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

Summary form only given. Gallarger´s decoding algorithm A is a typical algorithm to theoretically analyze the decoding of Low-Density Parity-Check (LDPC) code ensembles. It is the message passing algorithm with the simplest hard decision. It is well recognized that the performance of Gallarger´s decoding algorithm A cannot approach channel capacity because of the hard decision process. Although many papers have studied the message pass algorithm, LDPC code design and the property of capacity approaching. However, what is the optimal performance for the simple Gallarger´s decoding algorithm A is still unknown until now. 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. Our theoretical result is not difficult to understand through some simulations. When the average degrees of left nodes and right nodes are small, the performance of LDPC code ensembles is not good and has a big gap to channel capacity. However, when the average degrees of left nodes and right nodes are large, the effect of hard decision will become large. Although LDPC code ensembles have the pot- - ential to approach channel capacity, the hard decision causes big performance loss. Therefore, there exist theoretically optimal LDPC code ensembles for Gallarger´s decoding algorithm A in BSC.