Hamming code

[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 Binary 7,4,3 Hamming code.

Proof of uniqueness

proof

Construction

From quadratic residues

Let Ω =P1⁡𝔽7 =𝔽7 ∪{∞} be the projective line where 𝔽7 is a Galois field.¹² Ω divides neatly into quadratic residues and non-residues

Q={𝑥2:𝑥∈𝔽7}={0,1,2,4}N=Ω∖Q={3,5,6,∞}

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

C1=span𝕂2⁡{N+𝑖:𝑖∈𝔽7}C2=span𝕂2⁡{−N−𝑖:𝑖∈𝔽7}

satisfy

C1∩C2=𝔽2ΩC1+C2=E(Ω)𝑞(C1)=𝑞(C2)=0dim⁡C1=dim⁡C2=4

Both C1 and C2 are linearly equivalent to C.

Properties

• C is of FLM type II.
• C has weight enumerator 1 +14𝑞4 +𝑞8.

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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 ↩