  In this paper, we propose a new method to design low-density parity-check
Hadamard (LDPC-Hadamard) codes, a type of ultimate-Shannon-limit approaching
channel codes. The technique is based on applying Hadamard constraints to the
check nodes in a generalized protograph-based LDPC code, followed by lifting
the generalized protograph. We name the codes formed protograph-based LDPC
Hadamard (PLDPC-Hadamard) codes. We also propose a modified Protograph
Extrinsic Information Transfer (PEXIT) algorithm for analyzing and optimizing
PLDPC-Hadamard code designs. The proposed algorithm further allows the analysis
of PLDPC-Hadamard codes with degree-1 and/or punctured nodes. We find codes
with 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 (for
rate = 0.0494) to 0.16 dB (for rate = 0.003). Moreover, the error performance
of our codes is comparable to that of the traditional LDPC-Hadamard codes.
Finally, the BER performances of our codes after puncturing are simulated and
compared.
