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)

The Stability Condition of LDPC Codes Under MAP Decoding

Wei Liu, Rudiger L. Urbanke

Published Jun 1, 2018• 0 citations
Publisher

Abstract

We determine the stability condition of low-density parity-check codes under both bitwise and blockwise maximum a posteriori decoding. As a consequence, we prove that the stability condition determines an upper bound on both the bitwise and the blockwise maximum a posteriori threshold.
DOI
10.1109/isit.2018.8437939

Related papers

The Stability of Low-Density Parity-Check Codes and Some of its Consequences

Wei Liu, Rudiger L. Urbanke

We study the stability of low-density parity-check codes under blockwise or bitwise maximuma posterioridecoding, where transmission takes place over a binary-input memoryless output-symmetric channel. Our study stems from the consideration of constructing universal capacity-achieving codes under low-complexity decoding algorithms, where universality refers to the fact that we are considering a family of channels with equal capacity. Consider, e.g., the right-regular sequence by Shokrollahi and the heavy-tail Poisson sequence by Lubyet al. Both sequences are provably capacity-achieving under belief propagation decoding when transmission takes place over the binary erasure channel. In this paper we show that many existing capacity-achieving sequences of low-density parity-check codes are not universal under belief propagation decoding. We reveal that the key to showing this non-universality result is determined by the stability of the underlying codes. More concretely, for an ordered and complete channel family and a sequence of low-density parity-check code ensembles, we determine a stability threshold associated with them, which gives rise to a sufficient condition under which the sequence is not universal under belief propagation decoding. Moreover, we show that the same stability threshold applies to blockwise or bitwise maximuma posterioridecoding as well. We demonstrate how the stability threshold can determine an upper bound on the corresponding blockwise or bitwise maximuma posteriorithreshold, revealing the operational significance of the stability threshold.

arXiv:2009.08640•IEEE Transactions on Information Theory•Oct 11, 2021•4 citations

Doubly-Generalized LDPC Codes: Stability Bound Over the BEC

Enrico Paolini, M.P.C. Fossorier, Marco Chiani

The iterative decoding threshold of low-density parity-check (LDPC) codes over the binary erasure channel (BEC) fulfills an upper bound depending only on the variable and check nodes with minimum distance 2. This bound is a consequence of the stability condition, and is here referred to as stability bound. In this paper, a stability bound over the BEC is developed for doubly-generalized LDPC codes, where variable and check nodes can be generic linear block codes, assuming maximumaposteriorierasure correction at each node. It is proved that also in this generalized context the bound depends only on the variable and check component codes with minimum distance 2. A condition is also developed, namely, the derivative matching condition, under which the bound is achieved with equality. The stability bound leads to consider single parity-check codes used as variable nodes as an appealing option to overcome common problems created by generalized check nodes.

arXiv:0802.0823•IEEE Transactions on Information Theory•Feb 26, 2009•28 citations

Generalized Stability Condition for Generalized and Doubly-Generalized LDPC Codes

Enrico Paolini, M.P.C. Fossorier, Marco Chiani

In this paper, the stability condition for low-density parity-check (LDPC) codes on the binary erasure channel (BEC) is extended to generalized LDPC (GLDPC) codes and doubly-generalized LDPC (D-GLDPC) codes. It is proved that, in both cases, the stability condition only involves the component codes with minimum distance 2. The stability condition for GLDPC codes is always expressed as an upper bound to the decoding threshold. This is not possible for D-GLDPC codes, unless all the generalized variable nodes have minimum distance at least 3. Furthermore, a condition called derivative matching is defined in the paper. This condition is sufficient for a GLDPC or D- GLDPC code to achieve the stability condition with equality. If this condition is satisfied, the threshold of D-GLDPC codes (whose generalized variable nodes have all minimum distance at least 3) and GLDPC codes can be expressed in closed form.

arXiv:0704.3199•IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2007•7 citations

Bounds on the maximum-likelihood decoding error probability of low-density parity-check codes

G. Miller, David Burshtein

We derive both upper and lower bounds on the decoding error probability of maximum-likelihood (ML) decoded low-density parity-check (LDPC) codes. The results hold for any binary-input symmetric-output channel. Our results indicate that for various appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. However, the ensemble averaged decoding error probability decreases polynomially, and not exponentially. The lower and upper bounds coincide asymptotically, thus showing the tightness of the bounds. However, for ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing, with an error exponent that can be set arbitrarily close to the standard random coding exponent.

