Failures of the Gallager B Decoder: Analysis and Applications

In this paper, we study state transitions, induced by low-density parity-check codes, of the Gallager B algorithm in order to characterize failures of the decoder. The failures of the decoder depend on the properties of the underlying graph of the code. Two classes of sets, namely trapping sets and propagating sets, of variables of the code are defined to categorize failures of the decoder. Such a categorization helps to establish a connection between size and population of these sets in the code and the frame error rate performance of the decoder. This connection is explored to develop a semi-analytical method to estimate the frame error rate of a class of codes from balanced incomplete block designs.