From error-correcting codes through sphere packings to simple groups.

*(English)*Zbl 0545.94016
The Carus Mathematical Monographs, 21. Washington: The Mathematical Association of America. XIV, 228 p. $ 21.00 (1983).

The main aim of this book is to describe the history of a number of mathematical discoveries and their interconnections. The sequence starts with the origin of error-correcting codes (1947) and the subsequent discovery of perfect codes, in particular the binary Golay code (1949). From there the story proceeds via dense packings of spheres in high dimensions (1964) to the Leech lattice (1967). This leads to Conway’s work on simple groups and his three sporadic simple groups (1968). Simultaneously the book describes the mathematics in question, in general in such a way that very little mathematical background is needed to understand the exposition.

Through correspondence, taped interviews, and telephone conversations the author has tracked the history of these discoveries and recorded many events which influenced this wonderful area of mathematics. The first chapter describes how Hamming was led to the idea of error-correcting codes, after which Shannon’s fundamental paper on the mathematical theory of communication led to the rapid development of coding theory. The main topics in the first chapter are the Hamming codes, the idea of perfect codes, and the two Golay codes. It includes an interesting description of the Hamming-Golay priority controversy. The information on other perfect codes is sketchy and incomplete.

The second chapter starts with classical results on sphere packings, followed by a detailed description of Leech’s work on packings in 24- dimensional space. This led to the discovery of the Leech lattice. In the final chapter one learns how difficult it was for Leech to get somebody interested in finding the automorphism groups of his lattice. When Conway finally started to work on the problem, it took less than a day to discover his groups. The most detailed part of the book is the fifty page analysis of the structure of.0. In several appendices one finds among other things tables of dense sphere packings, the sporadic simple groups, a description of the Golay code and S(5,8,24), and one of the Mathieu group \(M_{24}\). In both aims, namely the description of the mathematics itself and showing its origin and evolution, the book is successful.

Through correspondence, taped interviews, and telephone conversations the author has tracked the history of these discoveries and recorded many events which influenced this wonderful area of mathematics. The first chapter describes how Hamming was led to the idea of error-correcting codes, after which Shannon’s fundamental paper on the mathematical theory of communication led to the rapid development of coding theory. The main topics in the first chapter are the Hamming codes, the idea of perfect codes, and the two Golay codes. It includes an interesting description of the Hamming-Golay priority controversy. The information on other perfect codes is sketchy and incomplete.

The second chapter starts with classical results on sphere packings, followed by a detailed description of Leech’s work on packings in 24- dimensional space. This led to the discovery of the Leech lattice. In the final chapter one learns how difficult it was for Leech to get somebody interested in finding the automorphism groups of his lattice. When Conway finally started to work on the problem, it took less than a day to discover his groups. The most detailed part of the book is the fifty page analysis of the structure of.0. In several appendices one finds among other things tables of dense sphere packings, the sporadic simple groups, a description of the Golay code and S(5,8,24), and one of the Mathieu group \(M_{24}\). In both aims, namely the description of the mathematics itself and showing its origin and evolution, the book is successful.

Reviewer: J.H.van Lint

##### MSC:

94B25 | Combinatorial codes |

05B40 | Combinatorial aspects of packing and covering |

20D08 | Simple groups: sporadic groups |

94-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

05-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics |