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International Symposium on Problems of Redundancy in Information and Control Systems (REDUNDANCY)

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

Emna Ben Yacoub

Published Oct 24, 2023• 1 citation
Publisher

Abstract

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.

DOI
10.1109/redundancy59964.2023.10330177

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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 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.
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.
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.
Among the recent advances in the area of low-density parity-check (LDPC) codes, protograph-based LDPC codes have the advantages of a simple design procedure and highly structured encoders and decoders. These advantages can also be exploited in the design of protograph-based generalized LDPC (G-LDPC) codes. In this dissertation we provide analytical tools which aid the design of protograph-based LDPC and G-LDPC codes. Specifically, we propose a method for computing the codeword-weight enumerators for finite-length protograph-based G-LDPC code ensembles, and then we consider the asymptotic case when the block-length goes to infinity. These results help the designer identify good ensembles of protograph-based G-LDPC codes in the minimum distance sense (i.e., ensembles which have minimum distances grow linearly with code length). Furthermore, good code ensembles can be characterized by good stopping set, trapping set, or pseudocodeword properties, which assist in the design of G-LDPC codes with low floors. We leverage our method for computing codeword-weight enumerators to compute stopping-set, and pseudocodeword enumerators for the finite-length and the asymptotic ensembles of protograph-based G-LDPC codes. Moreover, we introduce a method for computing trapping set enumerators for finite-length (and asymptotic) protograph-based LDPC code ensembles. Trapping set enumerators for G-LDPC codes represents a more complex problem which we do not consider here. Inspired by our method for computing trapping set enumerators for protograph-based LDPC code ensembles, we developed an algorithm for estimating the trapping set enumerators for a specific LDPC code given its parity-check matrix. We used this algorithm to enumerate trapping sets for several LDPC codes from communication standards. Finally, we study coded-modulation schemes with LDPC codes and pulse position modulation (LDPC-PPM) over the free-space optical channel. We present three different decoding schemes and compare their performances. In addition, we developed a new density evolution tool for use in the design of LDPC codes with good performances over this channel.