$[8, 4, 4] 2$ extended Hamming code

The $[8, 4, 4] 2$ (extended) Hamming code $C$ is the unique self-orthogonal doubly even binary linear code of length $8$ , and the extended code of the $[7, 4, 3] 2$ Hamming code.

Proof of uniqueness

#missing/proof

Construction

From quadratic residues

Let $Ω = P 1 𝔽 7 = 𝔽 7 \cup {\infty}$ be the projective line where $𝔽 7$ is a Galois field.¹² $Ω$ divides neatly into quadratic residues and non-residues

Q = {𝑥 2 : 𝑥 \in 𝔽 7} = {0, 1, 2, 4} N = Ω ∖ Q = {3, 5, 6, \infty}

defining the following subcodes of the even binary code $E (Ω)$

C 1 = s p a n 𝕂 2 {N + 𝑖 : 𝑖 \in 𝔽 7} C 2 = s p a n 𝕂 2 {- N - 𝑖 : 𝑖 \in 𝔽 7}

satisfy

C 1 \cap C 2 = 𝔽 2 Ω C 1 + C 2 = E (Ω) 𝑞 (C 1) = 𝑞 (C 2) = 0 d i m C 1 = d i m C 2 = 4

Both $C 1$ and $C 2$ are linearly equivalent to $C$ .

Properties

$C$ is of FLM type II.
$C$ has weight enumerator $1 + 14 𝑞 4 + 𝑞 8$ .

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

1988. Vertex operator algebras and the Monster, §10.1, p. 300 ↩
1988. Geometry of the Mathieu groups and Golay codes, pp. 155–156 ↩