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algebra analytic angle asymptotic axes axis BCH code binary block codes called codeword coefficient colours column components convolutional code coordinates corresponding crystallographic curvature curve cyclic code decoding defined DEFINITION denote differential digits dimension disk patterns dmin elements encoder enumeration equation equivalent error pattern EXAMPLE Figure finite fixed follows formula geodesic geometric Gilbert bound given gives group G Hamming codes Hamming weight Hence henomeric type homeomeric types integers inverse irreducible representations isometries isomorphic lattice length Lie group linear code linear recursion linearly independent mathematical matrix minimum distance multiplication multiset n-set n-tuples non-zero notation obtain orthogonal osculating plane parametrization parity-check parity-check matrix partition polynomial position problem properties PROPOSITION q-ary reflection rotation group satisfies sequence solution subgroup subsets subspace surface symmetric functions symmetry group symmetry type Table tangent vector Theorem theory topological group translation vector field vector space vertices zero