q-Polymatroids and Their Relation to Rank-Metric Codes

TL;DR

This paper investigates the representability of q-polymatroids by rank-metric codes, providing examples of non-representable cases, and explores operations like deletion, contraction, and flats to relate q-polymatroids to code properties.

Contribution

It introduces the concepts of deletion, contraction, and flats for q-polymatroids, linking their structure to rank-metric code properties and providing examples of non-representability.

Findings

Some q-polymatroids are not representable by any rank-metric code.

Deletion and contraction operations are dual and relate to code puncturing and shortening.

The flats of q-polymatroids determine the generalized rank weights of codes.

Abstract

It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a q-matroid that is not representable by any linear rank-metric code and, via a relation to paving matroids, provide examples of various q-matroids that are not representable by F_{q^m}-linear rank-metric codes. We then go on and introduce deletion and contraction for q-polymatroids and show that they are mutually dual and correspond to puncturing and shortening of rank-metric codes. Finally, we introduce a closure operator along with the notion of flats and show that the generalized rank weights of a rank-metric code are fully determined by the flats of the associated q-polymatroid.