Minimum Hamming distance or Minimum Hamming weight

Question

Why studying the minimum Hamming distance for linear codes over rings is interesting for coding theorists?(please, give some details aboute it, more than comments, thanks)

Answer 1

I cannot add much to the generic response in Gerry's comment (+1).

It depends on the ring and on the type of errors your application will encounter. If an error event will affect entire symbols, then the Hamming metric pops out naturally. In that case a code with minimum distance $d$ can cope with all patterns of $t$ errorneous symbols and $e$ erased symbols, if $d>2t+e$.

On the other hand when you internally represent the symbols using several bits, and the channel may affect individual bits (=partial symbols), then the Hamming metric loses its prominence. At least to some extent. A most interesting examle is to represent the elements of $\Bbb{Z}_4$ as pairs of bits (Gray coding), so $\overline{0}_4=00$, $\overline{1}_4=01$, $\overline{2}_4=11$ and $\overline{3}_4=10$. Here you want to take into account that it is (often) easier for channel to turn a $\overline{2}_4$ into $\overline{1}_4$ or $\overline{3}_4$ than into a $\overline{0}_4$, because in the last case you need to flip both the bits whereas in the other cases only one of the internal bits is in error. This suggest using the so called Lee-metric on $\Bbb{Z}_4$ as that corresponds better with the probabilities of the error events in question. Observe: in the case of $\Bbb{Z}_4$ the Lee-metric coincides with the Hamming distance of the internal presentation. This is not always the case with ring alphabets no matter how you try to Gray code them.

I guess my point is that you want your code design metric to correspond to the probabilities of the error events as well as possible, so that high minimum distance means that you can cope with the most common error events. For some of the ring alphabets that I have seen studied this is IMVHO a stretch. Some of those papers have the air that the said ring was studied by the authors simply because nobody else had bothered to look at that ring, yet. I'm exaggerating, but may be not by much. Some coding theory research seeks to advance to the direction of least resistance (I think that is paraphrasing von Neumann or some other sharp person).