binary Golay code
The binary Golay Code^{} ${\mathcal{G}}_{23}$ is a perfect linear binary [23,12,7]code with a plethora of different constructions.
Sample Constructions

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Lexicographic Construction: Let ${v}_{0}$ be the allzero word in ${\mathbb{F}}_{2}^{23}$, and inductively define ${v}_{j}$ to be the smallest word (smallest with respect to the lexicographic ordering on ${\mathbb{F}}_{2}^{23}$ that differs from ${v}_{i}$ in at least 7 places for all $$.

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Construction: ${\mathcal{G}}_{23}$ is the quadratic residue code of length 23.
The extended binary Golay Code ${\mathcal{G}}_{24}$ is obtained by appending a zerosum check digit to the end of every word in ${\mathcal{G}}_{23}$.
Both the binary Golay code and the extended binary Golay code have some remarkable .
Properties

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${\mathcal{G}}_{24}$ 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 ${\mathcal{G}}_{24}$ is the Mathieu group^{} ${M}_{24}$, one of the sporadic groups.

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The Golay Code is used to define the Leech Lattice^{}, one of the most efficient spherepackings 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 ${\mathcal{G}}_{24}$ form a $S(5,8,24)$ Steiner system^{}. In fact, this property uniquely determines the code.
Title  binary Golay code 

Canonical name  BinaryGolayCode 
Date of creation  20130322 14:23:39 
Last modified on  20130322 14:23:39 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  4 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 11T71 
Related topic  LeechLattice 
Related topic  Hexacode 
Defines  extended binary golay code 