© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2009.08640 · IEEE Transactions on Information Theory

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

Wei Liu, Rudiger L. Urbanke

Published Oct 11, 2021• 4 citations

Abstract

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.

DOI: 10.1109/tit.2021.3119392

Related papers

ON UNIVERSAL LDPC CODES OVER MEMORYLESS SYMMETRIC CHANNELS

Igal Sason, Boaz Shuval

A design of robust error-correcting codes that achieve reli able communication over various channels is of great theoretical and practical interest. Such codes are termed universal. This paper considers the universality of lowdensity parity-check (LDPC) code ensembles over families of memoryless binary-input output-symmetric (MBIOS) channels. Universality is considered both under belief-pr opagation (BP) and maximum-likelihood (ML) decoding. For the BP decoding case, we derive a density-evolution-based analytical method for designing LDPC code ensembles that are universal over various families of MBIOS channels. We also derive a necessary condition for universality of LDPC code ensembles under BP decoding; this condition is used to provide bounds on the universally achievable fraction of capacity. These results enable us to provide conditions for reliable / unreliable communications under BP decoding that are based on the Bhattacharyya parameter of the channel. For the ML decoding case, we prove that properly selected regular LDPC code ensembles are universally capacity-achieving for the set of equicapacity MBIOS channels and extend this result to punctured regular LDPC code ensembles. Index Terms Belief propagation (BP), Bhattacharyya parameter (B-parameter), density evolution (DE), linear programming (LP) bounds, low-density parity-check (LDPC) codes, maximum-likelihood (ML), memoryless binary-input outputsymmetric (MBIOS) channels, stability.

•Jan 1, 2011•1 citation

On Universal LDPC Code Ensembles Over Memoryless Symmetric Channels

Igal Sason, Boaz Shuval

A design of robust error-correcting codes that achieve reliable communication over various channels is of great theoretical and practical interest. Such codes are termed universal. This paper considers the universality of low-density parity-check (LDPC) code ensembles over families of memoryless binary-input output-symmetric (MBIOS) channels. Universality is considered both under belief-propagation (BP) and maximum-likelihood (ML) decoding. For the BP decoding case, we derive a density-evolution-based analytical method for designing LDPC code ensembles that are universal over various families of MBIOS channels. We also derive a necessary condition for universality of LDPC code ensembles under BP decoding; this condition is used to provide bounds on the universally achievable fraction of capacity. These results enable us to provide conditions for reliable/unreliable communications under BP decoding that are based on the Bhattacharyya parameter of the channel. For the ML decoding case, we prove that properly selected regular LDPC code ensembles are universally capacity-achieving for the set of equi-capacity MBIOS channels and extend this result to punctured regular LDPC code ensembles.

IEEE Transactions on Information Theory•Aug 1, 2011•20 citations

On the universality of LDPC code ensembles under belief propagation and ML decoding

Boaz Shuval, Igal Sason

A universal design of low-density parity-check (LDPC) code ensembles which enables to operate reliably over various channels is of great interest. This paper considers the universality of LDPC code ensembles over sets of memoryless binary-input output-symmetric (MBIOS) channels, addressing their universality under belief-propagation (BP) and maximum-likelihood (ML) decoding. Based on the density evolution approach, closed-form, easy-to-calculate, analytical results related to the universality of LDPC code ensembles under BP decoding are derived. LP Bounds on the achievable rate of LDPC code ensembles over a set of equi-capacity channels are also derived. This paper considers the universality under ML decoding of properly selected regular LDPC code ensembles for the set of equi-capacity MBIOS channels. The full paper version related to this work [13] provides further results, full proofs, and additional discussions on the theorems.

IEEE Convention of Electrical and Electronics Engineers in Israel•Nov 1, 2010•10 citations

Parity-check density versus performance of binary linear block codes over memoryless symmetric channels

Igal Sason, Rudiger L. Urbanke

We derive lower bounds on the density of parity-check matrices of binary linear codes which are used over memoryless binary-input output-symmetric (MBIOS) channels. The bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity; they are valid for every sequence of binary linear block codes if there exists a decoding algorithm under which the average bit-error probability vanishes. For every MBIOS channel, we construct a sequence of ensembles of regular low-density parity-check (LDPC) codes, so that an upper bound on the asymptotic density of their parity-check matrices scales similarly to the lower bound. The tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right-regular LDPC codes which was introduced by Shokrollahi, and which is known to achieve the capacity of this channel. Under iterative message-passing decoding, we show that this sequence of ensembles is asymptotically optimal (in a sense to be defined in this paper), strengthening a result of Shokrollahi. Finally, we derive lower bounds on the bit-error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs, and study their performance limitations over MBIOS channels. The latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error-correction codes.

IEEE Transactions on Information Theory•Jun 23, 2003•121 citations

The Stability Condition of LDPC Codes Under MAP Decoding

Wei Liu, Rudiger L. Urbanke

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.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2018•0 citations

On LDPC codes over symmetric channels

Hamid Saeedi, Hossein Pishro-Nik

In the past decade, there has been tremendous amount of research on the analysis and design of the Low-Density Parity-Check (LDPC) codes with belief propagation decoding over different types of Binary-Input Output-Symmetric Memoryless (BIOSM) channels. However, with the exception of the Binary Erasure Channel (BEC), analytical results on LDPC codes over such channels are limited and most results are based on numerical methods and optimization. In particular, systematic design of provably capacity achieving sequences of LDPC code ensembles over general class of BIOSM channels has remained a fundamental open problem. Such sequences have been designed only for the binary erasure channel (BEC). In this paper, we first prove some novel analytical properties of the LDPC code ensembles over BIOSM channels. In particular, we prove a result that generalizes the previously known stability condition over symmetric channels. This suggests that a modified version of the flatness condition, a property which has been shown to be critical for capacity achieving sequences over the BEC, can be used to devise capacity achieving sequences for general class of BIOSM channels. Based on this assumption, we propose a method which could result in the systematic design of such sequences over BIOSM channels. Numerical evidence is promising and provides a consistent convergence behavior to capacity over the considered BIOSM channels as the average check node degree increases.

