From error-correcting codes through sphere packings to semigroups

From error-correcting codes through sphere packings to semigroups

by Thompson, Thomas M.

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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 lead 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 neighbo…

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Title Statements

  • Full title :From error-correcting codes through sphere packings to semigroups / by Thomas M. Thompson.

Summary

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 lead 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 neighboring 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 some 10,000+ pages. And these connections, along with the fascinating history and the proof of the simplicity of one of those "sporatic" simple groups, are presented at an undergraduate mathematical level.

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Edition Statements

  • Publication Date:©1983

Publication Statements

The Physical Object

  • Description:1 online resource (244 pages) : : illustrations

Edition Identifiers

  • ISBN:9781614440215, 1614440212, 0883850370, 9780883850374
  • oclc:ocn880452868
  • LC Classification:QA268
  • Dewey Decimal:512.2/3

Table of Contents

Front Cover -- From Error-Correcting Codes Through Sphere Packings to Simple Groups -- Copyright Page -- Contents -- Preface -- Chapter 1. The Origin of Error-Correcting Codes -- 1. An introduction to coding -- 2. The work of Hamming -- 3. The Hamming-Holbrook patent -- 4. The Hamming codes are linear -- 5. The work of Golay -- 6. The priority controversy -- Chapter 2. From Coding to Sphere Packing -- 1. An introduction to sphere packing -- 2. The Leech connection -- 3. The origin of Leech's first packing in E24 -- 4. The matrix for Leech's first packing

5. The Leech latticeChapter 3. From Sphere Packing to New Simple Groups -- 1. Is there an interesting group in Leech's lattice? -- 2. The hard sell of a simple group -- 3. Twelve hours on Saturday, six on Wednesday -- 4. The structure of ·0 -- A. Introduction -- B. More on A -- C. The Mathieu groups -- D. The group Î? -- Î?. The order of.0. -- 5. 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 A2 Adjacent to One, Two, Three and Four Adjacent Sphe…

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Subjects

Codes correcteurs d'erreurs (Théorie de… Conference papers and proceedings Error-correcting codes (Information the… Error-correcting codes (Information the… Finite simple groups Groupes simples finis

Writers

Thompson, Thomas M.

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