Network Coding for $3$s$/n$t Sum-Networks

A sum-network is a directed acyclic network whereeach source independently generates one symbol from a givenfield F and each terminal wants to receive the sum (over F)of the source symbols. For sum-networks with two sources ortwo terminals, the solvability is characterized by the connectioncondition of each source-terminal pair [3]. A necessary andsufficient condition for the solvability of the 3-source 3-terminal(3s/3t) sum-networks was given by Shenvi and Dey [6]. However,the general case of arbitrary sources/sinks is still open. In thispaper, we investigate the sum-network with three sources andn sinks using a region decomposition method. A sufficient andnecessary condition is established for a class of 3s/nt sum-networks. As a direct application of this result, a necessary andsufficient condition of solvability is obtained for the special caseof 3s/3t sum-networks.