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 , #m/def/code and the extended code of the Perfect binary Golay code.1

Proof of uniqueness

#missing/proof

The codewords of weight eight are called octads, while the codewords of weight 12 are called dodecads. The octads form the Steiner system .

Construction

From a Hamming code

Let and be the two constructions of the 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 totally isotropic 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 , and we have the modules

with the faithful irreducible modules given by and .

#state/develop | #lang/en | #SemBr

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. ↩