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

The Stopping Set Property and the Iterative Decoding Perfor- mance of Binary Block Codes on BSC and AWGN Channel

Zhijie Ren

Published Sep 26, 2011• 1 citation
OpenAlex

Abstract

In the field of Error Correcting Coding (ECC), the concept of ''stopping set'' recently became a hot topic. A stopping set for a parity-check matrix is the set of bits which would cause the iterative decoding progress in Binary Erasure Channel (BEC) to ''stop'' when it is erased. It has already been proved in several papers that the decoding performance under iterative decoding algorithm of any linear block code in BEC is determined by their stopping set performance, in particular, their size and number. And it is commonly assumed by the coding researchers that those codes which have better stopping set performance would also perform better in other channels. In this thesis, the decoding performance of linear block codes, in particular, Hamming Codes, the original LDPC code proposed by Gallager and FG-LDPC codes is studied in Binary Symmetric Channel (BSC) and Additive White Gaussian Noise (AWGN) channel. Three iterative decoding algorithms, respectively Bit-Flipping algorithm, Weighted Bit-Flipping algorithm and Sum Product Algorithm are used. To emphasize the influence of stopping set property, the parity-check matrix for each code is modified by adding linearly dependent rows to the original parity-check matrix, or removing rows from it. These modifications are made in order to change the stopping set property of the codes. The performance of the modified parity-check matrices is thus analyzed, by comparing them to the original ones, using both theoretical analysis and simulation. The results show that the stopping set performance of linear block codes is not a crucial factor to their performances on other channels as it is in BEC. Sometimes the linear block codes of a better stopping set property do perform better in other channels, but sometimes the performance barely changes or even degrades. More precisely, the decoding performance of a certain parity-check matrix is determined by the type of codes, the type of channel and the decoding algorithm, rather than its stopping set property.

Related papers

Impact of Stopping Set Properties on Iterative Decoding of FG-LDPC Codes on the Binary Symmetric Channel

Zeyu Ren, Jos H. Weber, A.J. van Zanten

Stopping set properties determine the iterative decoding performance of any linear block code on the Binary Erasure Channel (BEC). Furthermore, there is a common conception that they are crucial factors on other channels, like the Binary Symmetric Channel (BSC), as well. In order to investigate whether this is true indeed, we study stopping set properties and iterative decoding performance on the BSC of Finite-Geometry Low-Density Parity-Check (FG-LDPC) codes, which have been chosen due to their eectiveness and the good structure of their parity-check matrices (PCMs). Also modied versions of these PCMs are considered. As a rather surprising result, we show that a PCM with better stopping set properties than another PCM of the same code does not necessarily lead a better decoding performance on the BSC. Also, PCMs of the same code with the same stopping set properties, do not always have the same decoding performance on the BSC.

•Jan 1, 2012•0 citations

Stopping Set Distributions of Some Linear Codes

Yong Jiang, Shu‐Tao Xia, Fang‐Wei Fu

Stopping sets and stopping set distribution of an low-density parity-check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let $C$ be a binary $[n,k]$ linear code with parity-check matrix $H$, where the rows of $H$ may be dependent. A stopping set $S$ of $C$ with parity-check matrix $H$ is a subset of column indices of $H$ such that the restriction of $H$ to $S$ does not contain a row of weight one. The stopping set distribution $\{T_i(H)\}_{i=0}^n$ enumerates the number of stopping sets with size $i$ of $C$ with parity-check matrix $H$. Note that stopping sets and stopping set distribution are related to the parity-check matrix $H$ of $C$. Let $H^{*}$ be the parity-check matrix of $C$ which is formed by all the non-zero codewords of its dual code $C^{\perp}$. A parity-check matrix $H$ is called BEC-optimal if $T_i(H)=T_i(H^*), i=0,1,..., n$ and $H$ has the smallest number of rows. On the BEC, iterative decoder of $C$ with BEC-optimal parity-check matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes and the extended Hamming codes.

arXiv:1003.0367•Mar 1, 2010•0 citations

On the probabilistic computation of the stopping redundancy of LDPC codes

Masanori Hirotomo, Yoshiho Konishi, Masakatu Morii

On the binary erasure channel (BEC), the performance of low-density parity-check (LDPC) codes decoded by iterative algorithms is estimated by the small stopping sets in the codes. The size of smallest stopping set is called the stopping distance. Since it depends on the structure of parity-check matrix for the code, we can construct the parity-check matrix by adding dependent rows so that the stopping distance is equal to the minimum distance. The stopping redundancy is defined as the smallest number of rows in such a parity-check matrix. In this paper, we propose a probabilistic algorithm for computing the stopping redundancy of LDPC codes. Additionally, we show numerical results of computing the stopping redundancy of (155,64), (504,252) and (1008,504) LDPC codes.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2009•1 citation

