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Problem Detail:

Let \$C\$ be a \$[n, k]\$ linear code over \$\mathbb{F}_q\$.

I want to calculate the covering radius of the Hamming codes.

I have thought the following:

Since the Hamming distance is \$3\$, the coverig radius will always be \$3\$.

Am I right?

The Hamming codes are perfect codes. This means that balls of radius \$(d-1)/2\$ (where \$d\$ is the minimal distance) centered around the codewords partition the space. In particular, the covering radius is \$(d-1)/2\$.