In this study, the authors explore a generalised scheme for the synchronous code division multiple access (CDMA). In this scheme, unlike the standard CDMA systems, each user has different codewords for communicating different messages. Two main problems are investigated. The first problem concerns whether uniquely detectable overloaded matrices (an injective matrix, i.e. the inputs and outputs are in one‐to‐one correspondence depending on the input alphabets) exist in the absence of additive noise, and if so, whether there are any practical optimum detectors for such input codewords. The second problem is about finding tight bounds for the sum channel capacity. In response to the first problem, the authors have constructed uniquely detectable matrices for the generalised scheme and the authors have developed practical maximum likelihood detection algorithms for such codes. In response to the second problem, lower bounds and conjectured upper bounds are derived. The results of this study are superior to other standard overloaded CDMA codes since the generalisation can support more users than the previous schemes.

In this paper, we explore some of the fundamentals of synchronous Code Division Multiple Access (CDMA) as applied to wireless and optical communication systems under very general settings (of any size) for the user symbols and the signature matrix entries. The channel is modeled by real/complex additive noise of arbitrary distribution. Two problems are addressed. The first problem concerns whether uniquely detectable overloaded matrices exist in the absence of additive noise under these general settings, and if so, whether there are any practical optimum detection algorithms. The second one is about the bounds for the sum channel capacity when user data and signature matrices employ any real or complex alphabets (finite or infinite). In response to the first problem, we have developed practical maximum likelihood detection 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, spreading gain, and signal-to-noise ratio can be derived. To show the power and utility of the main theorem, a number of sum capacity bounds for special cases are evaluated. 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 for signature matrices and input vectors with i.i.d. entries. Furthermore, upper and asymptotic bounds are derived and compared to other derivations.

Lower and upper bounds are derived for the sum capacity of synchronous CDMA where the signature matrix and input alphabets are binary or (2p + 1)-ary, in two cases of noiseless and noisy channels. The bounds are very tight in some regions. Interestingly, simulations show that the formulas for noisy systems tend to the ones for noiseless system as noise tends to 0 while it cannot be deduced easily from the formulas. The results give good insights about the extent of the number of users in which errorless communication is possible for a system with a given chip rate.