Sum-networks: Dependency on characteristic of the finite field under linear network coding

Sum-networks are networks where all the terminals demand the sum of the symbols generated at the sources. It has been shown that for any finite set/co-finite set of prime numbers, there exists a sum-network which has a rate 1 linear network coding solution if and only if the characteristic of the finite field belongs to the given set. It has also been shown that for any positive rational number k/n, there exists a sum-network which has capacity equal to k/n. It is a natural question whether, for any positive rational number k/n, and for any finite set/co-finite set of primes {p
₁
, p
₂
, ..., p
_l
}, there exists a sum-network which has a capacity achieving rate k/n fractional linear network coding solution if and only if the characteristic of the finite field belongs to the given set. We show that indeed there exists such a sum-network by giving an explicit construction.