# A Polymatroid Approach to Generalized Weights of Rank Metric Codes

## TL;DR

This paper introduces a polymatroid-based framework for defining and analyzing generalized weights of rank metric codes, establishing duality properties and connecting to existing code weight concepts.

## Contribution

It extends the notion of generalized weights to $(q,m)$-polymatroids and demi-polymatroids, and proves a duality theorem analogous to Wei duality for these weights.

## Key findings

- Established a duality for generalized weights of $(q,m)$-polymatroids.
- Derived results for Delsarte rank metric codes and relative generalized rank weights.
- Unified different approaches under a polymatroid framework.

## Abstract

We consider the notion of a $(q,m)$-polymatroid, due to Shiromoto, and the more general notion of $(q,m)$-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martinez-Penas and Matsumoto for relative generalized rank weights are derived as a consequence.