extended Hamming code

The (extended) Hamming code is the unique self-orthogonal doubly even binary linear code of length , and the extended code of the Hamming code.

Proof of uniqueness

#missing/proof

Construction

From quadratic residues

Let be the projective line where is a Galois field.¹² divides neatly into quadratic residues and non-residues

defining the following subcodes of the even binary code

satisfy

Both and are linearly equivalent to .

Properties

is of FLM type II.
has weight enumerator .

#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 ↩