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Abstract-It is shown that dominant trapping sets of regular LDPC codes, so called absorption sets, undergo a two-phased dynamic behavior in the iterative message-passing decoding algorithm. Using a linear dynamic model for the iteration behavior of these sets, it is shown that they undergo an initial geometric growth phase which stabilizes in a final bit-flipping behavior where the algorithm reaches a fixed point. This analysis is shown to lead to very accurate numerical calculations of the error floor bit error rates down to error rates that are inaccessible by simulation. The topology of the dominant absorption sets of an example code, the IEEE 802.3an (2048, 1723) regular LDPC code, are identified and tabulated using topological relationships in combination with search algorithms.Index Terms-absorption sets, error floor, Low-Density ParityCheck codes. I. SUMMARYT HE error floor in modern graph-based error control codes such as low-density parity-check codes is caused by inherent structural weaknesses in the code's interconnect network. The iterative message passing algorithm cannot overcome these weaknesses and gets trapped in error patterns which are easily identifiable as erroneous (in LDPC codes), and are thus not valid codewords, but difficult to overcome or correct [1], [2]. These weaknesses were termed trapping sets by Richardson in [3], a summary definition for the patterns on which the message passing algorithm fails for Gaussian channels. These trapping sets are dependent on the code, the channel used, and to a lesser degree also on the details of the decoding algorithm. Prior work in identifying the weaknesses of LDPC codes on erasure channels led to the definition of stopping sets in [4]. Stopping sets, being the weaknesses of LDPC codes on erasure channels, also play a role on Gaussian channels, but are not typically the dominant error mechanisms. In [5] the authors define absorption sets, which are the subgraphs of the code graph on which the Gallager bit-flipping decoding algorithms fail for binary symmetric channels. The authors observed that these absorption sets also show up as the dominant trapping sets in certain structured LDPC codes. In [6] they devise post-processing methods to reduce the effects of these absorption sets and lower the error floor of the codes in question.C. Schlegel and S. Zhang are with the High Capacity Digital Communications Laboratory (HCDC), Electrical and Computer Engineering Department, University of Alberta, Edmonton AB, T6G 2V4, CANADA (e-mail: {schlegel, szhang4}@ece.ualberta.ca).In this paper we present a linear algebraic approach to the dynamic behavior of absorption sets. We show that these sets follow a geometric growth phase during early iterations where messages inside the absorption set grow towards a largest eigenvector which characterizes the absorption set. The seemingly erratic behavior of the messages at early iterations is due to the decreasing influence of lesser eigenvectors. We define the gain of an absorption set and show how it affects the inf...

This paper presents a stochastic algorithm for iterative error control decoding. We show that the stochastic decoding algorithm is an approximation of the sum-product algorithm. When the code's factor graph is a tree, as with trellises, the algorithm approaches maximum a-posteriori decoding. We also demonstrate a stochastic approximations to the alternative update rule successive relaxation. Stochastic decoders have very simple digital implementations which have almost no RAM requirements. We present example stochastic decoders for a trellisbased Hamming code, and for a Block Turbo code constructed from Hamming codes.