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

•

Lexicographic Construction: Let $v_{0}$ be the all-zero 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 $i<j$ .
•

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 zero-sum 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 .

•

$\mathcal{G}_{24}$ has 4096 codewords: 1 of weight 0, 759 of weight 8, 2576 of weight 12, 759 of weight 18, and 1 of weight 24.
•

The automorphism group of $\mathcal{G}_{24}$ is the Mathieu group $M_{24}$ , 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 $\mathcal{G}_{24}$ form a $S(5,8,24)$ Steiner system. In fact, this property uniquely determines the code.