Capacity Theorems for the AWGN multi-way relay channel

The L-user additive white Gaussian noise multi-way relay channel is considered, where multiple users exchange information through a single relay at a common rate. Existing coding strategies, i.e., complete-decode-forward and compress-forward are shown to be bounded away from the cut-set upper bound at high signal-to-noise ratios (SNR). It is known that the gap between the compress-forward rate and the capacity upper bound is a constant at high SNR, and that between the complete-decode-forward rate and the upper bound increases with SNR at high SNR. In this paper, a functional-decode-forward coding strategy is proposed. It is shown that for L ≥ 3, complete-decode-forward achieves the capacity when SNR ≤ 0 dB, and functional-decode-forward achieves the capacity when SNR ≥ 0 dB. For L = 2, functional-decode-forward achieves the capacity asymptotically as SNR increases.