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Efficient Evaluation of Asymptotic Trapping Set Enumerators for Irregular LDPC Code Ensembles

Emna Ben Yacoub, Gianluigi Liva, Gerhard G. Kramer

Published Feb 26, 2020• 5 citations
Publisher

Abstract

The normalized logarithmic asymptotic distribution of elementary and general trapping sets for irregular low-density parity-check code ensembles is derived based on the generating functions approach. A numerical technique for its evaluation is presented, which requires solving a system of equations.
DOI
10.3929/ethz-b-000402705

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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.
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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.
Trapping sets represent subgraphs in the Tanner graph of a code that, for certain classes of channels, exhibit a strong influence on the height and point of onset of the error-floor. We compute the asymptotic normalized distributions of trapping sets in random, irregular, binary low-density parity-check (LDPC) code ensembles. Our derivations rely on techniques from large deviation theory and statistical methods for enumeracting random-like matrices. Similar methods can be used for computing the spectra of other combinatorial entities in LDPC code, such as subcodes and/or minimal codewords.
We evaluate the asymptotic normalized average distributions of a class of combinatorial configurations in random, regular and irregular, binary low-density parity-check (LDPC) code ensembles. Among the configurations considered are trapping and stopping sets. These sets represent subsets of variable nodes in the Tanner graph of a code that play an important role in determining the height and point of onset of the error-floor in its performance curve. The techniques used for deriving the spectra include large deviations theory and statistical methods for enumerating binary matrices with prescribed row and column sums. These techniques can also be applied in a setting that involves more general structural entities such as subcodes and/or minimal codewords, that are known to characterize other important properties of soft-decision decoders of linear block codes