Genie Chains and the Degrees of Freedom of the K-User MIMO Interference Channel

We explore the degrees of freedom (DoF) of the K > 3 user M-T x M-R MIMO Gaussian interference channel where each transmitter is equipped with M-T and each receiver is equipped with M-R antennas. Expressing the DoF characterization as a function of the ratio gamma = M/N, where M = min(M-T, M-R) and N = max(M-T, M-R), we find that when gamma <= gamma(o) = K-1/K(K-2) = gamma(o), the DoF value per user is piecewise linear depending on M and N alternately, similar to the DoF characterization for K = 3 which has been previously obtained. The regime gamma > gamma(o), which is the dominant regime for K > 3 users and is not encountered in the K = 3 user setting, is the main focus of this paper. Our DoF results in this regime are obtained through a novel "genie chains" approach, which is the main contribution of this work. It is a chain of mappings from genie signals provided to a receiver to the exposed signal spaces at that receiver, which then serve as the genie signals for the next receiver in the chain, until an acceptable genie with the required number of dimensions is obtained, essentially converting an information theoretic problem into a linear algebra problem.