Decoding LDPC Codes with Probabilistic Local Maximum Likelihood Bit Flipping

Low-density parity-check (LDPC) codes are high-performance linear error correcting codes with application to communication channels and digital storage media. LDPC codes are decoded using graph algorithms wherein a channel message sample is decoded with the aid of information from its adjacent graph neighborhood, called the syndrome. This work studies the conditional probability of a channel error given syndrome information at a particular decoding iteration to formulate a new algorithm called Probabilistic Local Maximum Likelihood Bit Flipping (PLMLBF). The PLMLBF algorithm uses a three dimensional Multi-iteration Probability Flip Matrix (MIPFM) to quantify the frequency of errors in a noise corrupted message frame being decoded using a specific LDPC code. The matrix is used to probabilistically decode noise corrupted message frames. The motivation for this work is to provide a theoretical framework for constructing probabilistic and noisy bit-flipping algorithms, such as the Noisy Gradient Descent Bit Flipping (NGDBF) algorithm, which up to now have been mainly heuristic in nature.