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We consider the symbol-synchronous code-division multiple-access (CDMA) channel in which every user is assigned a rate at which arbitrarily reliable transmission in the Shannon sense is to be guaranteed. For an overloaded system in which the number of active users exceeds the available processing gain, we optimally design the users' signature sequences and a power-control policy to minimize the required sum-power (i.e., sum of the users' powers) while meeting the rate-tuple constraint with a (joint) maximum-likelihood receiver. This result is extended to find the power-constrained capacity region of the system; this is the set of all achievable rate-tuples over all signature sequences and power-control policies whose sum-power is constrained. Furthermore, it is shown that this capacity region may be substantially and maximally expanded in those regions where there are oversized users whose rate requirements are relatively large compared to those of the other users; this is accomplished by allowing for the flexibility of multidimensional signaling in the sense of a user simultaneously transmitting several different scalar symbols, each modulated by its own signature sequence. From the viewpoint of resource efficiency, this means that a multicarrier approach is essential in systems that support multiple classes of users. Finally, we also address the dual problem of determining the region of valid power-control policies subject to a sum-capacity constraint on the system.