## 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. |

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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 |

Densest Known Sphere Packings | 176 |

Further Properties of the 1224 Golay | 187 |

A Calculation of the Number of Spheres | 193 |

The Mathieu Group M24 and the Order | 197 |

The Proof of Lemma 3 3 | 209 |

217 | |

225 | |

### Common terms and phrases

23 matrix 8-set in R(C adjacent binary calculations check digits codes of length codeword coding theory column congruent contact number Conway coordinates cube defined denote densest known density dimensions disjoint eight l's elements entries equivalent error-correcting Euclidean exactly example Figure finite form a Steiner four message digits Golay code Golay's Hamming code Hamming's integer lattice packing lattice points Leech lattice Lemma mathematics Mathieu group minimum distance modulo multiple of four nonzero normal subgroup Note number of l's packing in E24 pair paper parity check patent perfect permutation with structure positions proof quadratic residue Recall repetition code row space sends Shannon shown single error single orbit single-error-correcting codes sphere centers sphere of radius sphere packing sporadic simple groups Steiner system Steiner system 5(5 subset Suppose symmetry group Table Theorem upper bound vector in R(C vertex