The distribution of loop lengths in graphical models for turbo decoding

This correspondence analyzes the distribution of loop lengths in graphical models for turbo decoding. The properties of such loops are of significant interest in the context of iterative decoding algorithms based on belief propagation. We estimate the probability that there exist no loops of length less than or equal to c at a randomly chosen node in the acyclic directed graphical (ADC) model for turbo decoding, using a combination of counting arguments and approximations. When K, the number of information bits, is large, this probability is approximately e -2^c-1-4/K, for c⩾4, where nodes for input information bits are ignored for convenience. The analytical results are validated by simulations. For example, for turbo codes with K=64,000, a randomly chosen node has a less than 1% chance of being on a loop of length less than or equal to 10, but has a greater than 99.9% chance of being on a loop of length less than or equal to 20