Achievable degrees of freedom on K-user MIMO multi-way relay channel with common and private messages

This paper investigates the achievable total degrees of freedom (DoF) of the MIMO multi-way relay channel that consists of K users, where each user is equipped with M antennas, and a decode-and-forward relay equipped with N antennas. In this channel, each user wants to convey K - 1 private messages to the other users in addition to a common message to all of them. Due to the absence of direct links between the users, communication occurs through the relay in two phases: a multiple access channel (MAC) phase and a broadcast (BC) phase. We derive cut-set bounds on the total DoF of the network, and show that the network has DoF less than or equal to K min {N, M}. Achievability of the upper bound is shown by using signal space alignment for network coding in the MAC phase, and zero- forcing precoding in the BC phase. We show that introducing the common messages besides the private messages leads to achieving higher total DoF than using the private messages only.