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In this paper, we analyze the performance of irregular low-density parity-check (LDPC) codes with three belief propagation (BP) based decoding algorithms, namely, the uniformly most powerful (UMP) BP-based algorithm, the normalized BP-based algorithm, and the offset BP-based algorithm on a fast Rayleigh fading channel by using density evolution (DE). We also optimize the code construction of the irregular LDPC codes with three BP-based algorithms, and determine optimum parameters for the normalized BP-based algorithm and the offset BP-based algorithm on the fast Rayleigh fading channel. From the numerical results, we show that the performance of the irregular LDPC codes with offset BP-based algorithm can be very close to that with the BP algorithm on the fast Rayleigh fading channel. We also show that the irregular LDPC codes with offset BP-based algorithm can achieve a very good trade-off between the performance and decoding complexity.