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k Shannons fundamental theorem
k Threshold decoding
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algorithm basis vector BCH code binary cyclic code binary Golay code bits block length burst error called channel check polynomial code is perfect code of block code of length code over GF(q code word coefficients consider correcting code corresponding coset coset leader cycles cyclic code define DEFINITION denote dimension dual code element of GF(2 error pattern error probability example expected number extended code ftg(x H. F. Mattson Hamming code Hence idempotents integer lemma linear code linear subspace matrix H minimum distance minimum weight multiple number of errors order RM code orthogonal parity check matrix parity-check equations perfect codes perfect e-error-correcting code permutation positions primitive element primitive n-th root problem Proof prove quadratic residue r-th order RM remark root of unity rows of G Section sequence subcode syndrome ternary Golay code vector space weight enumerator words of weight zeros