On fractional linear network coding solution of multiple-unicast networks

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 present a generalization of this result for a linear network coding solution of any rate. Specifically, we show that for any non-zero positive rational number k/n, there exists a multiple-unicast network which has a rate 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.