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**Sphere packings, lattices and groups. With additional contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov.**
*(English)*
Zbl 0634.52002

Grundlehren der Mathematischen Wissenschaften, 290. New York etc.: Springer-Verlag. xxvii, 663 p., DM 178.00 (1988).

The main subject of this comprehensive monograph is the problem of packing spheres in \(n\)-dimensional Euclidean space and related problems as the kissing number problem asking how many spheres can be arranged so that they all touch one central sphere, the covering problem, and the quantizing problem having applications in data compression. In many cases the optimal solutions of these problems are provided by arrangements of spheres using lattices. Thus the study of lattices and the related classification of quadratic forms is another central theme in the book. Connections between these problems and error-correcting codes, Steiner systems, \(t\)-designs, and finite groups are also investigated. The geometrical problems treated in the book have numerous applications in other areas of mathematics, especially in number theory, as well as outside mathematics, as channel coding, chemistry (crystallography), and physics (dual theory and superstring theory).

The text is based on a series of articles, but a number of new facts which have not appeared in print before, are presented, including tables of the densest sphere packings, coverings and quantizers, lattices, spherical codes, new formulae for the theta series of many lattices, and a simple description for the Monster simple group. The book contains also many tables available before only in journals or conference proceedings, such as bounds for kissing numbers, the even and odd unimodular lattices in dimensions up to 24, tables of the best known codes, groups associated with the Leech lattice, simple groups related to the Monster, etc.

The text is organized in a preface, 30 chapters, index, and bibliography of about 1550 titles. The first three chapters provide a survey of the present state of the packing, kissing number, covering and quantizing problems. Chapter 4 describes a number of important lattices. In chapters 5–8 various constructions of sphere packings based on codes are described. Chapter 9 introduces bounds for codes and sphere packings obtained by techniques from harmonic analysis and linear programming. Chapter 10 and 11 study the Golay codes and their automorphism groups, as well as the group of the Leech lattice. Chapter 12 gives a proof of the uniqueness of the Leech lattice. Chapters 13 and 14 discuss the unique solutions of the kissing number problem in 8 and 24 dimensions. Chapters 15–19 deal with the classification of integral quadratic forms. The next four chapters 20–23 treat geometric properties of lattices and the evaluation of the covering radius of the Leech lattice. Chapter 24 gives various constructions of the Leech lattice based on Golay codes, the deep holes and Niemeier lattices. Chapter 25 contains a classification of the holes in the Leech lattice. Chapter 26 and 27 discuss Lorentzian forms for the Leech lattice and the automorphism group of the 26-dimensional Lorentzian lattice. Chapter 28 is about Leech roots and Vinberg groups. The last two chapters 29 and 30 describe a construction of the biggest sporadic simple group, the Monster, and a Lie algebra possibly related to the Monster.

The material is arranged in increasing order of difficulty and requiring background, so that the first chapters are supposed to be accessible even for undergraduates. The book is addressed to everybody interested in sphere packings and lattices, mathematicians interested in finite groups, quadratic forms, the geometry of numbers and combinatorics. It is also recommended to engineers working on channel coding and vector quantizers, chemists and physicists interested in n-dimensional crystallography.

The text is based on a series of articles, but a number of new facts which have not appeared in print before, are presented, including tables of the densest sphere packings, coverings and quantizers, lattices, spherical codes, new formulae for the theta series of many lattices, and a simple description for the Monster simple group. The book contains also many tables available before only in journals or conference proceedings, such as bounds for kissing numbers, the even and odd unimodular lattices in dimensions up to 24, tables of the best known codes, groups associated with the Leech lattice, simple groups related to the Monster, etc.

The text is organized in a preface, 30 chapters, index, and bibliography of about 1550 titles. The first three chapters provide a survey of the present state of the packing, kissing number, covering and quantizing problems. Chapter 4 describes a number of important lattices. In chapters 5–8 various constructions of sphere packings based on codes are described. Chapter 9 introduces bounds for codes and sphere packings obtained by techniques from harmonic analysis and linear programming. Chapter 10 and 11 study the Golay codes and their automorphism groups, as well as the group of the Leech lattice. Chapter 12 gives a proof of the uniqueness of the Leech lattice. Chapters 13 and 14 discuss the unique solutions of the kissing number problem in 8 and 24 dimensions. Chapters 15–19 deal with the classification of integral quadratic forms. The next four chapters 20–23 treat geometric properties of lattices and the evaluation of the covering radius of the Leech lattice. Chapter 24 gives various constructions of the Leech lattice based on Golay codes, the deep holes and Niemeier lattices. Chapter 25 contains a classification of the holes in the Leech lattice. Chapter 26 and 27 discuss Lorentzian forms for the Leech lattice and the automorphism group of the 26-dimensional Lorentzian lattice. Chapter 28 is about Leech roots and Vinberg groups. The last two chapters 29 and 30 describe a construction of the biggest sporadic simple group, the Monster, and a Lie algebra possibly related to the Monster.

The material is arranged in increasing order of difficulty and requiring background, so that the first chapters are supposed to be accessible even for undergraduates. The book is addressed to everybody interested in sphere packings and lattices, mathematicians interested in finite groups, quadratic forms, the geometry of numbers and combinatorics. It is also recommended to engineers working on channel coding and vector quantizers, chemists and physicists interested in n-dimensional crystallography.

Reviewer: Vladimir D. Tonchev (Houghton)

### MSC:

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

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

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

05B05 | Combinatorial aspects of block designs |

05B30 | Other designs, configurations |

94B75 | Applications of the theory of convex sets and geometry of numbers (covering radius, etc.) to coding theory |

20D08 | Simple groups: sporadic groups |

05B40 | Combinatorial aspects of packing and covering |