From Error-Correcting Codes through Sphere Packings to Simple Groups

Carus Mathematical Monographs #21

by Thomas M. Thompson

This book traces a remarkable path of mathematical connections through seemingly disparate topics. Frustrations with a 1940's electro-mechanical computer at a premier research laboratory begin this story. Subsequent mathematical methods of encoding messages to ensure correctness when transmitted over noisy channels led to discoveries of extremely efficient lattice packings of equal-radius balls, especially in 24-dimensional space. In turn, this highly symmetric lattice, with each point neighbouring exactly 196,560 other points, suggested the possible presence of new simple groups as groups of symmetries. Indeed, new groups were found and are now part of the 'Enormous Theorem' - the classification of all simple groups whose entire proof runs to some 10,000+ pages. And these connections, along with the fascinating history and the proof of the simplicity of one of those 'sporadic' simple groups, are presented at an undergraduate mathematical level.

Genres: Mathematics

244 pages, Paperback

First published January 1, 1984

Reviews

[Francis · Rating 5 out of 5]

The book under review is an expository gem that traces an interconnected set of mathematical developments across multiple fields. This book traces a fascinating mathematical journey that cuts across several disparate fields of mathematics and forms a key component of one of the greatest achievements of 20th century mathematics, the classification of finite simple groups. It begins in the 1940’s at Bell Labs, at the dawn of information theory, where Claude Shannon’s seminal work “A Mathematical Theory of Communication” established the unity of all information and laid the groundwork for essentially all digital communication. Shannon defined the “capacity” of a noisy channel (the maximum rate at which information can be transmitted across it) and proved that if one wants to send information across the channel at a rate beneath the capacity of the channel, then there exist "error correcting codes" that can transmit the information at that rate with arbitrarily low probability of transmission error.
This book requires a solid foundation in linear algebra, group theory (including the Sylow theorems, permutation groups, group actions on sets, and perhaps a little about linear groups), a little number theory (exposure to quadratic reciprocity would be good), and some combinatorics. There are reasonably self-contained introductions to several fundamental mathematical objects such as the Fano projective plane, finite fields, and linear groups, as well as very accessible and concrete introductions to the principal characters in the book such as error-correcting codes, sphere packings and Euclidean lattices. This book is a beautiful mix of linear algebra, combinatorics and group theory, and is highly recommended to all interested readers at the advanced undergraduate level and above.

This is an excerpt of a review posted on MAA Reviews at http://mathdl.maa.org/mathDL/19/ which may require a subscription.