Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Symposium on Applied Sciences in Biomedical and Communication Technologies

On the weight and stopping set size distributions of LDPC codes and their generalizations

Mark F. Flanagan, Enrico Paolini

Published Nov 1, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.1109/isabel.2009.5373632

Related papers

On the Growth Rate of the Weight Distribution of Irregular Doubly Generalized LDPC Codes

Mark F. Flanagan, Enrico Paolini, Marco Chiani +1 more

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.

IEEE Transactions on Information Theory•May 25, 2011•19 citations

On the Growth Rate of the Weight Distribution of Irregular Doubly-Generalized LDPC Codes

Mark F. Flanagan, Enrico Paolini, Marco Chiani +1 more

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.

arXiv:0903.1588•Mar 9, 2009•0 citations

Weight Distributions of Non-Binary Multi-Edge Type LDPC Code Ensembles: Analysis and Efficient Evaluation

Giuliano Garrammone, David Declercq, M.P.C. Fossorier

Non-binary multi-edge type ensembles of low-density parity-check codes are analyzed in terms of non-binary codeword weight distribution and its growth rate. In particular, an exact expression of the growth rate for small weights is developed. As a side result, the stopping set distributions of these ensembles are developed. Examples of weight distributions are provided, showing that the derived closed-form expressions can be easily evaluated. The obtained results can thus be exploited to analyze and design non-binary low-density parity-check codes that fall within the multi-edge type framework such as, but not limited to, protograph-based codes.

IEEE Transactions on Information Theory•Jan 4, 2017•7 citations

Weight Distributions of Regular Low-Density Parity-Check Codes Over Finite Fields

Shengtian Yang, Thomas Honold, Yan Chen +2 more

The average weight distribution of a regular low-density parity-check (LDPC) code ensemble over a finite field is thoroughly analyzed. In particular, a precise asymptotic approximation of the average weight distribution is derived for the small-weight case, and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved. Based on this analysis, a general result, including all previous results as special cases, is established for the minimum distance of individual codes in a regular LDPC code ensemble.

arXiv:1010.2030•IEEE Transactions on Information Theory•Aug 3, 2011•10 citations

Stopping-Set Enumerator Approximations for Finite-Length Protograph LDPC Codes

Kaiann Fu, Achilleas Anastasopoulos

Asymptotic analysis of low-density parity-check (LDPC) code weight and stopping-set enumerators, for codewords and stopping sets which grow linearly with codelength, has aided in designing codes with linear minimum distance and low error floors. However, the analysis cannot capture the behavior of codewords and stopping sets that grow sublinearly with codelength. Thus, it is unclear how well the analysis describes behavior for finite codelengths, particularly for short codes. In this paper, we provide another perspective on protograph-based and standard LDPC ensemble enumerators, based on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, protograph structure, and preceding. Using approximations to the stopping-set enumerators, we show that for stopping sets that grow at most logarithmically with codelength, the enumerators follow a polynomial relationship with codelength, unlike the exponential relationship for linearly- growing stopping sets. Further, we analyze for what stopping-set sizes and codelengths the approximations apply.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2007•11 citations

Weight distribution of non-binary LDPC codes

Kenta Kasai, Charly Poulliat, David Declercq +2 more

Weight distributions of binary low-density parity-check (LDPC) codes are well studied. We investigate the average distributions of symbol and binary weight for non-binary LDPC code ensemble. We derive the asymptotic growth rate and its linear approximation for small weight. Moreover, we show the typical minimum distance does not monotonically increase with the size of the field where the codes are defined.

International Symposium on Information Theory and its Applications•Dec 1, 2008•17 citations
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.
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.
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.
Non-binary multi-edge type ensembles of low-density parity-check codes are analyzed in terms of non-binary codeword weight distribution and its growth rate. In particular, an exact expression of the growth rate for small weights is developed. As a side result, the stopping set distributions of these ensembles are developed. Examples of weight distributions are provided, showing that the derived closed-form expressions can be easily evaluated. The obtained results can thus be exploited to analyze and design non-binary low-density parity-check codes that fall within the multi-edge type framework such as, but not limited to, protograph-based codes.
The average weight distribution of a regular low-density parity-check (LDPC) code ensemble over a finite field is thoroughly analyzed. In particular, a precise asymptotic approximation of the average weight distribution is derived for the small-weight case, and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved. Based on this analysis, a general result, including all previous results as special cases, is established for the minimum distance of individual codes in a regular LDPC code ensemble.
Asymptotic analysis of low-density parity-check (LDPC) code weight and stopping-set enumerators, for codewords and stopping sets which grow linearly with codelength, has aided in designing codes with linear minimum distance and low error floors. However, the analysis cannot capture the behavior of codewords and stopping sets that grow sublinearly with codelength. Thus, it is unclear how well the analysis describes behavior for finite codelengths, particularly for short codes. In this paper, we provide another perspective on protograph-based and standard LDPC ensemble enumerators, based on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, protograph structure, and preceding. Using approximations to the stopping-set enumerators, we show that for stopping sets that grow at most logarithmically with codelength, the enumerators follow a polynomial relationship with codelength, unlike the exponential relationship for linearly- growing stopping sets. Further, we analyze for what stopping-set sizes and codelengths the approximations apply.