Hamming code

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 Binary 7,4,3 Hamming code.

Proof of uniqueness

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 .

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §10.1, p. 300 ↩

2. 1988. Geometry of the Mathieu groups and Golay codes, pp. 155–156 ↩