Multicast Networks Solvable over Every Finite Field

TL;DR

This paper proves that any acyclic multicast network solvable over GF(2) is also solvable over all finite fields, introducing an algorithm and the concept of multicast matroid to establish this result.

Contribution

It introduces a novel algorithm to extend GF(2) solutions to all finite fields and the concept of multicast matroid for analyzing network solvability.

Findings

Acyclic multicast networks solvable over GF(2) are solvable over all finite fields.

Introduction of multicast matroid concept for network solvability analysis.

Algorithm for converting GF(2) solutions to arbitrary finite fields.

Abstract

In this work, it is revealed that an acyclic multicast network that is scalar linearly solvable over Galois Field of two elements, GF(2), is solvable over all higher finite fields. An algorithm which, given a GF(2) solution for an acyclic multicast network, computes the solution over any arbitrary finite field is presented. The concept of multicast matroid is introduced in this paper. Gammoids and their base-orderability along with the regularity of a binary multicast matroid are used to prove the results.