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In the bit-error rate (BER) analysis of code-division multiple-access (CDMA) systems, a Gaussian approximation is widely used to tackle the multiple access interference (MAI), although it does not always offer satisfactory accuracy. This paper investigates the BER performance of synchronous multicarrier (MC) CDMA systems over Nakagami-m-fading channels in a different way. We present an accurate and unified BER analysis for synchronous MC-CDMA systems. To facilitate our analysis, we assume a synchronous uplink, whose BER performance can be intuitively viewed as a lower BER bound of the more realistic asynchronous MC-CDMA. The basic idea is that, by using the Gauss-Chebyshev quadrature (GCQ) rule to perform inverse Laplace transform, an accurate BER can be numerically obtained from the moment generating function (MAG) of the output decision variable at a receiver, without any assumption about the MAI distribution. First, signals on all subcarriers of MC-CDMA systems are assumed to experience independent fading. Two standard diversity combining techniques, equal gain combining (EGC) and maximal ratio combining (MRC), are employed. The BER performance in both downlink and synchronous uplink is analyzed. We then consider a more general system model, in which signals on different subcarriers undergo correlated fading. The asymptotic (error floor) performance of downlink MC-CDMA with MRC is studied. In particular, we investigate the effects of spreading sequences and the delay spread of the channel on the system performance. Numerical examples are provided to show the main results of this paper. The accuracy of the GCQ and MGF based solution is verified by different approaches such as Monte Carlo integration and the exact residue method. In addition, the accuracy of the commonly used Gaussian approximation is also examined.