##
**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.
3rd ed.**
*(English)*
Zbl 0915.52003

Grundlehren der Mathematischen Wissenschaften. 290. New York, NY: Springer. lxxiv, 703 p. (1999).

For a review of the 1st ed. see (1988; Zbl 0634.52002) and of the 2nd ed. see (1993; Zbl 0785.11036).

The third edition of this well-known book shows the still growing interest in the subject matter of this book, which sometimes is called the bible of sphere packings. In fact the “Conway-Sloane” is indispensable for everybody working on this subject and related fields.

Compared with the first and second edition, the third one has changed in two essential points (besides some small corrections and improvements to the main text): The bibliography and the preface.

In particular the bibliography, which was already excellent in the first edition, has been enlarged and completed remarkably. It now consists of 106 pages. The supplementary bibliography (1988-1998) alone has 38 pages or more than 800 items.

Extremely useful is also the preface to the 3rd edition, which is a substantially expanded version of the preface to the 2nd edition. This 35-pages preface gives an excellent report on the essential developments during the last decade, i.e. since the appearance of the first edition. So the preface is an excellent survey itself. For the contents of the main part, which contains 30 chapters, we refer to the review of the first edition. Needless to say that they are updated and that they (different from the bible) contain also proofs.

Anybody interested in sphere packings, codes, lattices in \(n\)-space, Leech latticee geometry of numbers, finite groups or quadratic forms should buy this book. Of course also any mathematical institute and any library should buy it. But this timely book is also addressed to engineers working on coding theory, to physicists and chemists and crystallographers.

The price of this book is really moderate compared to the wealth and abundance of material and information. So it can also be recommended to students.

The third edition of this well-known book shows the still growing interest in the subject matter of this book, which sometimes is called the bible of sphere packings. In fact the “Conway-Sloane” is indispensable for everybody working on this subject and related fields.

Compared with the first and second edition, the third one has changed in two essential points (besides some small corrections and improvements to the main text): The bibliography and the preface.

In particular the bibliography, which was already excellent in the first edition, has been enlarged and completed remarkably. It now consists of 106 pages. The supplementary bibliography (1988-1998) alone has 38 pages or more than 800 items.

Extremely useful is also the preface to the 3rd edition, which is a substantially expanded version of the preface to the 2nd edition. This 35-pages preface gives an excellent report on the essential developments during the last decade, i.e. since the appearance of the first edition. So the preface is an excellent survey itself. For the contents of the main part, which contains 30 chapters, we refer to the review of the first edition. Needless to say that they are updated and that they (different from the bible) contain also proofs.

Anybody interested in sphere packings, codes, lattices in \(n\)-space, Leech latticee geometry of numbers, finite groups or quadratic forms should buy this book. Of course also any mathematical institute and any library should buy it. But this timely book is also addressed to engineers working on coding theory, to physicists and chemists and crystallographers.

The price of this book is really moderate compared to the wealth and abundance of material and information. So it can also be recommended to students.

Reviewer: Jörg M.Wills (Siegen)

### MathOverflow Questions:

Flat metrics on \(n\)-toruses, their systoles, and the ”shortest vector problem”### MSC:

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

52C07 | Lattices and convex bodies 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:

Leech lattice; kissing number; lattice packings; codes; spherical codes; designs; geometry of numbers; coverings; finite groups; sphere packings; lattices; quadratic forms
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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. 3rd ed. New York, NY: Springer (1999; Zbl 0915.52003)