Unlabeled equivalence for matroids representable over finite fields

TL;DR

This paper introduces a new equivalence concept for representable matroids over finite fields, enabling systematic generation of non-isomorphic matroids through geometric equivalence transformations.

Contribution

It defines geometric equivalence for matroid representations and provides a method to exhaustively generate all non-isomorphic matroids over finite fields.

Findings

Defines geometric equivalence for matroid representations

Provides a method for exhaustive generation of non-isomorphic matroids

Enables classification of matroids over finite fields

Abstract

We present a new type of equivalence for representable matroids that uses the automorphisms of the underlying matroid. Two $r\times n$ matrices $A$ and $A'$ representing the same matroid $M$ over a field $F$ are {\it geometrically equivalent representations} of $M$ if one can be obtained from the other by elementary row operations, column scaling, and column permutations. Using geometric equivalence, we give a method for exhaustively generating non-isomorphic matroids representable over a finite field $GF(q)$, where $q$ is a power of a prime.