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 $\mathbb{F}_2^{24}$. 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 $X = \{1,\ldots,24\}$, and the free vector space on it, identified with the power set of $X$. 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 $X$) has basis constructed as follows …

(see Wikipedia for the time being)

The automorphism group of the binary Golay code is the Mathieu group $M_{24}$. 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 $\mathbb{Z}/2$ which is not a group but a code loop, and can be used to construct the Monster group.

Related entries

• coding theory, error correcting code, quantum error correcting code

• linear code, binary linear code

• Mathieu group

• combinatorial design

• Leech lattice

• adinkra