Sphere Packings, Lattices and Groups

1. Front Matter

   Pages i-lxxiv

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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

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.

• Graph
• Group theory
• Lie algebra
• classification
• construction
• ring theory
• theory

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

• Mathematics Department, Princeton University, Princeton, USA

  J. H. Conway

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

  N. J. A. Sloane