Golay code

Extended binary Golay code

The (extended) Golay code is the unique self-orthogonal doubly-even code of length containing no elements of Hamming weight , code and the extended code of the Perfect binary Golay code.¹

Proof of uniqueness

proof

The codewords of weight eight are called octads, while the codewords of weight 12 are called dodecads. The octads form the Steiner system S(5,8,24).

Construction

From a Hamming code

Let and be the two constructions of the Binary 8,4,4 extended Hamming code From quadratic residues. Furthermore, let represent three disjoint copies of so that . Then let

where the -tuples denote the corresponding disjoint unions. This is the orthogonal² direct sum of 3 ^totallyIsotropic 4-dimensional subspaces of

and is hence 12-dimensional and totally isotropic, thus it is self-orthogonal and doubly-even, i.e. of FLM type II.

Proof of Golay code

Assume there exists with . Note

for some , so

It follows one of the summands must be zero, since each is even; say , whence . Thus are doubly even, so one must be zero, say . But this implies

so , a contradiction.

Properties

1. is of FLM type II.
2. is a quasi-perfect 3 error correcting code.
3. has weight enumerator .

Automorphisms

The automorphism group is the sporadic simple group Mathieu group M24, and we have the modules

with the faithful irreducible modules given by and .

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Footnotes

1. 1988. Vertex operator algebras and the Monster, p. 301 ↩

2. where the orthogonality of the first two follows from the self-orthogonality of and the orthogonality of either with the third follows from the fact that any nonzero result in one of the components appears twice. ↩