Feasible alphabets for communicating the sum of sources over a network

We consider directed acyclic sum-networks with m sources and n terminals where the sources generate symbols from an arbitrary alphabet field F, and the terminals need to recover the sum of the sources over F. We show that for any co-finite set of primes, there is a sum-network which is linearly solvable only over fields of characteristics belonging to that set. We further construct a sum-network where a scalar linear solution exists over all fields other than the binary field F₂. We also show that a sum-network is linearly solvable over a field if and only if its reverse network is linearly solvable over the same field.