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In this letter, we compute achievable information rates for finite-sized synchronous code-division multiple-access (CDMA) systems, and low-density parity-check (LDPC) codes are adopted to approach the achievable rates. The associated coding-spreading tradeoff problem is also considered using these results. We assume binary random spreading sequences, and the computed achievable rates are averaged over ensembles of spreading sequences. Unlike most prior papers, which analyze the spectral efficiency of large CDMA systems under Gaussianity assumptions (channel inputs and/or multiple-access interference), we make no such assumptions. In order to display the coding-spreading tradeoff, we plot the minimum required signal-to-noise ratio for reliable transmission as a function of information rate. It is shown that the coding-spreading tradeoff favors all coding (i.e., no spreading) when the optimal joint multiuser detector/decoder is employed, whereas for systems with a suboptimal multiuser detector and single-user decoders, there generally exists an optimal balance between coding and spreading. We also provide simulation results on the performance of LDPC-coded synchronous CDMA systems which approach the information-theoretic limits we have computed.