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The authors consider the design of LDPC codes for cooperative relay systems in the half-duplex mode. The capacity of halfduplex relay channels has been studied previously but the design of good channel codes for such channels remains a challenging problem. Employing an efficient relay protocol, we transform the half-duplex relay code design problem into a problem of ratecompatible LDPC code design where different code segments experience different SNRs. The density evolution with conventional Gaussian approximation for single user channels, which assumes invariant SNR within one codeword, is not capable of accurately predicting the code performance for this system. Here we develop a density evolution with a modified Gaussian approximation that takes into account the SNR variation in one received codeword as well as the rate-compatibility constraint. We then optimize the code ensemble using a modified differential evolution procedure. Extensive simulations are carried out to demonstrate that the proposed algorithm offers more accurate prediction of code performance in half-duplex relay channels than the conventional methods, and the optimized codes achieve a significant gain over existing codes.