Characterization of the Rate Region for Multi-Source Multicast Network Coding

The model of multi-source multicast network coding is investigated in this paper, where each sink node desires to decode correctly the source messages generated by multiple source nodes. For this model, the rate region is defined as the set of rate tuples of source nodes at which the source nodes can multicast source messages to all sink nodes for one use of the network on average. In this paper, we fully characterize the rate region and investigate the code construction for the model of multi-source multicast network coding. We first prove that for any subset of source nodes, the sum rate is upper bounded by the smallest minimum cut capacity separating a sink node from the subset of source nodes. We further develop a systematic construction of vector-linear network codes by using a transformation approach. This code construction is applicable to an arbitrary rate tuple that is achievable. Based on the upper bounds obtained and the developed code construction, we fully characterize the rate region for any model of multi-source multicast network coding problem.