It is shown that dominant trapping sets of regular LDPC codes, so called absorption sets, undergo a two-phase dynamic behavior in the iterative message-passing (MP) 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 802.3 an (2048, 1723) regular LDPC code, is identified and tabulated using topological relationships in combination with search algorithms.
It is shown that dominant trapping sets of regular LDPC codes, so called absorption sets, undergo a two-phase dynamic behavior in the iterative message-passing (MP) 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 802.3 an (2048, 1723) regular LDPC code, is identified and tabulated using topological relationships in combination with search algorithms.
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.
It is shown that dominant trapping sets of regular low-density parity-check (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.
The goal of error control coding is to encode information in such a way, that in the event that errors occur during transmission over a noisy communication channel or during storage in an unreliable memory, the receiver can correct the errors and recover the original transmitted information. Low-density parity-check (LDPC) codes are a kind of high performance linear block code, which are already used in many recent communication systems. Information rates guaranteed by these codes approach the theoretical Shannon capacity limit. In addition, LDPC codes are now widely used in practice and included in many communication standards. Therefore study of these codes is very important and practical approaches to their design and decoding techniques are of great interest. There exist very good decoding algorithms for LDPC codes with respect to different channel models. However, LDPC codes suffer from the infamous error floor problem at high signal-to-noise ratios. This problem is attributed to the trapping sets by Richardson. It is shown that the dominant trapping sets of regular LDPC codes, so called absorption sets, undergo a two-phase dynamic behavior in the iterative message-passing decoding algorithm. We present a linear dynamic model for the iteration behavior of these sets. It can be seen that they undergo an initial geometric growth phase which stabilizes in a final bit-flipping behavior where the algorithm reaches a fixed point. Our 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 topologies of the dominant absorption sets of two example codes, the IEEE 802.3an [2048,1723] regular (6,32) LDPC code and the Tanner [155,64,20] regular (3,5) LDPC code, are identified and tabulated by using topological features in combination with search algorithms, respectively. To make our analysis more complete, we provide more solid evidence showing that trapping sets are equivalent to absorption sets. In other words, we argue that absorption sets characterize all failure mechanisms. Some insights from our linear analysis can be borrowed to approach this problem. Another insight that this formula provides to us is the means to reduce the error floor, which is another important part of what we hoped to achieve by developing the formula in the first place. By allowing the log-likelihood ratios (LLRs) utilized by the message-passing decoding to grow bigger in precision length, the absorption sets as a trouble-causing structure will be successfully corrected. Therefore, the absorption sets, generically born with the code design, will no longer threaten the error correcting performance as long as the messages have enough room and time to grow. This translates to greater LLR clipping thresholds and more iterations, both of which are preset at the decoder. However, the actual settings are dependent on how much the subgraph of the dominant absorption set resembles that of a non-zero minimum-weight codeword, therefore the settings are empirical. In general more likeness in their topologies, the more effort it costs to correct the absorption set.
It is well-known that the performance of low-density parity-check codes is compromised by the emergence of an error floor, which is caused by so-called trapping sets. By identifying the dominant trapping sets, this error floor can be estimated analytically using a linear model. In this paper, this linear approach is improved to make the analysis more accurate. Then, guided by the error probability formula, a simple but effective method to improve the code performance in the error floor region is proposed, by introducing a boosting factor to the log-likelihood ratios returned from unsatisfied check nodes during the early decoding stages. It is shown that the effect of the dominant trapping sets can thus be reduced. The dominance of trapping sets can be further reduced by extending the computational range of the LLRs at the decoder. To illustrate these ideas, a short regular LDPC code constructed by Tanner, as well as the IEEE 802.3 LDPC code, are studied.
This paper presents an efficient algorithm for finding the dominant trapping sets of a low-density parity-check (LDPC) code. The algorithm can be used to estimate the error floor of LDPC codes or to be part of the apparatus to design LDPC codes with low error floors. For regular codes, the algorithm is initiated with a set of short cycles as the input. For irregular codes, in addition to short cycles, variable nodes with low degree and cycles with low approximate cycle extrinsic message degree (ACE) are also used as the initial inputs. The initial inputs are then expanded recursively to dominant trapping sets of increasing size. At the core of the algorithm lies the analysis of the graphical structure of dominant trapping sets and the relationship of such structures to short cycles, low-degree variable nodes and cycles with low ACE. The algorithm is universal in the sense that it can be used for an arbitrary graph and that it can be tailored to find other graphical objects, such as absorbing sets and Zyablov-Pinsker (ZP) trapping sets, known to dominate the performance of LDPC codes in the error floor region over different channels and for different iterative decoding algorithms. Simulation results on several LDPC codes demonstrate the accuracy and efficiency of the proposed algorithm. In particular, the algorithm is significantly faster than the existing search algorithms for dominant trapping sets.
The error floor phenomenon observed with LDPC codes and their graph-based, iterative, message-passing (MP) decoders is commonly attributed to the existence of error-prone substructures - variously referred to as near-codewords, trapping sets, absorbing sets, or pseudocodewords - in a Tanner graph representation of the code. Many approaches have been proposed to lower the error floor by designing new LDPC codes with fewer such substructures or by modifying the decoding algorithm. Using a theoretical analysis of iterative MP decoding in an idealized trapping set scenario, we show that a contributor to the error floors observed in the literature may be the imprecise implementation of decoding algorithms and, in particular, the message quantization rules used. We then propose a new quantization method - (q+1)-bit quasi-uniform quantization - that efficiently increases the dynamic range of messages, thereby overcoming a limitation of conventional quantization schemes. Finally, we use the quasi-uniform quantizer to decode several LDPC codes that suffer from high error floors with traditional fixed-point decoder implementations. The performance simulation results provide evidence that the proposed quantization scheme can, for a wide variety of codes, significantly lower error floors with minimal increase in decoder complexity.