Abstract-We propose Low Density Parity Check (LDPC) code designs for the half-duplex relay channel. Our designs are based on the information theoretic random coding scheme for decode-and-forward relaying. The source transmission is decoded with the help of side information in the form of additional parity bits from the relay. We derive the exact relationships that the component LDPC code profiles in the relay coding scheme must satisfy. These relationships act as constraints for the density evolution algorithm which is used to search for good relay code profiles. To speed up optimization, we outline a Gaussian approximation of density evolution for the relay channel. The asymptotic noise thresholds of the discovered relay code profiles are a fraction of a decibel away from the achievable lower bound for decode-and-forward relaying. With random component LDPC codes, the overall relay coding scheme performs within 1.2 dB of the theoretical limit.

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Abstract-The major drawback of the LDPC codes versus the turbo-codes is their comparative low convergence speed: 25-30 iterations vs. 8-10 iterations for turbo-codes. Recently, Hocevar showed by simulations that the convergence rate of the LDPC decoder can be accelerated by exploiting a 'turbo-scheduling' applied on the bit-node messages (rows of the parity check matrix). In this paper, we show analytically that the convergence rate for this type of scheduling is about two times increased for most of the regular LDPC codes. Second we prove that 'turbo-scheduling' applied on the rows of the parity check matrix is identical belief propagation algorithm as standard message passing algorithm. Furthermore, we propose two new message passing schedules: 1) a turbo-scheduling is applied on the checknode messages (columns of the parity check matrix); 2) a hybrid version of both previous schedules where the turbo-effect is applied on both check-nodes and bit-nodes. Frame error rate simulations validate the effectiveness of the proposed schedules.

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