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4-tuples algorithm basis BCH code binary code binary cyclic codes binary vectors called code of length codewords coefficients compute consider construct contain coordinate positions Corollary coset leader cyclic shift cyclotomic cosets decoding scheme defined denote dimension divides double-error-correcting doubly-even equations equivalent errors example f-design factor field F finite field g-value GF(p give Gleason polynomials Golay code Hamming code Hence idempotent integers irreducible polynomial Kmod Lemma linear code matrix H minimal polynomial minimum weight modulo monic multiplication nonzero elements number of vectors octad orthogonal parity check matrix permutation polynomial g(x polynomial of degree power moments Problem Proof QR code quadratic residue received vector reciprocal polynomial Reed-Muller codes roots rows of G self-dual code self-orthogonal set of vectors smallest solution Steiner system subspace Suppose syndrome Table ternary Theorem unique vector space vectors of weight weight distribution weight enumerator winning positions Z4 codes zero
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