## Sphere Packings, Lattices, and Groups |

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applying automorphism group basis binary bound called Chap chapter classes codewords column complete consists Construction contains coordinates corresponding covering deep holes defined definition denote density described determinant diagram digits dimensions distance elements equal equivalent example exist extended extremal fact Figure finite fixing follows function give given integral invariant kissing number known lattice packing layers least Leech lattice length linear log2 Math matrix maximal minimal norm minimal vectors modulo multiple n-dimensional obtained octad origin packing permutations points polynomial positive possible problem Proof quadratic forms radius reflection represented respectively root satisfies self-dual shown signs space sphere spherical square subgroup Table Theorem theory theta series Type unique values оо