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The $K$ -user multiple-input multiple-output (MIMO) multi-way relay channel (Y-channel) consisting of $K$ users with $M$ antennas each and a common relay node with $N$ antennas is studied in this paper. Each user wants to exchange messages with all the other users through the relay. A transmission strategy is proposed for this channel. The proposed strategy is based on two steps: 1) channel diagonalization and 2) cyclic communication. The channel diagonalization is applied by using zero-forcing beam-forming. After the channel diagonalization, the channel is decomposed into parallel sub-channels. Cyclic communication is then applied, where signal-space alignment for network-coding is used over each sub-channel. The proposed strategy achieves the optimal degree-of-freedom (DoF) region of the channel if $N\leq M$ . To prove this, a new DoFs outer bound is derived. As a by-product, we conclude that the MIMO Y-channel is not separable, i.e., independent coding on separate sub-channels is not enough, and one has to code jointly over several sub-channels. [ABSTRACT FROM AUTHOR]

A full-duplex wireless network with three users that want to establish full message exchange via a relay is considered. Thus, this network which is known as the Y-channel has a total of six messages, two outgoing, and two incoming at each user. The users are not physically connected, and thus the relay is essential for their communication. The deterministic Y-channel is considered first, its capacity region is characterized, and shown not to be given by the cut-set bounds. The capacity achieving scheme has three different components (strategies): 1) a bidirectional; 2) a cyclic; and 3) a unidirectional strategy. Network coding is used to realize the bidirectional and the cyclic strategies, and thus to prove the achievability of the capacity region. The result is then extended to the Gaussian Y-channel where the capacity region is characterized within a constant gap independent of the channel parameters. [ABSTRACT FROM AUTHOR]