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The binary Golay code is an abelian group which is a 12-dimensional subspace of the vector space . It is used in coding theory (as a binary linear code) and the theory of sporadic finite simple groups.
Consider the 24-element set , and the free vector space on it, identified with the power set of . The the binary Golay code (sometimes called the extended binary Golay code to distinguish it from the perfect binary Golay code, which uses only 23 elements of ) has basis constructed as follows …
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The automorphism group of the binary Golay code is the Mathieu group . Moreover, the other Mathieu group are obtained as stabiliser groups of various sets in the Golay code. There is a unique central extension of the binary Golay code by which is not a group but a code loop, and can be used to construct the Monster group.
Last revised on May 5, 2021 at 06:20:39. See the history of this page for a list of all contributions to it.