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Sum-networks are networks where all the terminals demand the sum of thesymbols generated at the sources. It has been shown that for any finiteset/co-finite set of prime numbers, there exists a sum-network which has avector linear solution if and only if the characteristic of the finite fieldbelongs to the given set. It has also been shown that for any positive rationalnumber $k/n$, there exists a sum-network which has capacity equal to $k/n$. Itis a natural question whether, for any positive rational number $k/n$, and forany finite set/co-finite set of primes $\{p_1,p_2,\ldots,p_l\}$, there exists asum-network which has a capacity achieving rate $k/n$ fractional linear networkcoding solution if and only if the characteristic of the finite field belongsto the given set. We show that indeed there exists such a sum-network byconstructing such a sum-network.

Sum-networks: Dependency on Characteristic of the Finite Field under Linear Network Coding