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

arXiv:cs/0609117 · IEEE International Symposium on Information Theory (ISIT)

Constructing LDPC Codes by 2-Lifts

Xudong Ma, En‐hui Yang

Published Jun 1, 2007• 3 citations
arXivHTMLPDF

Abstract

With the motivation of constructing Low-Density Parity-Check (LDPC) codes with low error floors, we propose a new code construction scheme based on random 2-lifts. An analysis on stopping set distributions of the proposed code ensembles is presented. The analysis shows that low-weight stopping sets in the resulting codes are typically the results of stopping sets with weak graph expansion properties in the base graphs. Based on this analysis, we propose a set of design criteria for constructing codes with low error floors. According to the design criteria, stopping sets with weak graph expansion properties should be avoided in the base graphs. We present numerical results on constructing capacity-approaching codes over Binary Erasure Channels (BEC). The numerical results show that the codes by the proposed scheme have significantly lower error floors compared with the codes by the standard construction.

DOI
10.1109/isit.2007.4557391

Related papers

Lowering the Error Floor of LDPC Codes Using Cyclic Liftings

Reza Asvadi, Amir H. Banihashemi, Mahmoud Ahmadian‐Attari

Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which are part of the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle ξ of lengthl(ξ) in the base graph such that the inverse image of ξ in the lifted graph consists of only cycles of length strictly larger thanl(ξ). The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions, and in some cases, the code rate. The constructed codes are quasi-cyclic and thus attractive from a practical point of view. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.

IEEE Transactions on Information Theory•Mar 15, 2011•101 citations

Lowering the error floor of LDPC codes using cyclic liftings

Reza Asvadi, Amir H. Banihashemi, Mahmoud Ahmadian‐Attari

Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which form the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle c of length ℓ(c) in the base graph such that the inverse image of c in the lifted graph consists of only cycles of length strictly larger than ℓ(c). The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2010•13 citations

Lowering the Error Floor of LDPC Codes Using Cyclic Liftings

Reza Asvadi, Amir H. Banihashemi, Mahmoud Ahmadian‐Attari

Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which form the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle $c$ of length $\ell(c)$ in the base graph such that the inverse image of $c$ in the lifted graph consists of only cycles of length strictly larger than $\ell(c)$. The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions, encoder and decoder structure, and in some cases, the code rate. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.

arXiv:1002.4311•Feb 23, 2010•0 citations

Generating Good Finite Length LDPC Codes Based on Lifted Graphs

Eran Sharon, Simon N. Litsyn

A method for generating good finite length Low-Density Parity-Check (LDPC) codes based on lifted graphs is described.The codes are constructed using a progressive edge growth technique by taking into account the influence that every introduced edge has on the error floor and waterfall threshold of the code.The generated code can be tailored according to prescribed specification, achieving the desirable error floor versus waterfall threshold trade-off.For the constructed codes, an analytical upper bound on the expected block error probability of the code over the Binary Erasure Channel (BEC) in the error floor region and an asymptotic Density Evolution (DE) waterfall threshold are obtained.

•Sep 27, 2006•0 citations

A novel algorithm of constructing LDPC codes with graph theory

Yan Cui, Xinhui Si, Yanan Shen

There are many algorithms to construct good low-density parity-check (LDPC) code, the most typical algorithms are bit-filling algorithm, randomly construction algorithm, and PEG (progressive edge growth) algorithm. As shown in [1], the error floor of LDPC Code is decreased by using PEG algorithm, but the error correction performance in waterfall region is compromised since a stopping set with small size will form the codeword with small Hamming weight over AWGN [2]. In this paper, we propose a novel algorithm to construct LDPC Codes. In our algorithm, we construct LDPC Code to avoid small girth and small stopping set by detecting the complete associated matrix of check node (defined in this paper) that converted from bipartite graph of LDPC Code based on the graph theory. Simulation shows that the LDPC code constructed by our algorithm has lower error floor than randomly constructed LDPC Code. The performance improvement of our algorithm is 0.1dB at BER of 10-3compared with PEG algorithm.

IEEE Conference on Cybernetics and Intelligent Systems•Sep 1, 2008•2 citations

Coset graphs for low-density parity check codes: performance on the binary erasure channel

Josef Lauri, Cen Jung Tjhai

The authors show that a popular way of constructing quasi-cyclic low-density parity check (LDPC) codes is a special case of a construction which is common in graph theory and group theory. It is shown that a generalisation of this construction as coset graphs produces (dv, dc)-regular LDPC codes that have an advantage in terms of the minimum stopping set size compared to quasi-cyclic LDPC codes. A (dv, dc)-regular quasi-cyclic LDPC code cannot have minimum stopping set size larger than (dv+1)!. However, by using coset graphs, a (3, 5)-regular LDPC code with minimum stopping set size of 28 and a (3, 4)-regular LDPC code with minimum stopping set size larger than 32 have been obtained. In addition, the idea of coset graphs also provides a compact algebraic way of describing bipartite graph and the associated parity-check matrix of an LDPC code. Simulation results of iterative decoding of the coset graphs LDPC codes over the binary erasure channel show that some of the codes converge well and based on the truncated stopping set distributions of the codes, which are exhaustively and efficiently enumerated, the error-floor of the codes at low probability of erasure is estimated.