Permutation Decoding and the Stopping Redundancy Hierarchy of Linear Block Codes

Thorsten Hehn, Olgica Milenković, Stefan Laendner +1 more

We investigate the stopping redundancy hierarchy of linear block codes and its connection to permutation decoding techniques. An element in the ordered list of stopping redundancy values represents the smallest number of possibly linearly dependent rows in any parity-check matrix of a code that avoids stopping sets of up to a given size. Redundant parity-check equations can be shown to have a similar effect on decoding performance as permuting the coordinates of the received codeword according to a selected set of automorphisms of the code. Based on this finding we develop new decoding strategies for data transmission over the binary erasure channel that combine iterative message passing and permutation decoding in order to avoid errors confined to stopping sets. We also introduce the notion of s-SAD sets, containing the smallest number of automorphisms of a code with the property that they move any set of not more than s erasures into positions that do not correspond to stopping sets within a judiciously chosen parity-check matrix.

arXiv:cs/0702050•IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2007•3 citations

An Iterative Scheme of Stopping Set Enlargement in The Design of Short-Length Irregular LDPC Codes

Jingyu Kang, Pingyi Fan, Zhigang Cao

Abstract: Small stopping sets have been proved to be harmful to the error floor performance of a finite-length irregular low-density parity-check (LDPC) code over the binary erasure channel (BEC). On the additive white gaussian noise (AWGN) channel, small stopping sets are expected to have a similar effect. This paper proposes an algorithm that iteratively removes the small-stopping-set-inducing patterns from the graph of a randomly generated irregular LDPC code by means of node exchange, aiming at enlarging the average size of stopping sets in this code. Computer simulation results show that the short-length LDPC codes after our processing achieve a considerable performance improvement in the low-BER region compared to the randomly generated codes. Key words: Error floor, irregular LDPC codes, iterative processing, small stopping set 1

•Jan 1, 2005•1 citation

Düşük yoğunluklu eşlik denetim kodları için kod çözme algoritmalarının karşılaştırılması.

Kolaylı, Mert

Low-density parity-check (LDPC) codes are a subclass of linear block codes. These codes have parity-check matrices in which the ratio of the non-zero elements to all elements is low. This property is exploited in defining low complexity decoding algorithms. Low-density parity-check codes have good distance properties and error correction capability near Shannon limits. In this thesis, the sum-product and the bit-flip decoding algorithms for low-density parity-check codes are implemented on Intel Pentium M 1,86 GHz processor using the software called MATLAB. Simulations for the two decoding algorithms are made over additive white gaussian noise (AWGN) channel changing the code parameters like the information rate, the blocklength of the code and the column weight of the parity-check matrix. Performance comparison of the two decoding algorithms are made according to these simulation results. As expected, the sum-product algorithm, which is based on soft-decision decoding, outperforms the bit-flip algorithm, which depends on hard-decision decoding. Our simulations show that the performance of LDPC codes improves with increasing blocklength and number of iterations for both decoding algorithms. Since the sum-product algorithm has lower error-floor characteristics, increasing the number of iterations is more effective for the sum-product decoder compared to the bit-flip decoder. By having better BER performance for lower information rates, the bit-flip algorithm performs according to the expectations; however, the performance of the sum-product decoder deteriorates for information rates below 0.5 instead of improving. By irregular construction of LDPC codes, a performance improvement is observed especially for low SNR values.

