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In this paper, we propose a new method to design low-density parity-checkHadamard (LDPC-Hadamard) codes, a type of ultimate-Shannon-limit approachingchannel codes. The technique is based on applying Hadamard constraints to thecheck nodes in a generalized protograph-based LDPC code, followed by liftingthe generalized protograph. We name the codes formed protograph-based LDPCHadamard (PLDPC-Hadamard) codes. We also propose a modified ProtographExtrinsic Information Transfer (PEXIT) algorithm for analyzing and optimizingPLDPC-Hadamard code designs. The proposed algorithm further allows the analysisof PLDPC-Hadamard codes with degree-1 and/or punctured nodes. We find codeswith decoding thresholds ranging from -1.53 dB to -1.42 dB. At a BER of 1e-5,the gaps of our codes to the ultimate-Shannon-limit range from 0.40 dB (forrate = 0.0494) to 0.16 dB (for rate = 0.003). Moreover, the error performanceof our codes is comparable to that of the traditional LDPC-Hadamard codes.Finally, the BER performances of our codes after puncturing are simulated andcompared.

Protograph-Based Low-Density Parity-Check Hadamard Codes