## From Error-correcting Codes Through Sphere Packings to Simple Groups, Volume 21This 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. -- Amazon.com. |

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### Contents

THE ORIGIN OF ERRORCORRECTING CODES | 1 |

FROM CODING TO SPHERE PACKING | 61 |

FROM SPHERE PACKING TO New SIMPLE GROUPS | 109 |

Copyright | |

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### Common terms and phrases

additional adjacent already appears binary calculations called check digits codeword column consider consists construction contact number contains Conway coordinates correct corresponding defined denote density differ digits dimensions disjoint distinct divides easy eight elements entries equivalent error exactly example exist fact field Figure five fixes four given Golay code Hamming Hamming code integer interest known later lattice lattice packing lattice points least Leech LEMMA length linear mathematics matrix means minimum distance modulo normal Note orbit origin packing pair parity check patent perfect permutation places plane points positions possible problem proof Recall received result sends seven shape shown simple groups single space sphere sphere packing Steiner system structure subgroup subset Suppose symmetry Table Theorem theory transitive unit vectors vertex vertices whole