Solution
= 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 has
$$
\binom70+\binom71=8
$$
<vectors>, and $16\cdot8=128=2^7$, 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.