The capacity gap calculation for multi-pair bidirectional Gaussian relay networks based on successive compute-and-forward strategy

In this work we obtain capacity gaps for a class of N-pair bidirectional Gaussian relay networks, where one relay can help communications between the corresponding user pairs. For the uplink, we apply the generalization of successive compute-and-forward strategy (SCAF) for decoding the linear combinations of the messages of each user pair at the relay. The downlink channel is considered as a broadcast network with N receiver groups. It is shown that for all channel gains, the achievable rate boundary lies within gaps of (N - 1 + log₂ N)=2N and (N + log₂ N)/2N bits/sec/Hz below the cut-set upper bound for restricted and non-restricted models, respectively. These gaps tend to 1/2 bits/sec/Hz per user as N goes to infinity. We first derive a comprehensive formulation for the N-step asymmetric SCAF and use it to derive the capacity result for our problem.