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IEEE Transactions on Communications

Asymptotic Distribution of Absorbing Sets and Fully Absorbing Sets for Regular Sparse Code Ensembles

Behzad Amiri, Chi‐Wei Lin, Lara Dolecek

Published Feb 1, 2013• 21 citations
Publisher

Abstract

It is well recognized that low-density parity-check (LDPC) codes can suffer from an error floor when decoded iteratively. This performance degradation is often attributed to the class of objects known as trapping sets. Past work has focused on characterizing the distribution of trapping sets for a variety of code ensembles, including regular, irregular and structured LDPC codes. As a subset of the trapping set collection, there exists a class of graphical structures called the absorbing sets. An absorbing set is a combinatorially-defined object; in particular a fully absorbing set is stable under bit-flipping decoding. By construction, there can exist trapping sets that are not stable under such a decoder. As a result, for finite-precision, iterative decoding algorithms used over additive channels, absorbing sets can describe decoding errors more accurately than the broader class of trapping sets. In this paper, we compute the normalized logarithmic asymptotic distributions of absorbing sets and fully absorbing sets, including elementary (fully) absorbing sets. The calculations are based on the trapping set enumeration method proposed by Milenkovic, Soljanin, and Whiting in [1]. We compare distributions of absorbing and trapping sets for representative code parameters of interest, and quantify the (lack of) discrepancies between the two. Good absorbing set properties are implied for known structured LDPC codes, including repeat accumulate codes and protograph-based constructions. Establishing the distribution of fully absorbing sets (especially when the discrepancy with the trapping set distribution is significant) allows one to further refine the estimates of the error rates under bit-flipping and related decoders.

DOI
10.1109/tcomm.2012.120512.110605

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It is well recognized that low-density parity-check (LDPC) codes can suffer from an error floor when decoded iteratively. This performance degradation is often attributed to the class of objects known as trapping sets. Past work has focused on characterizing the distribution of trapping sets for a variety of code ensembles, including regular, irregular and structured LDPC codes. As a subset of the trapping set collection, there exists a class of graphical structures called the absorbing sets. An absorbing set is a combinatorially-defined object; in particular a fully absorbing set is stable under bit-flipping decoding. By construction, there can exist trapping sets that are not stable under such a decoder. As a result, for finite-precision, iterative decoding algorithms used over additive channels, absorbing sets can describe decoding errors more accurately than the broader class of trapping sets. In this paper, we compute the normalized logarithmic asymptotic distributions of absorbing sets and fully absorbing sets, including elementary (fully) absorbing sets. The calculations are based on the trapping set enumeration method proposed by Milenkovic, Soljanin, and Whiting in [1]. We compare distributions of absorbing and trapping sets for representative code parameters of interest, and quantify the (lack of) discrepancies between the two. Good absorbing set properties are implied for known structured LDPC codes, including repeat accumulate codes and protograph-based constructions. Establishing the distribution of fully absorbing sets (especially when the discrepancy with the trapping set distribution is significant) allows one to further refine the estimates of the error rates under bit-flipping and related decoders.
It is well recognized that low-density parity-check (LDPC) codes can suffer from an error floor when decoded iteratively. This performance degradation is often attributed to the class of objects known as trapping sets. As a subset of the trapping set collection, there exists a class of graphical structures called the absorbing sets. An absorbing set is a combinatorially-defined object; in particular a fully absorbing set is stable under bit-flipping decoding. By construction, there can exist trapping sets that are not stable under such a decoder. As a result, for finite-precision, iterative decoding algorithms used over additive channels, absorbing sets can describe decoding errors more accurately than the broader class of trapping sets. In the first part of this thesis, we compute the normalized logarithmic asymptotic distributions of absorbing sets and fully absorbing sets, including elementary (fully) absorbing sets. We compare distributions of absorbing and trapping sets for representative code parameters of interest, and quantify the (lack of) discrepancies between the two. Good absorbing set properties are implied for known structured LDPC codes, including repeat accumulate codes and protograph-based constructions. Establishing the distribution of fully absorbing sets (especially when the discrepancy with the trapping set distribution is significant) allows one to further refine the estimates of the error rates under bit-flipping and related decoders.To reduce implementation complexity, the messages in a practical message passing decoder are necessarily quantized. Absorbing regions act as "decoding regions" around absorbing sets. In the second part of this thesis, we take a closer look at the interplay between quantization and absorbing regions. We provide a study of a range of quantization choices, the impact of quantization on the candidate absorbing regions, and derive guidelines for practical decoders. We show that, due to the non-linear dynamics of message passing decoders, coarser quantization may in fact perform better than finer quantization. Results of this type of work can be particularly useful in designing high performance decoders for very high-reliability storage systems, such as emerging data storage hard disk and solid state drives.
The definitions of (elementary) trapping and (fully) absorbing sets are extended to generalized low-density parity-check (GLDPC) codes. In particular, the fully absorbing sets are stable under the bit flipping algorithm. The finite-length (elementary) trapping and (fully) absorbing set enumerators for unstructured irregular GLDPC code ensembles are derived using generating functions. An efficient method to evaluate the asymptotic trapping and (fully) absorbing set distributions is presented, which requires solving a system of equations. Using this method, the asymptotic distribution of trapping and (fully) absorbing sets are evaluated for some example GLDPC code ensembles. Experimental results show that the proposed definitions yield graph structures that are harmful for bit flipping decoders. The generating function approach is also used to derive the finite-length and asymptotic trapping and (elementary) absorbing set enumerators for non-binary irregular LDPC code ensembles.
Past work focused on the asymptotic enumeration of trapping sets under random ensembles for specific codes. In this paper, we extend the asymptotic analysis of trapping sets to find the distribution of elementary absorbing sets and elementary fully absorbing sets, which are a special class of trapping sets. The asymptotic results for elementary (fully) absorbing sets are presented, and compared with the results of trapping sets. The distribution of fully absorbing sets allows one to determine the error rates for bit-flipping decoders more accurately.
The average finite-length (elementary) fully absorbing set enumerators for binary protograph-based low-density parity-check (LDPC) code ensembles are derived. The method relies on generating functions, and it provides an efficient mean to obtain the normalized logarithmic asymptotic distribution of (elementary) fully absorbing sets. Using these results, the asymptotic distribution of (elementary) fully absorbing sets is evaluated for some protograph-based LDPC code ensembles. Numerical results show how the distributions can be used to estimate the error floor performance of protograph LDPC codes.
Generalized low-density parity-check (GLDPC) codes have outstanding performance outperforming the standard low-density parity-check (LDPC) codes. However, the performance of these codes under iterative decoding can be degraded because of the presence of trapping, absorbing and fully absorbing sets in the code Tanner graphs. The (elementary) trapping and (fully) absorbing sets of protograph-based GLDPC codes are studied. The finite-length and asymptotic distributions of (elementary) trapping and (fully) absorbing sets for protograph-based GLDPC code ensembles are derived and evaluated. The asymptotic distributions can be efficiently computed by solving a system of equations. The relative minimum Δ-trapping/(fully) absorbing set size can be directly computed by adding one equation to the system of equations. This is useful for the code design where we can restrict the search to ensembles for which the minimum Δ-trapping and (fully) absorbing set sizes grow linearly with the blocklength. The normalized logarithmic asymptotic distributions of (elementary) trapping and (fully) absorbing sets for an example GLDPC code ensemble are evaluated.
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.