Sphere Packings, Lattices and Groups

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Springer Science & Business Media, Mar 9, 2013 - Mathematics - 682 pages
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The 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
Index
657
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