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This paper investigates the error performance and precoder design for a bandwidth-efficient bit-interleaved coded modulation (BICM) system over a non-orthogonal amplify-and-forward (NAF)half-duplex single-relay channel. A tight union bound on the bit error probability (BEP) is first derived for an arbitrary block length of 2N using a 2N × 2N precoder. This bound provides an useful tool to predict the error performance. Attention is then devoted to the system using 2×2 precoder, where a closed-form expression of the bound is obtained. Based on this expression, an optimal 2×2 precoder applicable to any modulation scheme is developed. Different from the optimal precoders designed for uncoded NAF systems, the derived precoder indicates that the source only needs to send the superposition of signals in the first time slot and being silent in the second time slot in order to achieve the best asymptotic performance. Analytical and simulation results show that the proposed precoder not only exploits full cooperative diversity but also offers a significant coding gain over optimal precoders for uncoded NAF systems.