Matroidal Structure of Skew Polynomial Rings with Application to Network Coding

TL;DR

This paper explores the matroidal structure of skew polynomial rings over finite fields, establishing their connection to linear independence and projective geometry, and applying this to network coding.

Contribution

It introduces a new representable matroid based on skew polynomials, linking it to projective geometry and enabling applications in network coding.

Findings

Defined the $qm[x;\sigma]$-matroid with a metric rank function

Established isometry between submatroids and projective geometry

Applied the matroidal structure to network coding scenarios

Abstract

Over a finite field $\mathbb{F}_{q^m}$, the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew polynomials to the familiar concept of linear independence for vector spaces. This relation allows for the definition of a representable matroid called the $\mathbb{F}_{q^m}[x;\sigma]$-matroid, with rank function that makes it a metric space. Specific submatroids of this matroid are individually bijectively isometric to the projective geometry of $\mathbb{F}_{q^m}$ equipped with the subspace metric. This isometry allows one to use the $\mathbb{F}_{q^m}[x;\sigma]$-matroid in a matroidal network coding application.