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2-design with parameters 2-transitive adjacency matrix affine plane arbitrary automorphism group axioms block designs block graph Bose classes code of length code words combinatorial configurations common points complementary consider consisting construction contains coordinates defined distinct divides dual code edges eigenvalue equality holds equation exactly example exists finite field Fisher inequality geometry GF(q given Golay code Hadamard matrices Hadamard matrix Hamming code hence implies incidence matrix integer isomorphic k-subsets LEMMA linear code lines Mathieu group matrix of order minimum distance minimum weight natural number nontrivial nonzero number of blocks number of errors number of points orbits of G pairs parity check matrix perfect codes permutation group plane of order polynomial projective plane PROOF Let quadratic residues quasi-symmetric design rank 3 graph repeated blocks satisfies self-dual Steiner system strongly regular graph subgraph subsets symmetric 2-(v symmetric design ternary Golay code total number unique vertices words of weight zero