text: The multiway relay channel is a multicast network where L users exchange data through a relay. In this paper, the capacity region of a class of multiway relay channels is derived, where the channel inputs and outputs take values over finite fields. The cut-set upper bound to the capacity region is derived and is shown to be achievable by our proposed functional-decode-forward coding strategy. More specifically, for the general case where the users can transmit at possibly different rates, functional-decode-forward, combined with rate splitting and joint source-channel decoding, is proved to achieve the capacity region; while for the case where all users transmit at a common rate, rate splitting and joint source-channel decoding are not required to achieve the capacity. That the capacity-achieving coding strategies do not utilize the users' received signals in the users' encoding functions implies that feedback does not increase the capacity region of this class of multiway relay channels.
  @cite_28: A 3-node lossy source coding problem for a 2-DMS (X 1 ,X 2 ) is considered. Source nodes 1 and 2 observe X 1 and X 2 , respectively, and each wishes to reconstruct the other source with a prescribed distortion. To achieve these goals, nodes 1 and 2 send descriptions of their sources to relay node 3. The relay node then broadcasts a joint description to the source nodes. A cutset outer bound and a compress-linear code inner bound are established and shown to coincide in several special cases. A compute-compress inner bound is then presented and shown to outperform the compress-linear code in some cases. An outer bound based on Kaspi's converse for the two-way source coding problem is shown to be strictly tighter than the cutset outer bound.
  @cite_18: Three dependent users are physically separated but communicate with each other via a satellite. Each user generates data which it stores locally. In addition, each user sends a message to the satellite. The satellite processes the messages received from the users and broadcasts one common message to all three users. Each user must be capable of reconstructing the data of the other two users based upon the broadcast message and its own stored data. Our problem is to determine the minimum amount of information which must be transmitted to and from the satellite. The solution to this problem is obtained for the case where subsequent data triples that are produced by the users are independent and identically distributed. The three symbols within each triple are assumed to be dependent. Crucial for the solution is an achievability proof that involves cascaded Slepian-Wolf (1973) source coding.
  @cite_12: The broadcast phase (downlink transmission) of the two-way relay network is studied in the source coding and joint source-channel coding settings. The rates needed for reliable communication are characterised for a number of special cases including: small distortions, deterministic distortion measures, and jointly Gaussian sources with quadratic distortion measures. The broadcast problem is also studied with common-reconstruction decoding constraints, and the rates needed for reliable communication are characterised for all discrete memoryless sources and per-letter distortion measures.
summary: The MWRC has also been studied from the point of view of source coding, where multiple users exchange possibly correlated data via a relay. In the source coding setting, the channel from the users to the relay and that from the relay to the users are assumed to be noiseless. The problem formulation is how many bits the users need to encode their respective messages to be sent to the relay; and after the relay receives these encoded messages, how many bits the relay needs to transmit to the users in order for each user to recover the messages of all other users. The three-user lossless case (where each user perfectly reconstructs the other two users' messages) was studied by Wyner @cite_18 , the two-user lossless case and lossy case (where each user reconstructs the other user's message with a prescribed distortion) was studied by Su and El Gamal @cite_28 , and the two-user lossy case with common reconstructions (where each user must also be able to determine the lossy reconstructed message of the other user) was studied by Timo @cite_12 .