Magnetization Performance of LDPC Reduced-Complexity Decoding Algorithms

Low-density parity-check (LDPC) codes are very efficient for communicating reliably through a noisy channel. N.Sourlas [1] showed that LDPC codes, which revolutionize the codes domain and used in many communications standards, can be mapped onto an Ising spin systems. Besides, it has been shown that the Belief-Propagation (BP) algorithm, the LDPC codes decoding algorithm, is equivalent to the Thouless-Anderson-Palmer (TAP) approach [2]. Unfortunately, no study has been made for the other decoding algorithms. In this paper, we develop the Log-Likelihood Ratios-Belief Propagation (LLR-BP) algorithm and its simplifications the BP-Based algorithm and the X-min algorithm with the TAP approach. We present the performance of these decoding algorithms using statistical physics argument i.e., we present the performance as function of the magnetization.