Golay code

Extended binary Golay code

The [24,12,8]2 (extended) Golay code C ≤E(Ω24) is the unique self-orthogonal doubly-even code of length 24 containing no elements of Hamming weight 4, code and the extended code of the [23,12,7]2 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 Ω =P1⁡𝕂7 and C1,C2 be the two constructions of the Binary 8,4,4 extended Hamming code From quadratic residues. Furthermore, let 3Ω represent three disjoint copies of Ω so that P(3Ω) =P(Ω)⊕3. Then let

C=⟨(𝑆,𝑆,∅),(𝑆,∅,𝑆),(𝑇,𝑇,𝑇):𝑆∈C1,𝑇∈C2⟩≤P(3Ω)

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

C=⟨(𝑆,𝑆,∅):𝑆∈C1⟩⦹⟨(𝑆,∅,𝑆):𝑆∈C2⟩⦹⟨(𝑇,𝑇,𝑇):𝑇∈C2⟩

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 𝐶 ∈C with |𝐶| =4. Note

𝐶=(𝑆,𝑆,∅)⦹(𝑆,∅,𝑆)⦹(𝑇,𝑇,𝑇)=(𝑆1+𝑇,𝑆2+𝑇,𝑆3+𝑇)

for some 𝑆1 +𝑆2 +𝑆3 =0, so

|𝑆1+𝑇|+|𝑆2+𝑇|+|𝑆3+𝑇|=4

It follows one of the summands must be zero, since each is even; say 𝑆3 +𝑇 =0, whence 𝑇 ∈𝕂2Ω. Thus |𝑆1+𝑇|,|𝑆2+𝑇| are doubly even, so one must be zero, say 𝑆1 +𝑇 =0. But this implies

𝑆1=𝑆1−2𝑇=𝑆1+𝑆2+𝑆3=0

so |𝑆1+𝑇| =|𝑇| ∈{0,8}, a contradiction.

Properties

1. C is of FLM type II.
2. C is a quasi-perfect 3 error correcting code.
3. C has weight enumerator 1 +759𝑞8 +2576𝑞12 +759𝑞16 +𝑞24.

Automorphisms

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

0≤𝕂2Ω≤C≤E(Ω)≤P(Ω)

with the faithful irreducible modules given by C/𝕂2Ω and E(Ω)/C.

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develop | 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 C1 and the orthogonality of either with the third follows from the fact that any nonzero result in one of the components appears twice. ↩