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

Conway, J. H.; Sloane, N. J. A.

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.

Reviewer: Jörg M.Wills (Siegen)

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

Citations:

Zbl 0634.52002; Zbl 0785.11036