Show that the binary Hamming Code is a perfect code.

Question

Since the minimum distance of a binary hamming code is 3, we have 1 as the radius of the sphere. How is it being centered about a codeword in a binary hamming code to be called as a perfect code? Can anyone show an explanation for this? Thank you!

Answer 1

Hint: A perfect code $C$ of the field with $q$ elements with parameters $(n,k,d)$ fulfills the Hamming or Sphere packing bound with equality:

$q^n = q^k\cdot \sum_{i=0}^t {n\choose i} (q-1)^i$,

where $t=\lceil (d-1)/2\rceil$ is the number of correctable errors.

Please check for Hamming codes!

Interpretation: The spheres of radiums 1 around the codewords cover the whole ambient space.