Minimal non-orientable matroids in a projective plane

TL;DR

This paper introduces a new family of minimal non-orientable matroids of rank three, some of which embed in Desarguesian projective planes, addressing a question about their existence over finite fields.

Contribution

It constructs minimal non-orientable matroids of rank three that embed in projective planes over finite fields, answering a question posed by Ziegler.

Findings

Constructed a new family of minimal non-orientable matroids

Some matroids embed in Desarguesian projective planes

Addresses Ziegler's question for all prime powers q

Abstract

We construct a new family of minimal non-orientable matroids of rank three. Some of these matroids embed in Desarguesian projective planes. This answers a question of Ziegler: for every prime power $q$, find a minimal non-orientable submatroid of the projective plane over the $q$-element field.