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IEEE Journal on Selected Areas in Communications

Predicting error floors of structured LDPC codes: deterministic bounds and estimates

Lara Dolecek, P. Lee, Zhengya Zhang, V. Anantharam, Borivoje Nikolić, Martin J. Wainwright

Published Jul 29, 2009• 94 citations
Publisher

Abstract

The error-correcting performance of low-density parity check (LDPC) codes, when decoded using practical iterative decoding algorithms, is known to be close to Shannon limits for codes with suitably large blocklengths. A substantial limitation to the use of finite-length LDPC codes is the presence of an error floor in the low frame error rate (FER) region. This paper develops a deterministic method of predicting error floors, based on high signal-to-noise ratio (SNR) asymptotics, applied to absorbing sets within structured LDPC codes. The approach is illustrated using a class of array-based LDPC codes, taken as exemplars of high-performance structured LDPC codes. The results are in very good agreement with a stochastic method based on importance sampling which, in turn, matches the hardware-based experimental results. The importance sampling scheme uses a mean-shifted version of the original Gaussian density, appropriately centered between a codeword and a dominant absorbing set, to produce an unbiased estimator of the FER with substantial computational savings over a standard Monte Carlo estimator. Our deterministic estimates are guaranteed to be a lower bound to the error probability in the high SNR regime, and extend the prediction of the error probability to as low as 10-30. By adopting a channel-independent viewpoint, the usefulness of these results is demonstrated for both the standard Gaussian channel and a channel with mixture noise.

