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IEEE International Symposium on Information Theory (ISIT)

Asymptotic Absorbing Set Enumerators for Non-Binary Protograph-Based LDPC Code Ensembles

Emna Ben Yacoub, Gianluigi Liva

Published Jun 1, 2020• 4 citations
Publisher

Abstract

The finite-length absorbing set enumerators for non-binary protograph based low-density parity-check (LDPC) ensembles are derived. An efficient method for the evaluation of the asymptotic absorbing set distributions is presented and evaluated.
DOI
10.1109/isit44484.2020.9174036

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The finite-length trapping and (elementary) absorbing set enumerators for nonbinary protograph-based LDPC code ensembles are derived. Both constrained and unconstrained edge labeling approaches are considered. The normalized logarithmic asymptotic distributions of trapping and (elementary) absorbing sets are obtained through an efficient method that requires solving a system of equations. Using these results, the asymptotic distributions of trapping and (elementary) absorbing sets are evaluated for some example nonbinary protograph-based LDPC code ensembles.
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.
Protograph-based LDPC codes have the advantages of a simple design (or search) procedure and highly structured encoders and decoders. These advantages have also been exploited in the design of protograph-based generalized LDPC (G-LDPC) codes. Recently, a technique for computing ensemble weight enumerators and stopping set enumerators for protograph-based LDPC codes has been published. In the current paper, we extend those results to protograph-based G-LDPC codes. That is, we first derive ensemble weight and stopping set enumerators for finite-length G-LDPC codes based on protographs, and then we consider the asymptotic case. In the weight enumerator case, the asymptotic results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length. In the stopping set enumerator case, the asymptotic results allows us to determine whether or not the typical smallest stopping set size grows linearly with codeword length.
This paper provides a comprehensive analysis of nonbinary low-density parity check (LDPC) codes built out of protographs. We consider both random and constrained edge-weight labeling, and refer to the former as the unconstrained nonbinary protograph-based LDPC codes (U-NBPB codes) and to the latter as the constrained nonbinary protograph-based LDPC codes (C-NBPB codes). Equipped with combinatorial definitions extended to the nonbinary domain, ensemble enumerators of codewords, trapping sets, stopping sets, and pseudocodewords are calculated. The exact enumerators are presented in the finite-length regime, and the corresponding growth rates are calculated in the asymptotic regime. An EXIT chart tool for computing the iterative decoding thresholds of protograph-based LDPC codes is presented, followed by several examples of finite-length U-NBPB and C-NBPB codes with high performance. Throughout this paper, we provide accompanying examples, which demonstrate the advantage of nonbinary protograph-based LDPC codes over their binary counterparts and over random constructions. The results presented in this paper advance the analytical toolbox of nonbinary graph-based codes.