Achievable degrees of freedom on K-user y channels

In this paper, we consider K-user Y channels where K users simultaneously exchange messages with each other via an intermediate relay. Degrees of freedom (DOF) of Y channels with multiple antennas is not known in general. Investigation of the feasibility conditions of signal space alignment for network coding is an initial step for addressing this open problem. We verify that when user i with M _i antennas sends K-1 independent messages to the other users through a relay with N antennas and each message achieves the DOF of d, the total DOF of dK(K-1) is attained if M _i d(K-1), N≥ dK(K-1)}/2 and N < min{ _i+M _j-d | forall i ≠j}. It is accomplished by adopting the signal space alignment for the network coding during both the multiple access phase and the broadcast phase. It is shown that the proposed scheme obtains not only a network coding gain but also an alignment gain in terms of the normalized DOF, as K → ∞. Also for Y channels where all nodes have a single antenna, we show that the DOF of 2 is achieved regardless of the number of users by using the rational dimension framework.