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Look Inside From Error-Correcting Codes through Sphere Packings to Simple Groups

From Error-Correcting Codes through Sphere Packings to Simple Groups

£24.99

Part of Carus Mathematical Monographs

  • Date Published: October 2004
  • availability: This item is not supplied by Cambridge University Press in your region. Please contact Mathematical Association of America for availability.
  • format: Paperback
  • isbn: 9780883850374

£ 24.99
Paperback

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About the Authors
  • 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.

    • A well received book now available in paperback
    • The title traces through a quarter-century of mathematicians and their theories, which describe the on-going interaction between applied and theoretical mathematics
    • The material has been written so that any mathematician with even a casual acquaintance with vector spaces and groups should be able to follow each mathematical step
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    Reviews & endorsements

    'Thompson provides an excellent example of the constant interaction between applied and theoretical mathematics. Thompson's work is first-rate historical research and good clear writing. As a mathematics teacher, he has gone to great pains to make the material understandable to anyone with even a casual acquaintance with vector spaces and groups.' Annals of the History of Computing

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    Product details

    • Date Published: October 2004
    • format: Paperback
    • isbn: 9780883850374
    • length: 244 pages
    • dimensions: 188 x 127 x 15 mm
    • weight: 0.235kg
    • contains: 64 b/w illus. 59 tables
    • availability: This item is not supplied by Cambridge University Press in your region. Please contact Mathematical Association of America for availability.
  • Table of Contents

    1. The origin of error-correcting codes
    an introduction to coding
    the work of Hamming
    the Hamming-Holbrook patent
    the Hamming codes are linear
    the work of Golay
    the priority controversy
    2. From coding to sphere packing
    an introduction to sphere packing
    the Leech connection
    the origin of Leech's first packing in E24
    the matrix for Leech's first packing
    the Leech lattice
    3. From sphere packing to new simple groups
    is there an interesting group in Leech's lattice?
    the hard sell of a simple group
    twelve hours on Saturday, six on Wednesday
    the structure of 0
    new simple groups
    Appendix 1. Densest known sphere packings
    Appendix 2. Further properties of the (12,24) Golay code and the related Steiner system s (5,8,24)
    Appendix 3. A calculation of the number of spheres with centers in Λ2
    Appendix 4. The Mathieu group M24 and the order of M22
    Appendix 5. The proof of lemma 3.3
    Appendix 6. The sporadic simple groups
    Bibliography
    Index.

  • Author

    Thomas M. Thompson, Walla Walla College, Washington

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