Tight Bounds on the Capacity of Binary Input Random CDMA Systems

In this paper, we consider code-division multiple-access (CDMA) communication over a binary input additive white Gaussian noise (AWGN) channel using random spreading. For a general class of symmetric distributions for spreading sequences, in the limit of a large number of users, we prove an upper bound to the capacity. The bound matches the formula obtained by Tanaka using the replica method. We also show concentration of various relevant quantities including mutual information and free energy. The mathematical methods are quite general and allow us to discuss extensions to other multiuser scenarios.