The binary Golay Code $\mc{G}_{23}$ is a perfect linear binary [23,12,7]-code with a plethora of different equivalent constructions.

Sample Constructions

• Lexicographic Construction: Let $v_0$ be the all-zero word in $\mb{F}_2^{23}$ and inductively define $v_j$ to be the smallest word (smallest with respect to the lexicographic ordering on $\mb{F}_2^{23}$ that differs from $v_i$ in at least 7 places for all $i<j$
• Quadratic Residue Construction: $\mc{G}_{23}$ is the quadratic residue code of length 23.

The extended binary Golay Code $\mc{G}_{24}$ is obtained by appending a zero-sum check digit to the end of every word in $\mc{G}_{23}$

Both the binary Golay code and the extended binary Golay code have some remarkable properties.

Properties

• $\mc{G}_{24}$ 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 $\mc{G}_{24}$ 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 $\mc{G}_{24}$ form a $S(5,8,24)$ Steiner system. In fact, this property uniquely determines the code.