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IEEE Transactions on Communications

Trapping and Absorbing Set Enumerators for Nonbinary Protograph-Based Low-Density Parity-Check Code Ensembles

Emna Ben Yacoub, Gianluigi Liva

Published Jan 31, 2023• 5 citations
Publisher

Abstract

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.

DOI
10.1109/tcomm.2023.3241149

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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.
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.