Moscow Mathematical Journal

Volume 15, Issue 4, October–December 2015 pp. 679–702.

On Linear Ordered Codes

Authors: Alexander Barg (1) and WooMyoung Park (2)
Author institution:(1) Dept. of Electrical and Computer Engineering and Institute for Systems Research, University of Maryland, College Park, MD 20742, USA, and Institute for Problems of Information Transmission, Russian Academy of Sciences, Moscow, Russia
(2) Samsung Electronics, Suwon, Gyeonggi-do, Korea. Research done while at Department of ECE and Institute for Systems Research, University of Maryland, College Park, MD

Summary:

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.

2010 Math. Subj. Class. 94B25.

Keywords: Ordered metrics, linear codes, poset matroids, binomial moments, higher poset weights, wiretap channel.

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