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In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics.〔See 〕 These codes are named in honor of Marcel J. E. Golay whose 1949 paper introducing them has been called, by E. R. Berlekamp, the "best single published page" in coding theory. There are two closely related binary Golay codes. The extended binary Golay code, ''G''24 (sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 7-bit errors can be detected. The other, the perfect binary Golay code, ''G''23, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit). In standard code notation the codes have parameters (12, 8 ) and (12, 7 ), corresponding to the length of the codewords, the dimension of the code, and the minimum Hamming distance between two codewords, respectively. == Mathematical definition == In mathematical terms, the extended binary Golay code, ''G''24 consists of a 12-dimensional subspace ''W'' of the space ''V''=F224 of 24-bit words such that any two distinct elements of ''W'' differ in at least eight coordinates. By linearity, the distance statement is equivalent to any non-zero element of ''W'' having at least eight non-zero coordinates. * The possible sets of non-zero coordinates as ''w'' ranges over ''W'' are called ''code words''. In the extended binary Golay code, all code words have the Hamming weights of 0, 8, 12, 16, or 24. * Up to relabeling coordinates, ''W'' is unique. The perfect binary Golay code, ''G''23 is a perfect code. That is, the spheres of radius three around code words form a partition of the vector space. The automorphism group of the perfect binary Golay code, ''G''23, is the Mathieu group . The automorphism group of the extended binary Golay code is the Mathieu group . The other Mathieu groups occur as stabilizers of one or several elements of ''W''. The supports of the Golay ''G''24 code words of weight eight are elements of the S(5,8,24) Steiner system. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Binary Golay code」の詳細全文を読む スポンサード リンク
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