##
**Sphere packings, lattices and groups. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov.
2. ed.**
*(English)*
Zbl 0785.11036

Grundlehren der Mathematischen Wissenschaften. 290. New York: Springer- Verlag. xliii, 679 p. (1993).

[For a review of the 1st ed. (1988) see Zbl 0634.52002.]

The second edition of this well-known treatise contains numerous minor corrections and improvements to the text. Extremely useful is the huge bibliography: The list of about 1.550 items in the first edition has been enlarged by a supplementary bibliography of approximately 350 items in the second edition covering mainly the period 1988 to 1992. The authors have also included a valuable 14 page preface to the second edition in which they summarize the progress obtained in the works of the supplementary bibliography. This report on recent developments is given in the order of the chapters of the work under review.

Summing up the authors have even improved the great usefulness of their work. Anyone interested in sphere packings, in lattices in \(n\)- dimensional space, in the Leech lattice, in finite groups, quadratic froms, the geometry of numbers, or combinatorics should buy this book, and of course any institutional library. Compared to the wealth of information given, the price is really moderate.

The authors are planning a sequel to the geometry of low-dimensional groups and lattices.

The second edition of this well-known treatise contains numerous minor corrections and improvements to the text. Extremely useful is the huge bibliography: The list of about 1.550 items in the first edition has been enlarged by a supplementary bibliography of approximately 350 items in the second edition covering mainly the period 1988 to 1992. The authors have also included a valuable 14 page preface to the second edition in which they summarize the progress obtained in the works of the supplementary bibliography. This report on recent developments is given in the order of the chapters of the work under review.

Summing up the authors have even improved the great usefulness of their work. Anyone interested in sphere packings, in lattices in \(n\)- dimensional space, in the Leech lattice, in finite groups, quadratic froms, the geometry of numbers, or combinatorics should buy this book, and of course any institutional library. Compared to the wealth of information given, the price is really moderate.

The authors are planning a sequel to the geometry of low-dimensional groups and lattices.

Reviewer: J.Elstrodt (Münster)

### MSC:

11H31 | Lattice packing and covering (number-theoretic aspects) |

52C07 | Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) |

52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |

11H06 | Lattices and convex bodies (number-theoretic aspects) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05B40 | Combinatorial aspects of packing and covering |

52-02 | Research exposition (monographs, survey articles) pertaining to convex and discrete geometry |

03B30 | Foundations of classical theories (including reverse mathematics) |

05B05 | Combinatorial aspects of block designs |

20D08 | Simple groups: sporadic groups |

11F27 | Theta series; Weil representation; theta correspondences |

94B99 | Theory of error-correcting codes and error-detecting codes |

### Keywords:

kissing number; exceptional group; Monster group; codes; modular forms; theta series; bibliography; sphere packings; lattices in \(n\)-dimensional space; Leech lattice; finite groups; quadratic froms; geometry of numbers; combinatorics### Citations:

Zbl 0634.52002
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\textit{J. H. Conway} and \textit{N. J. A. Sloane}, Sphere packings, lattices and groups. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. 2. ed. New York: Springer-Verlag (1993; Zbl 0785.11036)

### Online Encyclopedia of Integer Sequences:

Coordination sequence for hyperbolic tessellation 3^7 (from triangle group (2,3,7)).Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).

Theta series of lattice A_2 tensor E_8 (dimension 16, det. 6561, min. norm 4). Also theta series of Eisenstein version of E_8 lattice.

Theta series of 15-dimensional unimodular lattice A15+.

Theta series of odd Leech lattice (the unique unimodular 24-dimensional lattice with minimal norm 3).

Smallest dimension in which a unimodular lattice of minimal norm n occurs.

Smallest dimension in which a strictly odd unimodular lattice of minimal norm n occurs.

Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n.

Second coefficient of extremal theta series of even unimodular lattice in dimension 24n.

Theta series of 14-dimensional lattice Kappa_{14} with minimal norm 4.

16*a(n) gives theta series of the shadow of the 24-dimensional odd Leech lattice.

Number of points in Z^10 of norm <= n.

Theta series of Barnes-Wall lattice in 128 dimensions.

Theta series of Barnes-Wall lattice in 64 dimensions.

Weight distribution of [12,6,6]_3 ternary extended Golay code.

Weight distribution of [11,6,5]_3 ternary Golay perfect code.

Numbers n such that A008949(n) is a power of 2.

Theta series of Kagome net with respect to an atom.

Theta series of Kagome net with respect to a deep hole.

Represent the ring of Eisenstein integers E = {x+y*omega: x, y rational integers, omega = exp(2*Pi*i/3)} by the cells of a hexagonal grid; number the cells of the grid along a counterclockwise hexagonal spiral, with the cells 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

Represent the ring of Gaussian integers E = {x+y*i: x, y rational integers, i = sqrt(-1)} by the cells of a square grid; number the cells of the grid along a counterclockwise square spiral, with the cells representing the ring identities 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

Theta series of 15-dimensional lattice Kappa_15.

Theta series of 16-dimensional lattice Kappa_16.

Theta series of 17-dimensional lattice Kappa_17.

Theta series of 18-dimensional lattice Kappa_18.

Theta series of 19-dimensional lattice Kappa_19.

Theta series of 20-dimensional lattice Kappa_20.