On Cyclic Matroids and their Applications

TL;DR

This paper introduces cyclic matroids, explores their properties especially regarding basis sizes, and connects them to algebraic and geometric structures like cyclic codes and projective planes.

Contribution

It defines cyclic matroids, analyzes their properties using group actions, and links them to applications in algebra and geometry.

Findings

Cyclic matroids have automorphism groups containing an n-cycle.

The minimum size of bases in cyclic matroids is characterized.

Connections established between cyclic matroids, cyclic codes, and projective geometries.

Abstract

A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such as graphs, matrices, codes and projective geometries. In this work, we define cyclic matroids as matroids over a ground set of size $n$ whose automorphism group contains an $n$-cycle. We study the properties of such matroids, with special focus on the minimum size of their basis sets. For this, we broadly employ two different approaches: the multiple basis exchange property, and an orbit-stabilizer method, developed by analyzing the action of the cyclic group of order $n$ on the set of bases. We further present some applications of our theory to algebra and geometry, presenting connections to cyclic projective planes, cyclic codes and $k$-normal elements.