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

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 $w<m$.