On Shannon information theoretic capacity to coding distance metric translation

Question

Shannon theory says that given a channel source variable $X$ and received variable $Y$ and channel $Y/X$ there is a capacity associated with this channel. The notion of maximum likelihood leads from capacity statements to Euclidean distance over Gaussian channels and Hamming distance over finite field models.

1. Could there be other metrics that work over Gaussian channel and over finite field models?

2. Is there an example of a channel where two non-equivalent metrics have been helpful in attaining capacity?

Answer 1

Maximum likelihood does not have direct connection with the capacity of a channel, since achievability of the capacity is an attribute of the code, not the decoder.

ML decoders are optimal. Unfortunately, ML decoding is a NP-hard problem.

I suggest to take a look on Belief-propagetion(Message-passing) decoding.

Also, polar codes are a very good example (and the only) of capacity-achieving codes with sub-optimal decoding.