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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?

###### Answered By : Yuval Filmus

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 $(d-1)/2$.