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The Origin of ErrorCorrecting Codes
From Coding to Sphere Packing
From Sphere Packing to New Simple Groups
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23 matrix 8-set in R(C adjacent calculations check digits 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 n-blocks 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 transitive on A2 upper bound vectors in R(C vertex