what does the redundancy of a code means?
A linear error-correcting code encodes $m$ message bits using $w$ encoded bits. The redundancy is $r = w-m$. In other words, it is a collection of $M = 2^m$ codewords out of the possible $W = 2^w$ words of length $w$. We can extract the redundancy using the formula
$$ r = \log_2 \frac{W}{M}. $$
This formula makes sense for arbitrary error-correcting codes.

In your case, codewords are of length $n$, and there are at least $2^n/(6n-3)$ of them. Therefore the redundancy is at most
$$ \log_2 \frac{2^n}{2^n/(6n-3)} = \log_2 (6n-3). $$