Matroidal Structure of Skew Polynomial Rings with Application to Network Coding

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.