This thesis presents low complexity design and decoding schemes for low density parity check (LDPC) codes. First, we consider the iterative decoding of LDPC codes on multiple-input-multiple-output bit-interleaved coded modulation (MIMO-BICM) channels and two-way relay channels. More specifically, we study the log-likelihood ratio (LLR) calculation under MIMO-BICM channels when perfect channel information is known and LLR calculation for two-way relay channels when no channel information is

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... at the receiver. We propose the optimum piece-wise linear approximation in the sense of maximizing the achievable rate of the channel. Second, we introduce a novel "universal" LDPC code design method. We design universal LDPC codes based on our method and show that, compared to existing methods, a lager percentage of capacity is obtained. Then, we propose two conjectures about the extreme distributions under min-sum decoding based on numerical observations.