•Jan 1, 2006•0 citations
In the field of Error Correcting Coding (ECC), the concept of ''stopping set'' recently became a hot topic. A stopping set for a parity-check matrix is the set of bits which would cause the iterative decoding progress in Binary Erasure Channel (BEC) to ''stop'' when it is erased. It has already been proved in several papers that the decoding performance under iterative decoding algorithm of any linear block code in BEC is determined by their stopping set performance, in particular, their size and number. And it is commonly assumed by the coding researchers that those codes which have better stopping set performance would also perform better in other channels. In this thesis, the decoding performance of linear block codes, in particular, Hamming Codes, the original LDPC code proposed by Gallager and FG-LDPC codes is studied in Binary Symmetric Channel (BSC) and Additive White Gaussian Noise (AWGN) channel. Three iterative decoding algorithms, respectively Bit-Flipping algorithm, Weighted Bit-Flipping algorithm and Sum Product Algorithm are used. To emphasize the influence of stopping set property, the parity-check matrix for each code is modified by adding linearly dependent rows to the original parity-check matrix, or removing rows from it. These modifications are made in order to change the stopping set property of the codes. The performance of the modified parity-check matrices is thus analyzed, by comparing them to the original ones, using both theoretical analysis and simulation. The results show that the stopping set performance of linear block codes is not a crucial factor to their performances on other channels as it is in BEC. Sometimes the linear block codes of a better stopping set property do perform better in other channels, but sometimes the performance barely changes or even degrades. More precisely, the decoding performance of a certain parity-check matrix is determined by the type of codes, the type of channel and the decoding algorithm, rather than its stopping set property.
Stopping set properties determine the iterative decoding performance of any linear block code on the Binary Erasure Channel (BEC). Furthermore, there is a common conception that they are crucial factors on other channels, like the Binary Symmetric Channel (BSC), as well. In order to investigate whether this is true indeed, we study stopping set properties and iterative decoding performance on the BSC of Finite-Geometry Low-Density Parity-Check (FG-LDPC) codes, which have been chosen due to their eectiveness and the good structure of their parity-check matrices (PCMs). Also modied versions of these PCMs are considered. As a rather surprising result, we show that a PCM with better stopping set properties than another PCM of the same code does not necessarily lead a better decoding performance on the BSC. Also, PCMs of the same code with the same stopping set properties, do not always have the same decoding performance on the BSC.
Stopping sets and stopping set distribution of an low-density parity-check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let CCC be a binary [n,k][n,k][n,k] linear code with parity-check matrix HHH, where the rows of HHH may be dependent. A stopping set SSS of CCC with parity-check matrix HHH is a subset of column indices of HHH such that the restriction of HHH to SSS does not contain a row of weight one. The stopping set distribution {Ti(H)}i=0n\{T_i(H)\}_{i=0}^n{Ti​(H)}i=0n​ enumerates the number of stopping sets with size iii of CCC with parity-check matrix HHH. Note that stopping sets and stopping set distribution are related to the parity-check matrix HHH of CCC. Let H∗H^{*}H∗ be the parity-check matrix of CCC which is formed by all the non-zero codewords of its dual code C⊥C^{\perp}C⊥. A parity-check matrix HHH is called BEC-optimal if Ti(H)=Ti(H∗),i=0,1,...,nT_i(H)=T_i(H^*), i=0,1,..., nTi​(H)=Ti​(H∗),i=0,1,...,n and HHH has the smallest number of rows. On the BEC, iterative decoder of CCC with BEC-optimal parity-check matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes and the extended Hamming codes.
On the binary erasure channel (BEC), the performance of low-density parity-check (LDPC) codes decoded by iterative algorithms is estimated by the small stopping sets in the codes. The size of smallest stopping set is called the stopping distance. Since it depends on the structure of parity-check matrix for the code, we can construct the parity-check matrix by adding dependent rows so that the stopping distance is equal to the minimum distance. The stopping redundancy is defined as the smallest number of rows in such a parity-check matrix. In this paper, we propose a probabilistic algorithm for computing the stopping redundancy of LDPC codes. Additionally, we show numerical results of computing the stopping redundancy of (155,64), (504,252) and (1008,504) LDPC codes.
We investigate the stopping redundancy hierarchy of linear block codes and its connection to permutation decoding techniques. An element in the ordered list of stopping redundancy values represents the smallest number of possibly linearly dependent rows in any parity-check matrix of a code that avoids stopping sets of up to a given size. Redundant parity-check equations can be shown to have a similar effect on decoding performance as permuting the coordinates of the received codeword according to a selected set of automorphisms of the code. Based on this finding we develop new decoding strategies for data transmission over the binary erasure channel that combine iterative message passing and permutation decoding in order to avoid errors confined to stopping sets. We also introduce the notion of s-SAD sets, containing the smallest number of automorphisms of a code with the property that they move any set of not more than s erasures into positions that do not correspond to stopping sets within a judiciously chosen parity-check matrix.
Abstract: Small stopping sets have been proved to be harmful to the error floor performance of a finite-length irregular low-density parity-check (LDPC) code over the binary erasure channel (BEC). On the additive white gaussian noise (AWGN) channel, small stopping sets are expected to have a similar effect. This paper proposes an algorithm that iteratively removes the small-stopping-set-inducing patterns from the graph of a randomly generated irregular LDPC code by means of node exchange, aiming at enlarging the average size of stopping sets in this code. Computer simulation results show that the short-length LDPC codes after our processing achieve a considerable performance improvement in the low-BER region compared to the randomly generated codes. Key words: Error floor, irregular LDPC codes, iterative processing, small stopping set 1
Low-density parity-check (LDPC) codes are a subclass of linear block codes. These codes have parity-check matrices in which the ratio of the non-zero elements to all elements is low. This property is exploited in defining low complexity decoding algorithms. Low-density parity-check codes have good distance properties and error correction capability near Shannon limits. In this thesis, the sum-product and the bit-flip decoding algorithms for low-density parity-check codes are implemented on Intel Pentium M 1,86 GHz processor using the software called MATLAB. Simulations for the two decoding algorithms are made over additive white gaussian noise (AWGN) channel changing the code parameters like the information rate, the blocklength of the code and the column weight of the parity-check matrix. Performance comparison of the two decoding algorithms are made according to these simulation results. As expected, the sum-product algorithm, which is based on soft-decision decoding, outperforms the bit-flip algorithm, which depends on hard-decision decoding. Our simulations show that the performance of LDPC codes improves with increasing blocklength and number of iterations for both decoding algorithms. Since the sum-product algorithm has lower error-floor characteristics, increasing the number of iterations is more effective for the sum-product decoder compared to the bit-flip decoder. By having better BER performance for lower information rates, the bit-flip algorithm performs according to the expectations; however, the performance of the sum-product decoder deteriorates for information rates below 0.5 instead of improving. By irregular construction of LDPC codes, a performance improvement is observed especially for low SNR values.