Linear Fractional Network Coding and Representable Discrete Polymatroids

TL;DR

This paper explores the relationship between linear fractional network coding solutions and representable discrete polymatroids, extending known results and providing algorithms to construct networks with improved throughput using these concepts.

Contribution

It establishes a novel connection between linear FNC solutions and representability of discrete polymatroids, generalizing previous matroid-based network coding results.

Findings

Discrete polymatroids can be used to construct networks with FNC solutions.

Networks derived from discrete polymatroids may lack scalar or vector solutions.

FNC solutions with different message dimensions can achieve higher throughput.

Abstract

A linear Fractional Network Coding (FNC) solution over $\mathbb{F}_q$ is a linear network coding solution over $\mathbb{F}_q$ in which the message dimensions need not necessarily be the same and need not be the same as the edge vector dimension. Scalar linear network coding, vector linear network coding are special cases of linear FNC. In this paper, we establish the connection between the existence of a linear FNC solution for a network over $\mathbb{F}_q$ and the representability over $\mathbb{F}_q$ of discrete polymatroids, which are the multi-set analogue of matroids. All previously known results on the connection between the scalar and vector linear solvability of networks and representations of matroids and discrete polymatroids follow as special cases. An algorithm is provided to construct networks which admit FNC solution over $\mathbb{F}_q,$ from discrete polymatroids representable over $\mathbb{F}_q.$ Example networks constructed from discrete polymatroids using the algorithm are provided, which do not admit any scalar and vector solution, and for which FNC solutions with the message dimensions being different provide a larger throughput than FNC solutions with the message dimensions being equal.