-set center density Chap chapter classes code of length codewords column congruent Construction contains coordinates corresponding covering radius deep holes defined denote densest determinant diagram digits dimensions dodecad dual elements equivalent example Figure finite fixing follows functions given glue vectors Golay code hexacodeword hexads inner product invariant isomorphic kissing number laminated lattices lattice packing Leech lattice log2 Mac6 Math Mathieu group matrix maximal subgroups minimal distance minimal norm minimal vectors modulo n-dimensional N. J. A. Sloane Niemeier lattices nonlattice packings nonzero obtained octad orthogonal p-adic permutation PGIT polynomial polytopes problem Proof quadratic forms satisfies self-dual codes sextet shown in Fig simple group space sphere packings spherical codes spinor Steiner system Table tetrads Theorem theory theta series transitive unimodular lattices unique Voronoi cell weight enumerator