From Error-Correcting Codes Through Sphere Packings to Simple Groups

From Error-Correcting Codes Through Sphere Packings to Simple Groups

Thomas M. Thompson
Volume: 21
Copyright Date: 1983
Edition: 1
Pages: 244
https://www.jstor.org/stable/10.4169/j.ctt5hh9fv
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  • Book Info
    From Error-Correcting Codes Through Sphere Packings to Simple Groups
    Book Description:

    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.

    eISBN: 978-1-61444-021-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. PREFACE
    (pp. vii-xii)
    Thomas M. Thompson
  3. Table of Contents
    (pp. xiii-xv)
  4. CHAPTER 1 THE ORIGIN OF ERROR-CORRECTING CODES
    (pp. 1-60)

    Richard W. Hamming’s encounter with the Bell Telephone Laboratories’ mechanical relay computer in 1947 (quoted in the Preface) initiated what has come to be known as coding theory. In this chapter we will trace these origins of coding theory and develop enough of the theory itself to prepare for Leech’s work in sphere packing which appears in Chapter 2.

    Communication is imperfect. Even if a message is accurately stated, it may be garbled during transmission; and the consequences of a mistake in the interpretation of a financial, diplomatic, military or other message may be unfortunate. (Hamming’s work was goaded by...

  5. CHAPTER 2 FROM CODING TO SPHERE PACKING
    (pp. 61-108)

    In Euclideann-space,En, how may disjoint, open, congruentn-spheres be located to maximize the fraction of the volume ofEnthat then-spheres cover? That is the sphere packing problem, which goes back to a book review that Gauss wrote in 1831, in which he pointed out that a problem concerning the minimal nonzero value assumed by a positive definite quadratic form innvariables, first considered by Lagrange in 1773, could be translated into a problem on packing spheres (cf. C. A. Rogers [105, pp. 1, 106]).

    Though there might seem to be no connection between coding theory...

  6. CHAPTER 3 FROM SPHERE PACKING TO NEW SIMPLE GROUPS
    (pp. 109-175)

    What finite groups lie hidden in a lattice? For example, consider the standard integer lattice inE² pictured in Figure 3.1. LetGbe the family of distance-preserving maps of the plane which fix the origin and carry the lattice into itself. These maps are usually calledisometriesorEuclidean motions. They need not preserve orientation. The groupGhas order eight, since (1, 0) must and can be mapped to one of the four points (±1, 0) or (0, ±1), and then (0, 1) must and can be mapped to either of two of these points. Representative matrices for...

  7. APPENDIX 1. DENSEST KNOWN SPHERE PACKINGS
    (pp. 176-186)
  8. APPENDIX 2. FURTHER PROPERTIES OF THE (24, 12) GOLAY CODE AND THE RELATED STEINER SYSTEM S(5, 8, 24)
    (pp. 187-192)
  9. 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.
    (pp. 193-196)
  10. APPENDIX 4. THE MATHIEU GROUP M₂₄ AND THE ORDER OF M₂₂
    (pp. 197-208)
  11. APPENDIX 5. THE PROOF OF LEMMA 3.3
    (pp. 209-210)
  12. APPENDIX 6. THE SPORADIC SIMPLE GROUPS
    (pp. 211-216)
  13. BIBLIOGRAPHY
    (pp. 217-224)
  14. INDEX
    (pp. 225-228)