We consider communication over the Gaussian multiple-access channel in the regime where the number of users grows linearly with the codelength. We investigate coded CDMA schemes where each user's information is encoded via a linear code before being modulated with a signature sequence. We propose an efficient approximate message passing (AMP) decoder that can be tailored to the structure of the linear code, and provide an exact asymptotic characterization of its performance. Based on this result, we consider a decoder that integrates AMP and belief propagation and characterize the tradeoff between spectral efficiency and signal-to-noise ratio, for a given target error rate. Simulation results are provided to demonstrate the benefits of the concatenated scheme at finite lengths.