I have a basic question about Hamming distances, something confuse me in the book I'm reading about it.
Let's assume we have a codeword $y$ that received an error $e$. Thus we had the following event:
$$ y_0 \rightarrow y'=y_0+e $$
In Nielsen & Chuang, page 449, he says:
Provided the probability of a bit flip is less than 1/2, the most likely codeword to have been encoded is the codeword y which minimizes the number of bit flips needed to get from y to y',that is,which minimizes wt(e)= d(y, y').
Where $d(a,b)$ is the Hamming distance between $a$ and $b$, and $wt(e)$ is the Hamming weight of $e$, which is $d(e,0)$.
My question is the following:
I agree that if there is more probability of no bit flip than the probability to have one, then the most likely codeword is the closest one from $y'$, thus a codeword $y$ minimising $d(y,y')$.
However, as he writes $d(y,y')=wt(e)$, he seems to assume that the closest vector from $y'$ is necesseraly the one we encoded. Which for me is not true in general, the error could have put us closer to another encoded word than the one we encoded right ?
Where is my misunderstanding here ?