Additionally, the GaBP algorithm is shown to be immune to numerical problems of the preconditioned conjugate gradient method Syndrome-based BP decoding The previous description of BP algorithm is called the codeword-based decoding, which calculates the approximate marginal probability , given received codeword . There is an equivalent form, which calculate , where is the syndrome of the received codeword and is the decoded error. The decoded input vector is . This variation only change the interpretation of the mass function . Explicitly, the messages are where is the prior error probability on variable This syndrome-based decoder doesn't require information on the received bits, thus can be adapted to quantum codes, where the only information is the measurement syndrome. In the binary case, , those messages can be simplifies to cause an exponential reduction of in the complexity Define log-likelihood ratio , , then where The posterior log-likelihood ratio can be estimated as References Further reading Bickson, Danny. (2009). Gaussian Belief Propagation Resource Page —Webpage containing recent publications as well as Matlab source code. Coughlan, James. (2009). A Tutorial Introduction to Belief Propagation. Mackenzie, Dana (2005). "Communication Speed Nears Terminal Velocity", New Scientist. 9 July 2005. Issue 2507 (Registration required) Graph algorithms Graphical models Coding theory