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Sphere Packings, Lattices and Groups

  • Book
  • © 1988

Overview

Authors:
  1. J. H. Conway

    1. Mathematics Department, Princeton University, Princeton, USA

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    You can also search for this author in PubMed   Google Scholar

  2. N. J. A. Sloane

    1. Mathematical Science Department, AT&T Bell Laboratories, Murray Hill, USA

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 290)

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Table of contents (30 chapters)

  1. Front Matter

    Pages i-xxvii
    Download chapter PDF

  2. Sphere Packings and Kissing Numbers

    • J. H. Conway, N. J. A. Sloane
    Pages 1-30

  3. Coverings, Lattices and Quantizers

    • J. H. Conway, N. J. A. Sloane
    Pages 31-62

  4. Codes, Designs and Groups

    • J. H. Conway, N. J. A. Sloane
    Pages 63-93

  5. Certain Important Lattices and Their Properties

    • J. H. Conway, N. J. A. Sloane
    Pages 94-135

  6. Sphere Packing and Error-Correcting Codes

    • John Leech, N. J. A. Sloane
    Pages 136-156

  7. Laminated Lattices

    • J. H. Conway, N. J. A. Sloane
    Pages 157-180

  8. Further Connections Between Codes and Lattices

    • N. J. A. Sloane
    Pages 181-205

  9. Algebraic Constructions for Lattices

    • J. H. Conway, N. J. A. Sloane
    Pages 206-244

  10. Bounds for Codes and Sphere Packings

    • N. J. A. Sloane
    Pages 245-266

  11. Three Lectures on Exceptional Groups

    • J. H. Conway
    Pages 267-298

  12. The Golay Codes and The Mathieu Groups

    • J. H. Conway
    Pages 299-330

  13. A Characterization of the Leech Lattice

    • J. H. Conway
    Pages 331-336

  14. Bounds on Kissing Numbers

    • A. M. Odlyzko, N. J. A. Sloane
    Pages 337-339

  15. Uniqueness of Certain Spherical Codes

    • E. Bannai, N. J. A. Sloane
    Pages 340-351

  16. On the Classification of Integral Quadratic Forms

    • J. H. Conway, N. J. A. Sloane
    Pages 352-405

  17. Enumeration of Unimodular Lattices

    • J. H. Conway, N. J. A. Sloane
    Pages 406-420

  18. The 24-Dimensional Odd Unimodular Lattices

    • R. E. Borcherds
    Pages 421-426

  19. Even Unimodular 24-Dimensional Lattices

    • B. B. Venkov
    Pages 427-438

  20. Enumeration of Extremal Self-Dual Lattices

    • J. H. Conway, A. M. Odlyzko, N. J. A. Sloane
    Pages 439-442
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Keywords

About this book

The 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.

Authors and Affiliations

  • Mathematics Department, Princeton University, Princeton, USA

    J. H. Conway

  • Mathematical Science Department, AT&T Bell Laboratories, Murray Hill, USA

    N. J. A. Sloane

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