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We study coding over a class of two-user broadcast channels with additive white Gaussian noise and fading known at the receivers only. Joint decoding of low-density parity-check codes is analyzed. The message update rule at the mapping node linking the users' codes is derived and is shown to exhibit an interesting soft interference cancellation property. Good degree distributions are found using the differential evolution optimization technique and extrinsic information transfer analysis. The corresponding codes have rates close to the boundary of the achievable region for binary constrained input channels, both with and without fading. Simulation results for moderate blocklength show that the optimized codes operate within 1 dB of their thresholds.