Theoretically Optimal Low-Density Parity-Check Code Ensemble for Gallager's Decoding Algorithm A

For a class of low-density parity-check (LDPC) code ensembles with right node degrees as binomial distribution, this paper proves that the theoretically optimal LDPC code ensemble should be regular for a binary-symmetric channel (BSC) and Gallager’s decoding algorithm A. Our proof consists of two steps. First, with the assumption of right edge degrees as binomial, we prove that the LDPC threshold of single left edge degree is larger than that of multiple left edge degrees. Second, we verify that the LDPC threshold is the largest when binomial distribution of right node degrees degrades to single value. Very interestingly, although both right and left edge degrees are unique in the theoretically optimal LDPC code ensemble, they are floating values. When the floating degrees are approximated by a two-term binomial distribution, the threshold at half rate is exactly the same as Bazzi’s result via linear programming. It verifies our proof from another angle