# Matroids and Codes with the Rank Metric

## TL;DR

This paper explores the connection between q-analogue matroids and rank metric codes, establishing identities that link their combinatorial and algebraic properties, with implications for coding theory.

## Contribution

It introduces a Greene type identity for rank generating functions and provides a combinatorial proof of a MacWilliams identity for Delsarte rank-metric codes.

## Key findings

- Proved a Greene type identity for rank generating functions.
- Established a MacWilliams type identity for Delsarte rank-metric codes.
- Linked matroid theory with rank metric code properties.

## Abstract

We study the relationship between a q-analogue of matroids and linear codes with the rank metric in the vector space of matrices with entries in a finite field. We prove a Greene type identity for the rank generating function of these matroidal structures and the rank weight enumerator of these linear codes. As an application, we give a combinatorial proof of a MacWilliams type identity for Delsarte rank-metric codes.