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From Error-Correcting Codes Through Sphere Packings to Simple Groups
 
From Error-Correcting Codes Through Sphere Packings to Simple Groups
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-5460-9
Product Code:  CAR/21
List Price: $55.00
MAA Member Price: $41.25
AMS Member Price: $41.25
eBook ISBN:  978-1-61444-021-5
Product Code:  CAR/21.E
List Price: $50.00
MAA Member Price: $37.50
AMS Member Price: $37.50
Softcover ISBN:  978-1-4704-5460-9
eBook: ISBN:  978-1-61444-021-5
Product Code:  CAR/21.B
List Price: $105.00 $80.00
MAA Member Price: $78.75 $60.00
AMS Member Price: $78.75 $60.00
From Error-Correcting Codes Through Sphere Packings to Simple Groups
Click above image for expanded view
From Error-Correcting Codes Through Sphere Packings to Simple Groups
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-5460-9
Product Code:  CAR/21
List Price: $55.00
MAA Member Price: $41.25
AMS Member Price: $41.25
eBook ISBN:  978-1-61444-021-5
Product Code:  CAR/21.E
List Price: $50.00
MAA Member Price: $37.50
AMS Member Price: $37.50
Softcover ISBN:  978-1-4704-5460-9
eBook ISBN:  978-1-61444-021-5
Product Code:  CAR/21.B
List Price: $105.00 $80.00
MAA Member Price: $78.75 $60.00
AMS Member Price: $78.75 $60.00
  • Book Details
     
     
    The Carus Mathematical Monographs
    Volume: 211983; 228 pp
    MSC: Primary 94; Secondary 01; 11; 20; 52
    Recipient of the Mathematical Association of America's Beckenbach Book Prize in 1989!

    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.

  • Table of Contents
     
     
    • Chapters
    • Chapter 1. The origin of error-correcting codes
    • Chapter 2. From coding to sphere packing
    • Chapter 3. From sphere packing to new simple groups
    • Appendix 1. Densest known sphere packings
    • Appendix 2. Further properties of the (24, 12) Golay code and the related Steiner system $S(5,8,24)$
    • Appendix 3. A calculation of the number of spheres with centers in $\Lambda _2$ adjacent to one, two, three and four adjacent spheres with centers in $\Lambda _2$
    • Appendix 4. The Mathieu group $M_{24}$ and the order of $M_{22}$
    • Appendix 5. The proof of lemma 3.3
    • Appendix 6. The sporadic simple groups
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 211983; 228 pp
MSC: Primary 94; Secondary 01; 11; 20; 52
Recipient of the Mathematical Association of America's Beckenbach Book Prize in 1989!

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.

  • Chapters
  • Chapter 1. The origin of error-correcting codes
  • Chapter 2. From coding to sphere packing
  • Chapter 3. From sphere packing to new simple groups
  • Appendix 1. Densest known sphere packings
  • Appendix 2. Further properties of the (24, 12) Golay code and the related Steiner system $S(5,8,24)$
  • Appendix 3. A calculation of the number of spheres with centers in $\Lambda _2$ adjacent to one, two, three and four adjacent spheres with centers in $\Lambda _2$
  • Appendix 4. The Mathieu group $M_{24}$ and the order of $M_{22}$
  • Appendix 5. The proof of lemma 3.3
  • Appendix 6. The sporadic simple groups
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.