  In this paper, we explore the mystery of synchronous CDMA as applied to
wireless and optical communication systems under very general settings for the
user symbols and the signature matrix entries. The channel is modeled with
real/complex additive noise of arbitrary distribution. Two problems are
addressed. The first problem concerns whether overloaded error free codes exist
in the absence of additive noise under these general settings, and if so
whether there are any practical optimum decoding algorithms. The second one is
about the bounds for the sum channel capacity when user data and signature
codes employ any real or complex alphabets (finite or infinite). In response to
the first problem, we have developed practical Maximum Likelihood (ML) decoding
algorithms for overloaded CDMA systems for a large class of alphabets. In
response to the second problem, a general theorem has been developed in which
the sum capacity lower bounds with respect to the number of users and spreading
gain and Signal-to-Noise Ratio (SNR) can be derived as special cases for a
given CDMA system. To show the power and utility of the main theorem, a number
of sum capacity bounds for special cases are simulated. An important conclusion
of this paper is that the lower and upper bounds of the sum capacity for
small/medium size CDMA systems depend on both the input and the signature
symbols; this is contrary to the asymptotic results for large scale systems
reported in the literature (also confirmed in this paper) where the signature
symbols and statistics disappear in the asymptotic sum capacity. Moreover,
these questions are investigated for the case when not all users are active.
Furthermore, upper and asymptotic bounds are derived and numerically evaluated
and compared to other derivations.
