Protograph-based LDPC-Hadamard Codes

We propose a novel type of ultimate-Shannon-limit-approaching codes, namely protograph-based low-density parity-check Hadamard (PLDPC-Hadamard) codes in this paper. We also propose a systematic way of analyzing such codes using Protograph EXtrinsic Information Transfer (PEXIT) charts. Using the analytical technique we have found a code of rate about 0.05 having a theoretical threshold of -1.42dB. At a BER of $10^{-5}$, the gaps of our code to the Shannon capacity for R=0.05 and to the ultimate Shannon limit are 0.25 dB and 0.40 dB, respectively.