Extended binary Golay code

The $[24, 12, 8] 2$ (extended) Golay code $C \leq E (Ω 24)$ is the unique self-orthogonal doubly-even code of length $24$ containing no elements of Hamming weight $4$ , #m/def/code and the extended code of the $[23, 12, 7] 2$ 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 $S (5, 8, 24)$ .

Construction

From a Hamming code

Let $Ω = P 1 𝕂 7$ and $C 1, C 2$ be the two constructions of the $[8, 4, 4] 2$ extended Hamming code From quadratic residues. Furthermore, let $3 Ω$ represent three disjoint copies of $Ω$ so that $P (3 Ω) = P (Ω) \oplus 3$ . Then let

C = ⟨ (𝑆, 𝑆, \emptyset), (𝑆, \emptyset, 𝑆), (𝑇, 𝑇, 𝑇) : 𝑆 \in C 1, 𝑇 \in C 2 ⟩ \leq P (3 Ω)

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

C = ⟨ (𝑆, 𝑆, \emptyset) : 𝑆 \in C 1 ⟩ ⦹ ⟨ (𝑆, \emptyset, 𝑆) : 𝑆 \in C 2 ⟩ ⦹ ⟨ (𝑇, 𝑇, 𝑇) : 𝑇 \in C 2 ⟩

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 $𝐶 \in C$ with $| 𝐶 | = 4$ . Note

𝐶 = (𝑆, 𝑆, \emptyset) ⦹ (𝑆, \emptyset, 𝑆) ⦹ (𝑇, 𝑇, 𝑇) = (𝑆 1 + 𝑇, 𝑆 2 + 𝑇, 𝑆 3 + 𝑇)

for some $𝑆 1 + 𝑆 2 + 𝑆 3 = 0$ , so

| 𝑆 1 + 𝑇 | + | 𝑆 2 + 𝑇 | + | 𝑆 3 + 𝑇 | = 4

It follows one of the summands must be zero, since each is even; say $𝑆 3 + 𝑇 = 0$ , whence $𝑇 \in 𝕂 2 Ω$ . Thus $| 𝑆 1 + 𝑇 |, | 𝑆 2 + 𝑇 |$ are doubly even, so one must be zero, say $𝑆 1 + 𝑇 = 0$ . But this implies

𝑆 1 = 𝑆 1 - 2 𝑇 = 𝑆 1 + 𝑆 2 + 𝑆 3 = 0

so $| 𝑆 1 + 𝑇 | = | 𝑇 | \in {0, 8}$ , a contradiction.

Properties

$C$ is of FLM type II.
$C$ is a quasi-perfect 3 error correcting code.
$C$ has weight enumerator $1 + 759 𝑞 8 + 2576 𝑞 12 + 759 𝑞 16 + 𝑞 24$ .

Automorphisms

The automorphism group $A u t C$ is the sporadic simple group $M 24$ , and we have the modules

0 \leq 𝕂 2 Ω \leq C \leq E (Ω) \leq P (Ω)

with the faithful irreducible modules given by $C / 𝕂 2 Ω$ and $E (Ω) / C$ .

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

1988. Vertex operator algebras and the Monster, p. 301 ↩
where the orthogonality of the first two follows from the self-orthogonality of $C 1$ and the orthogonality of either with the third follows from the fact that any nonzero result in one of the components appears twice. ↩