An analysis of the joint compute-and-forward decoder for the binary-input two-way relay channel

We provide an information-theoretic analysis of reliable decoding of functions of transmitted signals for the class of binary-input discrete memoryless two-way relay channels. Two popular decoding paradigms for reliable physical layer network coding, decode-and-forward and compute-and-forward, are each suboptimal for the computation problem depending on the channel parameters. Adaptively using the better of these two schemes depending on the channel parameters provides the best known achievable information rates for reliable physical layer network coding. If the transmitting nodes use identical linear codes for encoding, recently, it has been empirically shown that the joint compute-and-forward decoder outperforms the compute-and-forward decoder. We show that the joint compute-and-forward decoder naturally achieves the best of decode-and-forward and compute-and-forward but cannot achieve any information rates higher than those provided by decode-and-forward and compute-and-forward when random cosets of an identical linear code are employed at the transmitters. The converse result is obtained by using the results of Bandemer, El Gamal and Kim on simultaneous non-unique decoding. Further, we show that when joint compute-and-forward decoding is used, the use of random cosets of an identical linear code at the two users does not cause any rate loss compared to choosing the codes independently for both users.