IEEE Transactions on Information Theory•Jan 1, 2001•157 citations

Density evolution for low-density parity-checkcodes under Max-Log-MAP decoding

X. Wei, Ali N. Akansu

A density evolution procedure for low-density parity-check (LDPC) codes under Max-Log-MAP decoding is presented. Using this technique, the precise convergence threshold for LDPC code could be easily derived.

Electronics Letters•Aug 30, 2001•29 citations

Asymptotic MAP upper bounds for LDPC codes

David Matas, Meritxell Lamarca

This paper aims at computing tight upper bounds for the maximum a posteriori threshold of low-density parity check codes in the asymptotic blocklength regime for the transmission over binary-input memoryless symmetric-output channels. While these bounds are already known, we propose a novel derivation based on a completely different approach: based solely on the concept of the chain rule and the conditional entropy, resorting to the concentration theorem for the code ensemble to compute the syndrome entropy with low complexity employing density evolution.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2016•0 citations
We study the stability of low-density parity-check codes under blockwise or bitwise maximuma posterioridecoding, where transmission takes place over a binary-input memoryless output-symmetric channel. Our study stems from the consideration of constructing universal capacity-achieving codes under low-complexity decoding algorithms, where universality refers to the fact that we are considering a family of channels with equal capacity. Consider, e.g., the right-regular sequence by Shokrollahi and the heavy-tail Poisson sequence by Lubyet al. Both sequences are provably capacity-achieving under belief propagation decoding when transmission takes place over the binary erasure channel. In this paper we show that many existing capacity-achieving sequences of low-density parity-check codes are not universal under belief propagation decoding. We reveal that the key to showing this non-universality result is determined by the stability of the underlying codes. More concretely, for an ordered and complete channel family and a sequence of low-density parity-check code ensembles, we determine a stability threshold associated with them, which gives rise to a sufficient condition under which the sequence is not universal under belief propagation decoding. Moreover, we show that the same stability threshold applies to blockwise or bitwise maximuma posterioridecoding as well. We demonstrate how the stability threshold can determine an upper bound on the corresponding blockwise or bitwise maximuma posteriorithreshold, revealing the operational significance of the stability threshold.
The iterative decoding threshold of low-density parity-check (LDPC) codes over the binary erasure channel (BEC) fulfills an upper bound depending only on the variable and check nodes with minimum distance 2. This bound is a consequence of the stability condition, and is here referred to as stability bound. In this paper, a stability bound over the BEC is developed for doubly-generalized LDPC codes, where variable and check nodes can be generic linear block codes, assuming maximumaposteriorierasure correction at each node. It is proved that also in this generalized context the bound depends only on the variable and check component codes with minimum distance 2. A condition is also developed, namely, the derivative matching condition, under which the bound is achieved with equality. The stability bound leads to consider single parity-check codes used as variable nodes as an appealing option to overcome common problems created by generalized check nodes.
In this paper, the stability condition for low-density parity-check (LDPC) codes on the binary erasure channel (BEC) is extended to generalized LDPC (GLDPC) codes and doubly-generalized LDPC (D-GLDPC) codes. It is proved that, in both cases, the stability condition only involves the component codes with minimum distance 2. The stability condition for GLDPC codes is always expressed as an upper bound to the decoding threshold. This is not possible for D-GLDPC codes, unless all the generalized variable nodes have minimum distance at least 3. Furthermore, a condition called derivative matching is defined in the paper. This condition is sufficient for a GLDPC or D- GLDPC code to achieve the stability condition with equality. If this condition is satisfied, the threshold of D-GLDPC codes (whose generalized variable nodes have all minimum distance at least 3) and GLDPC codes can be expressed in closed form.
We derive both upper and lower bounds on the decoding error probability of maximum-likelihood (ML) decoded low-density parity-check (LDPC) codes. The results hold for any binary-input symmetric-output channel. Our results indicate that for various appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. However, the ensemble averaged decoding error probability decreases polynomially, and not exponentially. The lower and upper bounds coincide asymptotically, thus showing the tightness of the bounds. However, for ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing, with an error exponent that can be set arbitrarily close to the standard random coding exponent.
This paper aims at computing tight upper bounds for the maximum a posteriori threshold of low-density parity check codes in the asymptotic blocklength regime for the transmission over binary-input memoryless symmetric-output channels. While these bounds are already known, we propose a novel derivation based on a completely different approach: based solely on the concept of the chain rule and the conditional entropy, resorting to the concentration theorem for the code ensemble to compute the syndrome entropy with low complexity employing density evolution.