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binary Golay code
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(Definition)
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The binary Golay Code
 is a perfect linear binary [23,12,7]-code with a plethora of different equivalent constructions.
- 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  .
- Quadratic Residue 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.
-
 has 4096 codewords: 1 of weight 0, 759 of weight 8, 2576 of weight 12, 759 of weight 18, and 1 of weight 24.
- The automorphism group of
 is the Mathieu group  , one of the sporadic groups.
- The Golay Code is used to define the Leech Lattice, one of the most efficient sphere-packings known to date.
- 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.
- The words of weight 8 in
 form a  Steiner system. In fact, this property uniquely determines the code.
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"binary Golay code" is owned by mathcam.
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(view preamble)
| Also defines: |
extended binary golay code |
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Cross-references: property, Steiner system, current, game, strategy, lattice, group, automorphism group, weight, digit, length, code, quadratic residue, places, lexicographic ordering, binary, perfect
There are 5 references to this entry.
This is version 1 of binary Golay code, born on 2004-06-04.
Object id is 5891, canonical name is BinaryGolayCode.
Accessed 5167 times total.
Classification:
| AMS MSC: | 11T71 (Number theory :: Finite fields and commutative rings :: Algebraic coding theory; cryptography) |
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Pending Errata and Addenda
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