nLab
binary Golay code

Contents

table of contents

Idea

The binary Golay code is an abelian group which is a 12-dimensional subspace of the vector space 𝔽 2 24\mathbb{F}_2^{24}. It is used in coding theory (as a binary linear code) and the theory of sporadic finite simple groups.

Definition

Consider the 24-element set X={1,,24}X = \{1,\ldots,24\}, and the free vector space on it, identified with the power set of XX. The the binary Golay code (sometimes called the extended binary Golay code to distinguish it from the perfect binary Golay code, which uses only 23 elements of XX) has basis constructed as follows …

(see Wikipedia for the time being)

The automorphism group of the binary Golay code is the Mathieu group M 24M_{24}. Moreover, the other Mathieu group are obtained as stabiliser groups of various sets in the Golay code. There is a unique central extension of the binary Golay code by /2\mathbb{Z}/2 which is not a group but a code loop, and can be used to construct the Monster group.