## 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.

### 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

Zbl 0634.52002