  In this paper, we obtain a family of lower bounds for the sum capacity of
Code Division Multiple Access (CDMA) channels assuming binary inputs and binary
signature codes in the presence of additive noise with an arbitrary
distribution. The envelope of this family gives a relatively tight lower bound
in terms of the number of users, spreading gain and the noise distribution. The
derivation methods for the noiseless and the noisy channels are different but
when the noise variance goes to zero, the noisy channel bound approaches the
noiseless case. The behavior of the lower bound shows that for small noise
power, the number of users can be much more than the spreading gain without any
significant loss of information (overloaded CDMA). A conjectured upper bound is
also derived under the usual assumption that the users send out equally likely
binary bits in the presence of additive noise with an arbitrary distribution.
As the noise level increases, and/or, the ratio of the number of users and the
spreading gain increases, the conjectured upper bound approaches the lower
bound. We have also derived asymptotic limits of our bounds that can be
compared to a formula that Tanaka obtained using techniques from statistical
physics; his bound is close to that of our conjectured upper bound for large
scale systems.