IET Communications•Mar 25, 2011•0 citations
With the motivation of constructing Low-Density Parity-Check (LDPC) codes with low error floors, we propose a new code construction scheme based on random 2-lifts. An analysis on stopping set distributions of the proposed code ensembles is presented. The analysis shows that low-weight stopping sets in the resulting codes are typically the results of stopping sets with weak graph expansion properties in the base graphs. Based on this analysis, we propose a set of design criteria for constructing codes with low error floors. According to the design criteria, stopping sets with weak graph expansion properties should be avoided in the base graphs. We present numerical results on constructing capacity-approaching codes over Binary Erasure Channels (BEC). The numerical results show that the codes by the proposed scheme have significantly lower error floors compared with the codes by the standard construction.
Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which are part of the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle ξ of lengthl(ξ) in the base graph such that the inverse image of ξ in the lifted graph consists of only cycles of length strictly larger thanl(ξ). The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions, and in some cases, the code rate. The constructed codes are quasi-cyclic and thus attractive from a practical point of view. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.
Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which form the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle c of length ℓ(c) in the base graph such that the inverse image of c in the lifted graph consists of only cycles of length strictly larger than ℓ(c). The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.
Cyclic liftings are proposed to lower the error floor of low-density parity-check (LDPC) codes. The liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which form the trapping sets. We derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle ccc of length ℓ(c)\ell(c)ℓ(c) in the base graph such that the inverse image of ccc in the lifted graph consists of only cycles of length strictly larger than ℓ(c)\ell(c)ℓ(c). The proposed method is universal in the sense that it can be applied to any LDPC code over any channel and for any iterative decoding algorithm. It also preserves important properties of the base code such as degree distributions, encoder and decoder structure, and in some cases, the code rate. The proposed method is applied to both structured and random codes over the binary symmetric channel (BSC). The error floor improves consistently by increasing the lifting degree, and the results show significant improvements in the error floor compared to the base code, a random code of the same degree distribution and block length, and a random lifting of the same degree. Similar improvements are also observed when the codes designed for the BSC are applied to the additive white Gaussian noise (AWGN) channel.
A method for generating good finite length Low-Density Parity-Check (LDPC) codes based on lifted graphs is described.The codes are constructed using a progressive edge growth technique by taking into account the influence that every introduced edge has on the error floor and waterfall threshold of the code.The generated code can be tailored according to prescribed specification, achieving the desirable error floor versus waterfall threshold trade-off.For the constructed codes, an analytical upper bound on the expected block error probability of the code over the Binary Erasure Channel (BEC) in the error floor region and an asymptotic Density Evolution (DE) waterfall threshold are obtained.
There are many algorithms to construct good low-density parity-check (LDPC) code, the most typical algorithms are bit-filling algorithm, randomly construction algorithm, and PEG (progressive edge growth) algorithm. As shown in [1], the error floor of LDPC Code is decreased by using PEG algorithm, but the error correction performance in waterfall region is compromised since a stopping set with small size will form the codeword with small Hamming weight over AWGN [2]. In this paper, we propose a novel algorithm to construct LDPC Codes. In our algorithm, we construct LDPC Code to avoid small girth and small stopping set by detecting the complete associated matrix of check node (defined in this paper) that converted from bipartite graph of LDPC Code based on the graph theory. Simulation shows that the LDPC code constructed by our algorithm has lower error floor than randomly constructed LDPC Code. The performance improvement of our algorithm is 0.1dB at BER of 10-3compared with PEG algorithm.
The authors show that a popular way of constructing quasi-cyclic low-density parity check (LDPC) codes is a special case of a construction which is common in graph theory and group theory. It is shown that a generalisation of this construction as coset graphs produces (dv, dc)-regular LDPC codes that have an advantage in terms of the minimum stopping set size compared to quasi-cyclic LDPC codes. A (dv, dc)-regular quasi-cyclic LDPC code cannot have minimum stopping set size larger than (dv+1)!. However, by using coset graphs, a (3, 5)-regular LDPC code with minimum stopping set size of 28 and a (3, 4)-regular LDPC code with minimum stopping set size larger than 32 have been obtained. In addition, the idea of coset graphs also provides a compact algebraic way of describing bipartite graph and the associated parity-check matrix of an LDPC code. Simulation results of iterative decoding of the coset graphs LDPC codes over the binary erasure channel show that some of the codes converge well and based on the truncated stopping set distributions of the codes, which are exhaustively and efficiently enumerated, the error-floor of the codes at low probability of erasure is estimated.