Low-rate generalized low-density parity-check codes with hadamard constraints

We consider the design and analysis of generalized low-density parity-check (GLDPC) codes specified by a bipartite Tanner graph, as with standard LDPC codes, but with the single parity-check constraints replaced by general coding constraints. In particular, we consider imposing Hadamard code constraints at the check nodes for a low-rate approach, termed LDPC-Hadamard codes. The achievable capacity with the GLDPC codes is then discussed. A modified LDPC-Hadamard code graph is also proposed. We then optimize the LDPC-Hadamard code ensemble using a low-complexity optimization method based on approximating the density evolution by a one-dimensional dynamic system represented by an extrinsic mutual information transfer (EXIT) chart. Simulation results show that a rate-0.003 LDPC-Hadamard code with large block length can achieve a bit-error-rate (BER) performance of 10^-5 at -1.44 dB, only 0.15 dB away from the ultimate Shannon limit (-1.592 dB)