Rank-Metric Codes and $q$-Polymatroids

TL;DR

This paper introduces $q$-polymatroids as a new combinatorial framework to analyze rank-metric codes, linking algebraic properties of the codes to combinatorial invariants.

Contribution

It develops the theory of $q$-polymatroids and shows how they encode key properties of rank-metric codes, including weights, optimality, and duality.

Findings

$q$-polymatroids capture code invariants

Association of $q$-polymatroids with rank-metric codes

Structural properties of codes are reflected in $q$-polymatroids

Abstract

This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce $q$-polymatroids, the $q$-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.