DOI
10.1109/jsac.2009.090809

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The error-correcting performance of low-density parity check (LDPC) codes, when decoded using practical iterative decoding algorithms, is known to be close to Shannon limits for codes with suitably large blocklengths. A substantial limitation to the use of finite-length LDPC codes is the presence of an error floor in the low frame error rate (FER) region. This paper develops a deterministic method of predicting error floors, based on high signal-to-noise ratio (SNR) asymptotics, applied to absorbing sets within structured LDPC codes. The approach is illustrated using a class of array-based LDPC codes, taken as exemplars of high-performance structured LDPC codes. The results are in very good agreement with a stochastic method based on importance sampling which, in turn, matches the hardware-based experimental results. The importance sampling scheme uses a mean-shifted version of the original Gaussian density, appropriately centered between a codeword and a dominant absorbing set, to produce an unbiased estimator of the FER with substantial computational savings over a standard Monte Carlo estimator. Our deterministic estimates are guaranteed to be a lower bound to the error probability in the high SNR regime, and extend the prediction of the error probability to as low as 10-30. By adopting a channel-independent viewpoint, the usefulness of these results is demonstrated for both the standard Gaussian channel and a channel with mixture noise.
Abstract — The error-correcting performance of low-density parity check (LDPC) codes, when decoded using practical iterative decoding, is known to approach Shannon limits in the asymptotic limit of large blocklengths. A substantial limitation to the use of finite-length LDPC codes is the presence of an error floor in the low frame error rate (FER) region. This paper develops a method, based on importance sampling and high SNR asymptotics as applied to suitably defined absorbing structures within the LDPC code, to predict error floors. Our results are in very close agreement with hardware-based experimental results, and moreover extend the prediction of the error probability to even lower regions. We compute both importance sampling estimates of error probabilities and deterministic estimates that are guaranteed to lower bound the error probability in the high SNR regime. I.
We present an importance sampling method for the evaluation of the low frame error rate (FER) performance of LDPC codes under iterative decoding. It relies on a combinatorial characterization of absorbing sets, which are the dominant cause of decoder failure in the low FER region. The biased density in the importance sampling scheme is a mean-shifted version of the original Gaussian density, which is suitably centered between a codeword and a dominant absorbing set. This choice of biased density yields an unbiased estimator for the FER with a variance lower by several orders of magnitude than the standard Monte Carlo estimator. Using this importance sampling scheme in software, we obtain good agreement with the experimental results obtained from a fast hardware emulator of the decoder.
Ever since the publication of Shannon's seminal work in 1948, the search for capacity achieving codes has led to many interesting discoveries in channel coding theory. Low-density parity-check (LDPC) codes originally proposed in 1963 were largely forgotten and rediscovered recently. The significance of LDPC codes lies in their capacity approaching performance even when decoded using low complexity sub-optimal decoding algorithms. Iterative decoders are one such class of decoders that work on a graphical representation of a code known as the Tanner graph. Their properties have been well understood in the asymptotic limit of the code length going to infinity. However, the behavior of various decoders for a given finite length code remains largely unknown.An understanding of the failures of the decoders is vital for the error floor analysis of a given code. Broadly speaking, error floor is the abrupt degradation in the frame error rate (FER) performance of a code in the high signal-to-noise ratio domain. Since the error floor phenomenon manifests in the regions not reachable by Monte-Carlo simulations, analytical methods are necessary for characterizing the decoding failures. In this work, we consider hard decision decoders for transmission over the binary symmetric channel (BSC).For column-weight-three codes, we provide tight upper and lower bounds on the guaranteed error correction capability of a code under the Gallager A algorithm by studying combinatorial objects known as trapping sets. For higher column weight codes, we establish bounds on the minimum number of variable nodes that achieve certain expansion as a function of the girth of the underlying Tanner graph, thereby obtaining lower bounds on the guaranteed error correction capability. We explore the relationship between a class of graphs known as cage graphs and trapping sets to establish upper bounds on the error correction capability.We also propose an algorithm to identify the most probable noise configurations, also known as instantons, that lead to error floor for linear programming (LP) decoding over the BSC. With the insight gained from the above analysis techniques, we propose novel code construction techniques that result in codes with superior error floor performance.
The class of low-density parity-check (LDPC) codes is attractive, since such codes can be decoded using practical message-passing algorithms, and their performance is known to approach the Shannon limits for suitably large block lengths. For the intermediate block lengths relevant in applications, however, many LDPC codes exhibit a so-called “error floor,” corresponding to a significant flattening in the curve that relates signal-to-noise ratio (SNR) to the bit-error rate (BER) level. Previous work has linked this behavior to combinatorial substructures within the Tanner graph associated with an LDPC code, known as (fully) absorbing sets. These fully absorbing sets correspond to a particular type of near-codewords or trapping sets that are stable under bit-flipping operations, and exert the dominant effect on the low BER behavior of structured LDPC codes. This paper provides a detailed theoretical analysis of these (fully) absorbing sets for the class ofCp,γC_{p, \gamma}Cp,γ​array-based LDPC codes, including the characterization of all minimal (fully) absorbing sets for the array-based LDPC codes forγ=2,3,4\gamma = 2,3,4γ=2,3,4, and moreover, it provides the development of techniques to enumerate them exactly. Theoretical results of this type provide a foundation for predicting and extrapolating the error floor behavior of LDPC codes.
Error correcting codes are used in virtually all communication systems to ensure reliable transmission of information. In 1948, Shannon established an upper-bound on the maximum rate at which information can be transmitted reliably over a noisy channel. Reliably transmitting information with a rate close to this theoretical limit, known as the channel capacity, has been the goal of channel coding scientists for the last five decades. The rediscovery of low-density parity-check (LDPC) codes in the 1990s added much-renewed excitement in the coding community. LDPC codes are interesting because they can approach channel capacity under sub-optimum decoding algorithms whose complexity is linear in the code length. Unsurprisingly, LDPC codes quickly attained their popularity in practical applications such as magnetic storage, wireless and optical communications. One, if not the most, important and challenging problem in LDPC code research is the study and analysis of the error floor phenomenon. This phenomenon is described as an abrupt degradation in the frame error rate performance of LDPC codes in the high signal-to-noise ratio region. Error floor is harmful because its presence prevents the LDPC decoder from reaching very low probability of decoding failure, an important requirement for many applications. Not long after the rediscovery of LDPC codes, scientists established that error floor is caused by certain harmful structures, most commonly known as trapping sets, in the Tanner representation of a code. Since then, the study of error floor mostly consists of three major problems: 1) estimating error floor; 2) constructing LDPC codes with low error floor and 3) designing decoders that are less susceptible to error floor. Although some parts of this dissertation can be used as important elements in error floor estimation, our main contributions are a novel method for constructing LDPC codes with low error floor and a novel class of low complexity decoding algorithms that can collectively alleviate error floor. These contributions are summarized as follows. A method to construct LDPC codes with low error floors on the binary symmetric channel is presented. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the Trapping Set Ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum-product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets. We apply this method to construct structured LDPC codes. To facilitate the discussion, we give a new description of structured LDPC codes whose parity-check matrices are arrays of permutation matrices. This description uses Latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the Tanner graph of a code to be free of four-cycles. A new class of bit flipping algorithms for LDPC codes over the binary symmetric channel is proposed. Compared to the regular (parallel or serial) bit flipping algorithms, the proposed algorithms employ one additional bit at a variable node to represent its "strength." The introduction of this additional bit allows an increase in the guaranteed error correction capability. An additional bit is also employed at a check node to capture information which is beneficial to decoding. A framework for failure analysis and selection of two-bit bit flipping algorithms is provided. The main component of this framework is the (re)definition of trapping sets, which are the most "compact" Tanner graphs that cause decoding failures of an algorithm. A recursive procedure to enumerate trapping sets is described. This procedure is the basis for selecting a collection of algorithms that work well together. It is demonstrated that decoders which employ a properly selected group of the proposed algorithms operating in parallel can offer high speed and low error floor decoding.
The performance of low-density parity-check (LDPC) codes at high signal-to-noise ratios (SNRs) is known to be limited by the presence of certain sub-graphs that exist in the Tanner graph representation of the code, for example trapping sets and absorbing sets. This paper derives a lower bound on the frame error rate (FER) of any LDPC code containing a given problematic sub-graph, assuming a particular message passing decoder and decoder quantization. A crucial aspect of the lower bound is that it is code-independent, in the sense that it can be derived based only on a problematic sub-graph and then applied to any code containing it. Due to the complexity of evaluating the exact bound, assumptions are proposed to approximate it, from which we can estimate decoder performance. Simulated results obtained for both the quantized sum-product algorithm (SPA) and the quantized min-sum algorithm (MSA) are shown to be consistent with the approximate bound and the corresponding performance estimates. Different classes of LDPC codes, including both structured and randomly constructed codes, are used to demonstrate the robustness of the approach.