binary Golay code
The binary Golay Code is a perfect linear binary [23,12,7]-code with a plethora of different constructions.
Sample Constructions
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Lexicographic Construction: Let be the all-zero word in , and inductively define to be the smallest word (smallest with respect to the lexicographic ordering on that differs from in at least 7 places for all .
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Construction: is the quadratic residue code of length 23.
The extended binary Golay Code is obtained by appending a zero-sum check digit to the end of every word in .
Both the binary Golay code and the extended binary Golay code have some remarkable .
Properties
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has 4096 codewords: 1 of weight 0, 759 of weight 8, 2576 of weight 12, 759 of weight 18, and 1 of weight 24.
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The automorphism group
of is the Mathieu group
, one of the sporadic groups.
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The Golay Code is used to define the Leech Lattice
, one of the most efficient sphere-packings known to date.
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The optimal strategy to the mathematical game called Mogul is to always revert the current position to one corresponding to a word of the Golay code.
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The words of weight 8 in form a Steiner system
. In fact, this property uniquely determines the code.