 Solution

The original binary Hamming code has parameters $[7,4,3]$, so it has $2^4=16$ codewords. Its minimum distance three makes the radius-one balls about its codewords disjoint. Each ball hasvectors, and $16\cdot 8=128=2^7$, the size of the ambient space. Equality in the Hamming bound therefore shows that these balls partition ${\mathbb{F}}_2^7$: the code is perfect.