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We investigate the spectral efficiency achievable by random synchronous code-division multiple access (CDMA) with quaternary phase-shift keying (QPSK) modulation and binary error-control codes, in the large system limit where the number of users, the spreading factor, and the code block length go to infinity. For given codes, we maximize spectral efficiency assuming a minimum mean-square error (MMSE) successive stripping decoder for the cases of equal rate and equal power users. In both cases, the maximization of spectral efficiency can be formulated as a linear program and admits a simple closed-form solution that can be readily interpreted in terms of power and rate control. We provide examples of the proposed optimization methods based on off-the-shelf low-density parity-check (LDPC) codes and we investigate by simulation the performance of practical systems with finite code block length.