From error-correcting codes through sphere packings to simple groups / by Thomas M. Thompson.

Format:

Book

Author/Creator:

Stein, Sherman K., author.

Contributor:

Mathematical Association of America, issuing body.

Series:

Carus mathematical monographs ; Number 21.

Carus Mathematical Monographs ; Number 21

Language:

English

Subjects (All):

Error-correcting codes (Information theory).

Finite simple groups.

Physical Description:

1 online resource (xiv, 228 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Washington : Mathematical Association of America, 1983.

Language Note:

English

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.

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 lattice""""Chapter 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 Spheres with Centers in A2""""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""

Notes:

Title from publisher's bibliographic system (viewed on 02 Oct 2015).

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-61444-021-2