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 Transactions on Information Theory

Finite-Dimensional Bounds on ${\BBZ}_m$ and Binary LDPC Codes With Belief Propagation Decoders

Chih-Chun Wang, Sanjeev R. Kulkarni, H. Vincent Poor

Published Jan 1, 2007• 30 citations
Publisher

Abstract

This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zopfmand binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For ZopfmLDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for ZopfmLDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with nonequiprobably distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional noniterative bound, the latter of which is the best known bound that is tight for binary-symmetric channels (BSCs), and is a strict improvement over the existing bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for nonsymmetric channels, which coincides with the existing bound for symmetric channels

DOI
10.1109/tit.2006.887085

Related papers

Finite-Dimensional Bounds on Zm and Binary LDPC Codes with Belief Propagation Decoders

Chih-Chun Wang, Sanjeev R. Kulkarni, H. Vincent Poor

This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zm and binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For Zm LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for Zm LDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with non-equiprobable distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional non-iterative bound, the latter of which is the best known bound that is tight for binary symmetric channels (BSCs), and is a strict improvement over the bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for non-symmetric channels, which coincides with the existing bound for symmetric channels.

arXiv:cs/0610052•Oct 10, 2006•4 citations

Bounds on the Belief Propagation Threshold of Non-Binary LDPC Codes

Leonid Geller, David Burshtein

We consider low-density parity-check (LDPC) code ensembles over non-binary Galois fields when used for transmission over arbitrary discrete memoryless channels. Belief propagation decoding for these codes has been shown to achieve excellent results. However, computing the decoding threshold using density evolution is usually impractical, since one needs to propagate multi-dimensional probability distributions, and Monte Carlo simulations are required instead. By considering the evolution of the message Bhattacharyya parameter and the message expected value parameter, we derive a simple lower bound on the performance of the algorithm. This bound applies for both regular and irregular non-binary LDPC ensembles.

IEEE Transactions on Information Theory•Mar 9, 2016•7 citations

Bounds on the performance of belief propagation decoding

David Burshtein, G. Miller

We consider Gallager's (1963) soft-decoding (belief propagation) algorithm for decoding low-density parity-check (LDPC) codes, when applied to an arbitrary binary-input symmetric-output channel. By considering the expected values of the messages, we derive both lower and upper bounds on the performance of the algorithm. We also derive various properties of the decoding algorithm, such as a certain robustness to the details of the channel noise. Our results apply both to regular and irregular LDPC codes.

IEEE Transactions on Information Theory•Jan 1, 2002•42 citations

Bounds on the belief propagation threshold of non-binary LDPC codes

Leonid Geller, David Burshtein

We consider LDPC code ensembles over non-binary Galois fields, when used for transmission over arbitrary memoryless channels. Belief propagation decoding for these codes has been shown to achieve excellent results. However, computing the decoding threshold using density evolution is usually impractical, since one needs to propagate multi-dimensional probability distributions, and Monte Carlo simulations are required instead. By considering the evolution of the message Bhattacharyya parameter and the message expected value parameter we derive a simple lower bound on the performance of the algorithm. This bound applies for both regular and irregular non-binary LDPC ensembles.

IEEE Information Theory Workshop (ITW)•Sep 1, 2012•1 citation

Iterative Decoding Performance Bounds for LDPC Codes on Noisy Channels

Chun-Hao Hsu, Achilleas Anastasopoulos

The asymptotic iterative decoding performances of low-density parity-check (LDPC) codes using min-sum (MS) and sum-product (SP) decoding algorithms on memoryless binary-input output-symmetric (MBIOS) channels are analyzed in this paper. For MS decoding, the analysis is done by upper bounding the bit error probability of the root bit of a tree code by the sequence error probability of a subcode of the tree code assuming the transmission of the all-zero codeword. The result is a recursive upper bound on the bit error probability after each iteration. For SP decoding, we derive a recursively determined lower bound on the bit error probability after each iteration. This recursive lower bound recovers the density evolution equation of LDPC codes on the binary erasure channel (BEC) with inequalities satisfied with equalities. A significant implication of this result is that the performance of LDPC codes under SP decoding on the BEC is an upper bound of the performance on all MBIOS channels with the same uncoded bit error probability. All results hold for the more general multi-edge type LDPC codes.

