Defining the q-analogue of a matroid

TL;DR

This paper introduces the concept of q-matroids, exploring their properties, definitions, and connections to rank metric codes, aiming to extend matroid theory into a q-analogue framework.

Contribution

It formally defines q-matroids, analyzes their properties, and links them to rank metric codes, providing foundational insights into their structure and cryptomorphism issues.

Findings

Q-matroids can be defined with various properties and dualities.

Rank metric codes induce q-matroids, establishing a new connection.

Discussion on cryptomorphism highlights differences from classical matroids.

Abstract

This paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) cryptomorphic. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid.