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Network coding is becoming more and more popular in recent years for its ability to improve network throughput and robustness. In this paper, we propose a scheme of joint network coding and channel coding based on Turbo-BICM over multiple-access channel (MAC). The scheme with only a turbo decoder significantly reduces the complexity of system. Moreover, it can offer the best error performance in the high SNR regime. These conclusions are validated by both theoretical analysis and simulation results. Interesting, we also find that a detached scheme, which first operates channel coding then network coding, approaches the upper bound in the low SNR regime.