Maybe I am the one who is not thinking clearly but I think there is a typo in a question from Introduction to Error Correcting Codes with Applications by Scott A. Vanstone. Its question 5 on page 96 which asks the following:
Question: What is the smallest integer n for which a single-error correcting binary linear code of block length $n$ carrying $62$ bits of information per codeword can be constructed.
As this is a single error correcting code, then it is clearly a Hamming code since the minimum distance of the code is 3. I think the question should be to determine the smallest number $r$. Clearly $n=62$ so I don't see why this was asked again. And by calculation the smallest value of $r$ is $6$
The problem with the suggestion $r=6$ is that the Hamming code of length $2^6-1=63$ has six check bits and hence a single block carries only $63-6=57$ bits of information.
Meaning that you really need $r=7$. For the price of seven check bits you can go up to blocks carrying $(2^7-1)-7=120$ bits worth of information using a longer Hamming code. Provided that in your application this does not uncomfortably increase the risk of there being two bit errors within a block.