On Fractional Linear Network Coding Solution of Multiple-Unicast Networks

TL;DR

This paper demonstrates that for any positive rational rate, there exists a multiple-unicast network with a fractional linear network coding solution precisely characterized by the field's characteristic belonging to a specific set of primes.

Contribution

It generalizes previous results by establishing the existence of networks with fractional rates corresponding to any positive rational number, linked to the field characteristic.

Findings

Existence of networks with fractional rate solutions for any positive rational number.

Solution existence depends on the characteristic of the finite field.

Characterization of solution existence via finite or co-finite prime sets.

Abstract

It is known that there exists a multiple-unicast network which has a rate $1$ linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes. In this paper, we show that for any non-zero positive rational number $\frac{k}{n}$, there exists a multiple-unicast network which has a rate $\frac{k}{n}$ fractional linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes.