Linear Sum Capacity for Gaussian Multiple Access Channels with Feedback

The capacity region of the N -sender additive white Gaussian noise (AWGN) multiple access channel (MAC) with feedback is not known in general, despite significant contributions by Cover, Leung, Ozarow, Thomas, Pombra, Ordentlich, Kramer, and Gastpar. This paper studies the class of generalized linear feedback codes that includes (nonlinear) nonfeedback codes at one extreme and the linear feedback codes by Schalkwijk and Kailath, Ozarow, and Kramer at the other extreme. The linear sum capacity CL(N,P ), the maximum sum rate achieved by generalized linear feedback codes, is characterized under symmetric block power constraints P for all the senders. In particular, it is shown that Kramer’s linear code achieves this linear sum capacity. The proof involves the dependence balance condition introduced by Hekstra and Willems and extended by Kramer and Gastpar. This condition is not convex in general, and the corresponding nonconvex optimization problem is carefully analyzed via Lagrange dual formulation. Based on the properties of the conditional maximal correlation—an extension of the Hirschfeld–Gebelein–Renyi maximal correlation—it is further conjectured that Kramer’s linear code achieves not only the linear sum capacity, but also the (general) sum capacity, i.e., the maximum sum rate achieved by arbitrary feedback codes.