## 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. |

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### Contents

Chapter | 1 |

Chapter 2 | 31 |

Lattices Quadratic Forms and Number Theory | 41 |

Quantizers | 56 |

Chapter 3 | 63 |

ErrorCorrecting Codes | 75 |

tDesigns Steiner Systems and Spherical tDesigns | 88 |

Chapter 4 | 94 |

Further Constructions for M2 | 327 |

Bounds on Kissing Numbers | 337 |

Chapter 15 | 352 |

Rational Invariants of Quadratic Forms | 370 |

The Classification of Positive Definite Forms | 396 |

Computational Complexity | 402 |

The Mass Formulae for Lattices | 408 |

Chapter 17 | 421 |

Notation Theta Functions | 101 |

The nDimensional Lattices D and D | 117 |

The 24Dimensional Leech Lattice A2 | 131 |

Other Constructions from Codes | 146 |

Construction C | 150 |

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 |

Examples of Construction E | 238 |

Chapter 9 | 245 |

The Linear Programming Bounds | 257 |

Other Bounds | 265 |

Chapter 23 | 268 |

Chapter 11 | 299 |

Completing Octads from 5 of their Points | 305 |

The Octad Group 2 As 3 11 | 311 |

The Octern Group 3 18 | 318 |

Even Unimodular 24Dimensional Lattices | 427 |

Construction of the Niemeier Lattices | 434 |

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 |

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 |

Supplementary Bibliography | 640 |

657 | |

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### Common terms and phrases

automorphism group binary center density Chap chapter classes code of length codewords column congruent Construction contains coordinates corresponding cosets covering radius deep holes defined denote determinant diag digits dimension dodecad dual elements entries equivalent example Figure finite fixing follows function given glue vector group of order hexacode hexacodeword hexads icosian inner product invariant involutions isomorphic kissing number laminated lattices lattice packing Leech lattice log2 Macé Mathieu group matrix maximal subgroups minimal norm minimal vectors modulo multiple n-dimensional Niemeier lattices nonzero notation O-O O-O obtained octad orbits orthogonal p-adic pairs permutation points polynomial prime problem Proof quadratic forms root system satisfies self-dual self-dual codes sextet space sphere packings spherical codes spinor square stabilizer Steiner system subgroups of M24 sublattice subset symbol Table tetrads Theorem theta series trio unimodular lattices unique vectors of norm Voronoi cell weight enumerator