When is a minimum distance decoder also a maximum likelihood decoder? It is well known that if we have a binary symmetric channel with crossover probability $\epsilon<0.5$ and we send a word $x$ through it, the most likely word is the one with minimum hamming distance to the received word $y$, i.e. the received word itself. If we send a word from a codebook $C$ and we choose it uniformly at random then our best guess is the codeword closest in hamming distance to the received word: 
$\hat x =\arg \min_{x\in C}\|x-y\|_1$
The same argumentation follows with the Gaussian channel but instead of minimizing the hamming distance the minimization is with respect to the $l_2$ distance. Then my question is: 
Is it known for what binary input channels a minimum distance decoder is equivalent to a maximum likelihood decoder?

If the probability of error on a discrete memoryless channel $P(E)$ is strictly less than one half, then minimum distance decoding is equivalent to maximum likelihood decoding
$$P(E) = \sum r P(E|r)P(r)= \sum r P\left(v’=\frac{v}{r}\right)P(r)<0.5$$