arXiv:cs/0607020•Jul 6, 2006•1 citation

Bounds on the performance of belief propagation decoding

D. Burshtein, G. Miller

We consider Gallager's (1963) soft decoding (belief propagation) algorithm for decoding low density parity check (LDPC) codes, when applied to an arbitrary binary-input symmetric-output channel. By considering the expected values of the messages, we derive both lower and upper bounds on the performance of the algorithm. We also derive various properties of the decoding algorithm, such as robustness to the initial values of the messages. Our results apply both to regular and irregular LDPC codes.

IEEE International Symposium on Information Theory (ISIT)•Nov 13, 2002•2 citations
This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zopfmand binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For ZopfmLDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for ZopfmLDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with nonequiprobably distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional noniterative bound, the latter of which is the best known bound that is tight for binary-symmetric channels (BSCs), and is a strict improvement over the existing bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for nonsymmetric channels, which coincides with the existing bound for symmetric channels
This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zm and binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For Zm LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for Zm LDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with non-equiprobable distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional non-iterative bound, the latter of which is the best known bound that is tight for binary symmetric channels (BSCs), and is a strict improvement over the bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for non-symmetric channels, which coincides with the existing bound for symmetric channels.
We consider low-density parity-check (LDPC) code ensembles over non-binary Galois fields when used for transmission over arbitrary discrete memoryless channels. Belief propagation decoding for these codes has been shown to achieve excellent results. However, computing the decoding threshold using density evolution is usually impractical, since one needs to propagate multi-dimensional probability distributions, and Monte Carlo simulations are required instead. By considering the evolution of the message Bhattacharyya parameter and the message expected value parameter, we derive a simple lower bound on the performance of the algorithm. This bound applies for both regular and irregular non-binary LDPC ensembles.
We consider Gallager's (1963) soft-decoding (belief propagation) algorithm for decoding low-density parity-check (LDPC) codes, when applied to an arbitrary binary-input symmetric-output channel. By considering the expected values of the messages, we derive both lower and upper bounds on the performance of the algorithm. We also derive various properties of the decoding algorithm, such as a certain robustness to the details of the channel noise. Our results apply both to regular and irregular LDPC codes.
We consider LDPC code ensembles over non-binary Galois fields, when used for transmission over arbitrary memoryless channels. Belief propagation decoding for these codes has been shown to achieve excellent results. However, computing the decoding threshold using density evolution is usually impractical, since one needs to propagate multi-dimensional probability distributions, and Monte Carlo simulations are required instead. By considering the evolution of the message Bhattacharyya parameter and the message expected value parameter we derive a simple lower bound on the performance of the algorithm. This bound applies for both regular and irregular non-binary LDPC ensembles.
The asymptotic iterative decoding performances of low-density parity-check (LDPC) codes using min-sum (MS) and sum-product (SP) decoding algorithms on memoryless binary-input output-symmetric (MBIOS) channels are analyzed in this paper. For MS decoding, the analysis is done by upper bounding the bit error probability of the root bit of a tree code by the sequence error probability of a subcode of the tree code assuming the transmission of the all-zero codeword. The result is a recursive upper bound on the bit error probability after each iteration. For SP decoding, we derive a recursively determined lower bound on the bit error probability after each iteration. This recursive lower bound recovers the density evolution equation of LDPC codes on the binary erasure channel (BEC) with inequalities satisfied with equalities. A significant implication of this result is that the performance of LDPC codes under SP decoding on the BEC is an upper bound of the performance on all MBIOS channels with the same uncoded bit error probability. All results hold for the more general multi-edge type LDPC codes.
We consider Gallager's (1963) soft decoding (belief propagation) algorithm for decoding low density parity check (LDPC) codes, when applied to an arbitrary binary-input symmetric-output channel. By considering the expected values of the messages, we derive both lower and upper bounds on the performance of the algorithm. We also derive various properties of the decoding algorithm, such as robustness to the initial values of the messages. Our results apply both to regular and irregular LDPC codes.