The ordered Hamming metric and ordered symmetric channels

The ordered Hamming metric is a generalization of the usual Hamming distance derived from a partial order on the set of coordinates. So far, coding theory in ordered spaces has primarily focused on combinatorial aspects. The main object of this paper is to develop the relation of the ordered Hamming space to the context of information transmission. Using the models in previous works of Rosenbloom and Tsfasman (1997) and Tavildar and Viswanath (2006) as a starting point, we define the ordered symmetric channel and the ordered erasure channel which are counterparts of the q-ary symmetric channel and the q-ary erasure channel, respectively. We establish a set of basic results for these channels as well as their relation to linear ordered codes.