A necessary and sufficient condition for solvability of a 3s/3t sum-network

We consider a directed acyclic network with three sources and three terminals such that each source independently generates one symbol from a given field F and each terminal wants to receive the sum (over F) of the source symbols. Each link is error-free, delay-free and can carry one symbol from the field in each use. We call such a network a 3-source 3-terminal (3s/3t) sum-network. We give a necessary and sufficient condition for a 3s/3t sum-network to be solvable over any field. Some lemmas provide interesting simpler sufficient conditions for the same. We show that linear codes, and in most cases XOR codes, are sufficient for this problem for 3s/3t though they are known to be insufficient for arbitrary number of sources and terminals. We also prove a recent conjecture that the capacity of a 3s/3t sum-network is either 0, 2/3 or ≥ 1.