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IEEE Journal on Selected Areas in Communications

On asymptotic ensemble weight enumerators of LDPC-like codes

Chung-Li Wang, M.P.C. Fossorier

Published Jul 29, 2009• 6 citations
Publisher

Abstract

For LDPC-like codes such as LDPC, GLDPC, and DGLDPC codes, it is well known that the error floor can be caused by the codewords of small weights or stopping sets of small sizes. In this paper, we investigate the computation of asymptotic weight enumerators such that it becomes a convenient tool to determine a good distribution of code ensembles. In addition, by analyzing the first order approximation, we derive a condition to obtain a negative asymptotic growth rate of the codewords of small linear-sized weights, which is an important constraint for distribution optimization. Also the weight enumerators of turbo and repeat-accumulate codes are investigated. Furthermore, we extend our results to nonbinary DGLDPC codes. Generalization to N-layer and convolutional code based LDPC-like codes are also developed.

DOI
10.1109/jsac.2009.090808

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For LDPC-like codes such as LDPC, GLDPC, and DGLDPC codes, it is well known that the error floor can be caused by the codewords of small weights or stopping sets of small sizes. In this paper, we investigate the computation of asymptotic weight enumerators such that it becomes a convenient tool to determine a good distribution of code ensembles. In addition, by analyzing the first order approximation, we derive a condition to obtain a negative asymptotic growth rate of the codewords of small linear-sized weights, which is an important constraint for distribution optimization. Also the weight enumerators of turbo and repeat-accumulate codes are investigated. Furthermore, we extend our results to nonbinary DGLDPC codes. Generalization to N-layer and convolutional code based LDPC-like codes are also developed.
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.
In this paper, the asymptotic growth rate of the weight distribution of irregular doubly generalized LDPC (D-GLDPC) codes is derived. The analysis yields a compact expression which accurately approximates the growth rate function for the case of small linear-weight codewords. This paper generalizes existing results for LDPC and generalized LDPC (GLDPC) codes. Ensembles with smallest check or variable node minimum distance greater than 2 are shown to have good growth-rate behavior, while for other ensembles a fundamental parameter is identified which discriminates between an asymptotically small and an asymptotically large expected number of small linear-weight codewords. Also, in the latter case it is shown that the growth rate depends only on the check and variable nodes with minimum distance 2. An important connection between this new result and the stability condition of D-GLDPC codes over the BEC is highlighted. Such a connection, previously observed for LDPC and GLDPC codes, is now extended to the case of D-GLDPC codes. Finally, it is shown that the analysis may be extended to include the growth rate of the stopping set size distribution of irregular D-GLDPC codes.
Protograph-based doubly generalized LDPC (DGLDPC) codes are explored in this paper. We extend (and in the process simplify) the technique for computing the ensemble weight enumerators of protograph-based LDPC and GLDPC codes to DGLDPC codes. We find that with careful design, protograph-based DGLDPC codes can have a better asymptotic growth rate of minimum distance than that of the protograph-based LDPC and GLDPC codes. Simulation results confirm that protograph-based DGLDPC codes have a low error floor.
In this paper, an expression for the asymptotic growth rate of the number of small linear-weight codewords of irregular doubly-generalized LDPC (D-GLDPC) codes is derived. The expression is compact and generalizes existing results for LDPC and generalized LDPC (GLDPC) codes. Ensembles with check or variable node minimum distance greater than 2 are shown to be have good growth rate behavior, while for other ensembles a fundamental parameter is identified which discriminates between an asymptotically small and an asymptotically large expected number of small linear-weight codewords. Also, in the latter case it is shown that the growth rate depends only on the check and variable nodes with minimum distance 2. An important connection between this new result and the stability condition of D-GLDPC codes over the BEC is highlighted. Such a connection, previously observed for LDPC and GLDPC codes, is now extended to the case of D-GLDPC codes. Finally, it is shown that the analysis may be extended to include the growth rate of the stopping set size distribution of irregular D-GLDPC codes.
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This paper overviews and connects together recent results by the authors regarding the weight and stopping set size distributions of generalizations of low-density parity-check (LDPC) codes. We begin by reviewing a result which allows for evaluation of the growth rate of the weight distribution for codewords of any weight (linear in the codeword length). We then show how asymptotic analysis of this general result can lead to a result which encapsulates the behaviour of the growth rate of the weight distribution for the case of small linear-weight codewords. We then show that many existing results in the literature along this line may be viewed as special cases of this general result.