binary Golay code

The binary Golay CodeMathworldPlanetmath 𝒢23 is a perfect linear binary [23,12,7]-code with a plethora of different constructions.

Sample Constructions

  • Lexicographic Construction: Let v0 be the all-zero word in 𝔽223, and inductively define vj to be the smallest word (smallest with respect to the lexicographic ordering on 𝔽223 that differs from vi in at least 7 places for all i<j.

  • Construction: 𝒢23 is the quadratic residue code of length 23.

The extended binary Golay Code 𝒢24 is obtained by appending a zero-sum check digit to the end of every word in 𝒢23.

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


  • 𝒢24 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 groupMathworldPlanetmathPlanetmath of 𝒢24 is the Mathieu groupMathworldPlanetmath M24, one of the sporadic groups.

  • The Golay Code is used to define the Leech LatticeMathworldPlanetmath, 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 𝒢24 form a S(5,8,24) Steiner systemMathworldPlanetmath. In fact, this property uniquely determines the code.

Title binary Golay code
Canonical name BinaryGolayCode
Date of creation 2013-03-22 14:23:39
Last modified on 2013-03-22 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