IEEE Information Theory Workshop (ITW)•Jun 1, 2009•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.

A design of robust error-correcting codes that achieve reli able communication over various channels is of great theoretical and practical interest. Such codes are termed universal. This paper considers the universality of lowdensity parity-check (LDPC) code ensembles over families of memoryless binary-input output-symmetric (MBIOS) channels. Universality is considered both under belief-pr opagation (BP) and maximum-likelihood (ML) decoding. For the BP decoding case, we derive a density-evolution-based analytical method for designing LDPC code ensembles that are universal over various families of MBIOS channels. We also derive a necessary condition for universality of LDPC code ensembles under BP decoding; this condition is used to provide bounds on the universally achievable fraction of capacity. These results enable us to provide conditions for reliable / unreliable communications under BP decoding that are based on the Bhattacharyya parameter of the channel. For the ML decoding case, we prove that properly selected regular LDPC code ensembles are universally capacity-achieving for the set of equicapacity MBIOS channels and extend this result to punctured regular LDPC code ensembles. Index Terms Belief propagation (BP), Bhattacharyya parameter (B-parameter), density evolution (DE), linear programming (LP) bounds, low-density parity-check (LDPC) codes, maximum-likelihood (ML), memoryless binary-input outputsymmetric (MBIOS) channels, stability.

A design of robust error-correcting codes that achieve reliable communication over various channels is of great theoretical and practical interest. Such codes are termed universal. This paper considers the universality of low-density parity-check (LDPC) code ensembles over families of memoryless binary-input output-symmetric (MBIOS) channels. Universality is considered both under belief-propagation (BP) and maximum-likelihood (ML) decoding. For the BP decoding case, we derive a density-evolution-based analytical method for designing LDPC code ensembles that are universal over various families of MBIOS channels. We also derive a necessary condition for universality of LDPC code ensembles under BP decoding; this condition is used to provide bounds on the universally achievable fraction of capacity. These results enable us to provide conditions for reliable/unreliable communications under BP decoding that are based on the Bhattacharyya parameter of the channel. For the ML decoding case, we prove that properly selected regular LDPC code ensembles are universally capacity-achieving for the set of equi-capacity MBIOS channels and extend this result to punctured regular LDPC code ensembles.

A universal design of low-density parity-check (LDPC) code ensembles which enables to operate reliably over various channels is of great interest. This paper considers the universality of LDPC code ensembles over sets of memoryless binary-input output-symmetric (MBIOS) channels, addressing their universality under belief-propagation (BP) and maximum-likelihood (ML) decoding. Based on the density evolution approach, closed-form, easy-to-calculate, analytical results related to the universality of LDPC code ensembles under BP decoding are derived. LP Bounds on the achievable rate of LDPC code ensembles over a set of equi-capacity channels are also derived. This paper considers the universality under ML decoding of properly selected regular LDPC code ensembles for the set of equi-capacity MBIOS channels. The full paper version related to this work [13] provides further results, full proofs, and additional discussions on the theorems.

We derive lower bounds on the density of parity-check matrices of binary linear codes which are used over memoryless binary-input output-symmetric (MBIOS) channels. The bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity; they are valid for every sequence of binary linear block codes if there exists a decoding algorithm under which the average bit-error probability vanishes. For every MBIOS channel, we construct a sequence of ensembles of regular low-density parity-check (LDPC) codes, so that an upper bound on the asymptotic density of their parity-check matrices scales similarly to the lower bound. The tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right-regular LDPC codes which was introduced by Shokrollahi, and which is known to achieve the capacity of this channel. Under iterative message-passing decoding, we show that this sequence of ensembles is asymptotically optimal (in a sense to be defined in this paper), strengthening a result of Shokrollahi. Finally, we derive lower bounds on the bit-error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs, and study their performance limitations over MBIOS channels. The latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error-correction codes.

In the past decade, there has been tremendous amount of research on the analysis and design of the Low-Density Parity-Check (LDPC) codes with belief propagation decoding over different types of Binary-Input Output-Symmetric Memoryless (BIOSM) channels. However, with the exception of the Binary Erasure Channel (BEC), analytical results on LDPC codes over such channels are limited and most results are based on numerical methods and optimization. In particular, systematic design of provably capacity achieving sequences of LDPC code ensembles over general class of BIOSM channels has remained a fundamental open problem. Such sequences have been designed only for the binary erasure channel (BEC). In this paper, we first prove some novel analytical properties of the LDPC code ensembles over BIOSM channels. In particular, we prove a result that generalizes the previously known stability condition over symmetric channels. This suggests that a modified version of the flatness condition, a property which has been shown to be critical for capacity achieving sequences over the BEC, can be used to devise capacity achieving sequences for general class of BIOSM channels. Based on this assumption, we propose a method which could result in the systematic design of such sequences over BIOSM channels. Numerical evidence is promising and provides a consistent convergence behavior to capacity over the considered BIOSM channels as the average check node degree increases.