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

IEEE International Symposium on Information Theory (ISIT)

Weight distributions of multi-edge type LDPC codes

Kenta Kasai, Tomoharu Awano, David Declercq, Charly Poulliat, Kohichi Sakaniwa

Published Jun 1, 2009• 10 citations
Publisher

Abstract

For a (lambda(x); rho(x)) standard irregular LDPC code ensemble, the growth rate of the average weight distribution for small relative weight omega is given by log(lambda'(0)rho'(1))omega + O(omega2) in the limit of code length n. If lambda'(0)rho'(1) < 1, there exist exponentially few code words of small linear weight, as n tends to infinity. It is known that the condition coincides with the stability condition of density evolution over the erasure channels with the erasure probability 1. In this paper, we show that this is also the case with multi-edge type LDPC (MET-LDPC) codes. MET-LDPC codes are generalized structured LDPC codes introduced by Richardson and Urbanke. The parameter corresponding lambda'(0)rho'(1) appearing in the conditions for MET-LDPC codes is given by the spectral radius of the matrix defined by extended degree distributions.

DOI
10.1109/isit.2009.5205696

Related papers

Weight Distributions of Multi-Edge Type LDPC Codes

Kenta Kasai, Tomoharu Awano, David Declercq +2 more

The multi-edge type LDPC codes, introduced by Richardson and Urbanke, present the general class of structured LDPC codes. In this paper, we derive the average weight distributions of the multi-edge type LDPC code ensembles. Furthermore, we investigate the asymptotic exponential growth rate of the average weight distributions and investigate the connection to the stability condition of the density evolution.

arXiv:1009.1137•IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Jan 1, 2010•13 citations

Analysis of weight distributions of two-edge type LDPC codes by Hayman approximation

Tomoharu Awano, Kenta Kasai, Tomoharu Shibuya +1 more

Multi-edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles have better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with small linear weight codewords exponentially decreasing in code length. Nakasendo et al. derived the condition in which code ensembles have exponentially decreasing small linear weight codewords for two-edge type LDPC code ensembles which is simple example of multi-edge type LDPC code ensembles. Our conclusion is the same as that of Nakasendo et al., and although Nakasendopsilas method is difficult to apply to more than three edge-types, there is a possibility that our method using Hayman approximation as Di et al. do derives the condition for multi-edge type LDPC codes ensembles.

International Symposium on Turbo Codes and Iterative Information Processing•Sep 1, 2008•2 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, 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

Two-edge type LDPC code ensembles with exponentially few codewords with linear small weight

Tsuyoshi Nakasendo, Kenta Kasai, Tomoharu Shibuya +1 more

Multi-Edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles has better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with minimum distance linearly increasing in code length. We derive the condition corresponding to Orlitsky’s condition for two-edge type LDPC code ensembles which is simple example of Multi-Edge type LDPC code ensembles.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2008•2 citations

Average Coset Weight Distribution of Multi-Edge Type LDPC Code Ensembles

K. KASAI, Yuji Shimoyama, T. SHIBUYA +1 more

Multi-Edge type Low-Density Parity-Check codes (MET-LDPC codes) introduced by Richardson and Urbanke are generalized LDPC codes which can be seen as LDPC codes obtained by concatenating several standard (ir)regular LDPC codes. We prove in this paper that MET-LDPC code ensembles possess a certain symmetry with respect to their Average Coset Weight Distributions (ACWD). Using this symmetry, we drive ACWD of MET-LDPC code ensembles from ACWD of their constituent ensembles.

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Oct 1, 2006•8 citations

Three-edge type LDPC code ensembles with exponentially few codewords with linear small weight

Tomoharu Awano, Kenta Kasai, Tomoharu Shibuya +1 more

Multi-edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles have better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with minimum distance linearly increasing in code length. Nakasendo et al. derived the condition that code ensembles have exponentially decreasing small linear weight codewords for two-edge type LDPC code ensembles which is simple example of multi-edge type LDPC code ensembles. In this paper, we derive the condition for three-edge type LDPC code ensembles whose edge-types does not share any variable node and does share all of the check nodes with exponentially decreasing small linear weight codewords. The condition is necessary for the existence of the average relative minimum distance of ensembles. Our method is expected to derive the condition for multi-edge type LDPC codes.

International Symposium on Information Theory and its Applications•Dec 1, 2008•6 citations
For a (lambda(x); rho(x)) standard irregular LDPC code ensemble, the growth rate of the average weight distribution for small relative weight omega is given by log(lambda'(0)rho'(1))omega + O(omega2) in the limit of code length n. If lambda'(0)rho'(1) < 1, there exist exponentially few code words of small linear weight, as n tends to infinity. It is known that the condition coincides with the stability condition of density evolution over the erasure channels with the erasure probability 1. In this paper, we show that this is also the case with multi-edge type LDPC (MET-LDPC) codes. MET-LDPC codes are generalized structured LDPC codes introduced by Richardson and Urbanke. The parameter corresponding lambda'(0)rho'(1) appearing in the conditions for MET-LDPC codes is given by the spectral radius of the matrix defined by extended degree distributions.
Multi-edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles have better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with small linear weight codewords exponentially decreasing in code length. Nakasendo et al. derived the condition in which code ensembles have exponentially decreasing small linear weight codewords for two-edge type LDPC code ensembles which is simple example of multi-edge type LDPC code ensembles. Our conclusion is the same as that of Nakasendo et al., and although Nakasendopsilas method is difficult to apply to more than three edge-types, there is a possibility that our method using Hayman approximation as Di et al. do derives the condition for multi-edge type LDPC codes ensembles.
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.
Multi-edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles have better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with minimum distance linearly increasing in code length. Nakasendo et al. derived the condition that code ensembles have exponentially decreasing small linear weight codewords for two-edge type LDPC code ensembles which is simple example of multi-edge type LDPC code ensembles. In this paper, we derive the condition for three-edge type LDPC code ensembles whose edge-types does not share any variable node and does share all of the check nodes with exponentially decreasing small linear weight codewords. The condition is necessary for the existence of the average relative minimum distance of ensembles. Our method is expected to derive the condition for multi-edge type LDPC codes.