The capacity region of three user Gaussian inverse-compute-and-forward channels

We consider a three user multiple access channel where transmitter m has access to the linear equation u_m = Σ³_{l = 1} f_ml w_l of independent messages w₁ ϵ F_p^k1, w2 ϵ F_p^k2, w3 ϵ F_p^k3 (and f_ml ϵ F_p), and the destination wishes to recover all three messages. This problem is motivated as the last hop in a network where relay nodes employ the Compute-and-Forward strategy and decode linear equations of messages; we seek to do the reverse and extract messages from sums over a multiple access channel. An achievable rate region for the two user problem was previously derived; here we extend and strengthen this work to show capacity for the two and three user Gaussian channel models subject to invertability conditions on the matrix of coefficients describing the given linear equations of messages. The optimal transmission scheme is not to independently send the three equations u_m over the MAC but rather to exploit their special correlation structure.