On Cyclic Matroids and their Applications

Abstract

A matroid is a combinatorial structure which captures and generalizes the 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 the n-cycle. We study the properties of such matroids, with special focus on the minimum size of their bases 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 group Z n on the set of bases. We further present some applications of our theory to algebra and geometry, by studying the relation between cyclic matroids and cyclic projective planes, cyclic codes and k-normal elements. Finally, a general counting argument for the number of orbits of the cyclic group action is also provided.