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Sphere Packings, Lattices and Groups

  • Textbook
  • © 1999

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. Information Sciences Research, AT&T Labs — Research, Florham Park, USA

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    You can also search for this author in PubMed   Google Scholar

  • A timely and definitive book on this widely applicable subject New edition has been long awaitedsecond edition had been out of stock for some time Describes modern applications to areas such as number theory, coding theory, analog-to-digital conversion and data compression, n-dimensional crystallography, and superstring theory in physics New edition includes a report on recent developments in the field and an updated and enlarged supplementary bibliography with over 800 items Written by two very well known researchers

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-lxxiv
    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 421-428

  19. Even Unimodular 24-Dimensional Lattices

    • B. B. Venkov
    Pages 429-440

  20. Enumeration of Extremal Self-Dual Lattices

    • J. H. Conway, A. M. Odlyzko, N. J. A. Sloane
    Pages 441-444
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Keywords

About this book

We now apply the algorithm above to find the 121 orbits of norm -2 vectors from the (known) nann 0 vectors, and then apply it again to find the 665 orbits of nann -4 vectors from the vectors of nann 0 and -2. The neighbors of a strictly 24 dimensional odd unimodular lattice can be found as follows. If a norm -4 vector v E II . corresponds to the sum 25 1 of a strictly 24 dimensional odd unimodular lattice A and a !-dimensional lattice, then there are exactly two nonn-0 vectors of ll25,1 having inner product -2 with v, and these nann 0 vectors correspond to the two even neighbors of A. The enumeration of the odd 24-dimensional lattices. Figure 17.1 shows the neighborhood graph for the Niemeier lattices, which has a node for each Niemeier lattice. If A and B are neighboring Niemeier lattices, there are three integral lattices containing A n B, namely A, B, and an odd unimodular lattice C (cf. [Kne4]). An edge is drawn between nodes A and B in Fig. 17.1 for each strictly 24-dimensional unimodular lattice arising in this way. Thus there is a one-to-one correspondence between the strictly 24-dimensional odd unimodular lattices and the edges of our neighborhood graph. The 156 lattices are shown in Table 17 .I. Figure I 7. I also shows the corresponding graphs for dimensions 8 and 16.

Reviews

Third Edition

J.H. Conway and N.J.A. Sloane

Sphere Packings, Lattices and Groups

"This is the third edition of this reference work in the literature on sphere packings and related subjects. In addition to the content of the preceding editions, the present edition provides in its preface a detailed survey on recent developments in the field, and an exhaustive supplementary bibliography for 1988-1998. A few chapters in the main text have also been revised."—MATHEMATICAL REVIEWS

Authors and Affiliations

  • Mathematics Department, Princeton University, Princeton, USA

    J. H. Conway

  • Information Sciences Research, AT&T Labs — Research, Florham Park, USA

    N. J. A. Sloane

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