On Achievability of an $(r,l)$ Fractional Linear Network Code

TL;DR

This paper generalizes previous results on network solvability, demonstrating that for any positive integers, there exist networks with specific fractional linear solutions that do not extend to smaller ratios, highlighting limitations in network coding.

Contribution

It introduces a new class of networks with fractional linear solutions that cannot be scaled down, extending prior work on vector linear solutions in network coding.

Findings

Existence of networks with (mk,mn) fractional solutions but no (wk,wn) solutions for w<m.

Generalization of previous results on scalar and vector linear solvability.

Highlights limitations of fractional linear network coding scalability.

Abstract

It is known that there exists a network, called as the M-network, which is not scalar linearly solvable but has a vector linear solution for message dimension two. Recently, a generalization of this result has been presented where it has been shown that for any integer $m\geq 2$, there exists a network which has a $(m,m)$ vector linear solution, but does not have a $(w,w)$ vector linear solution for $w<m$. This paper presents a further generalization. Specifically, we show that for any positive integers $k,n,$ and $m\geq 2$, there exists a network which has a $(mk,mn)$ fractional linear solution, but does not have a $(wk,wn)$ fractional linear solution for $w<m$.