On Linear Ordered Codes

We consider linear codes in the metric space with the Niederreiter-Rosenbloom-Tsfasman (NRT) metric, calling them linear ordered codes. In the first part of the paper we examine a linear-algebraic perspective of linear ordered codes, focusing on the distribution of “shapes” of codevectors. We define a multivariate Tutte polynomial of the linear code and prove a duality relation for the Tutte polynomial of the code and its dual code. We further relate the Tutte polynomial to the distribution of support shapes of linear ordered codes, and find this distribution for ordered MDS codes. Using these results as a motivation, we consider ordered matroids defined for the NRT poset and establish basic properties of their Tutte polynomials. We also discuss connections of linear ordered codes with simple models of information transmission channels.