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:1808.05258

Edge Coloring Technique to Remove Small Elementary Trapping Sets from Tanner Graph of QC-LDPC Codes with Column Weight 4

Mohammad‐Reza Sadeghi, Farzane Amirzade

Published Aug 15, 2018• 3 citations
arXivHTMLPDF

Abstract

One of the phenomena that causes high decoding failure rates is trapping sets. Characterization of $(a,b)$ elementary trapping sets (ETSs), their graphical properties and the lower bounds on their size in variable regular LDPC codes with column weights 3, 4, 5 and 6, where $a$ is the size of the ETS and $b$ is the number of degree-one check nodes, have been an interesting subject among researchers. Although progressive-edge-growth method (PEG) to construct LDPC codes free of an specific ETS has been proposed in the literature, it is mostly applied to LDPC codes with column weight 3. In this paper, we focus on constructing QC-LDPC codes with column weight 4 whose Tanner graphs are free of small ETSs. Using coloring the edges of the variable node (VN) graph corresponding to an ETS, we provide the sufficient conditions to obtain QC-LDPC codes with column weight 4, girth 6 and free of $(5,b)$ ETSs, where $b\leq4$, and $(6,b)$ ETs, where $b\leq2$. Moreover, for $(4,n)$-regular QC-LDPC codes with girth 8, we present a method to remove $(7,4)$ ETSs from Tanner graphs.

DOI
10.48550/arxiv.1808.05258

Related papers

Efficient Search of QC-LDPC Codes with Girths 6 and 8 and Free of Elementary Trapping Sets with Small Size

Farzane Amirzade, Mohammad‐Reza Sadeghi

One of the phenomena that influences significantly the performance of low-density parity-check codes is known as trapping sets. An $(a,b)$ elementary trapping set, or simply an ETS where $a$ is the size and $b$ is the number of degree-one check nodes and $\frac{b}{a}<1$, causes high decoding failure rate and exert a strong influence on the error floor. In this paper, we provide sufficient conditions for exponent matrices to have fully connected $(3,n)$-regular QC-LDPC codes with girths 6 and 8 whose Tanner graphs are free of small ETSs. Applying sufficient conditions on the exponent matrix to remove some 8-cycles results in removing all 4-cycles, 6-cycles as well as some small elementary trapping sets. For each girth we obtain a lower bound on the lifting degree and present exponent matrices with column weight three whose corresponding Tanner graph is free of certain ETSs.

arXiv:1803.08141•Mar 21, 2018•4 citations

On Characterization of Elementary Trapping Sets of Variable-Regular LDPC Codes

Karimi, Mehdi, Amir H. Banihashemi

In this paper, we study the graphical structure of elementary trapping sets (ETS) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree $d_l$ and girth $g$, we identify all the non-isomorphic structures of an arbitrary class of $(a,b)$ ETSs, where $a$ is the number of variable nodes and $b$ is the number of odd-degree check nodes in the induced subgraph of the ETS. Our study leads to a simple characterization of dominant classes of ETSs (those with relatively small values of $a$ and $b$) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set ${\cal S}$ in the class is a layered superset (LSS) of a short cycle, where the term "layered" is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate ${\cal S}$. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for $d_l = 3, 4, 5, 6$, $g = 6, 8$ and $a, b \leq 10$ in this paper. The results of this paper can be used for the error floor analysis and for the design of LDPC codes with low error floors.

arXiv:1308.1259•Aug 6, 2013•0 citations

QC-LDPC Codes With Large Column Weight and Free of Small Size ETSs

Farzane Amirzade, Mohammad‐Reza Sadeghi, Daniel Panario

An approach to improve the performance of QC-LDPC codes is the removal of harmful trapping sets by increasing the girth. However, constructing these LDPC codes with large column weights and girth more than 8 is not easy. We are concerned with protograph-based LDPC codes with large column weights and free of small size trapping sets. We use the edge-coloring technique and some concepts from graph theory such as rainbow cycles to show that for large column weights the removal of all 8-cycles but the ones we call$\textit {rainbow 8-cycle}$causes the elimination of several small size trapping sets. We provide a detailed theoretical analysis of these harmful trapping sets. Then, we apply them to array-based LDPC codes to significantly simplify and optimize the necessary and sufficient conditions to eliminate those 8-cycles from the Tanner graph. The given exponent matrices and simulation results show the impact of this simplification and the removal of the above mentioned 8-cycles.

IEEE Communications Letters•Dec 27, 2021•11 citations

QC-LDPC construction free of small size elementary trapping sets based on multiplicative subgroups of a finite field

Farzane Amirzade, Mohammad‐Reza Sadeghi, Daniel Panario

Trapping sets significantly influence the performance of low-density parity-check codes. An $ (a, b) $ elementary trapping set (ETS) causes high decoding failure rate and exert a strong influence on the error floor of the code, where $ a $ and $ b $ denote the size and the number of unsatisfied check-nodes in the ETS, respectively. The smallest size of an ETS in $ (3, n) $-regular LDPC codes with girth 6 is 4. In this paper, we provide sufficient conditions to construct fully connected $ (3, n) $-regular algebraic-based QC-LDPC codes with girth 6 whose Tanner graphs are free of $ (a, b) $ ETSs with $ a\leq5 $ and $ b\leq2 $. We apply these sufficient conditions to the exponent matrix of a new algebraic-based QC-LDPC code with girth at least 6. As a result, we obtain the maximum size of a submatrix of the exponent matrix which satisfies the sufficient conditions and yields a Tanner graph free of those ETSs with small size. Some algebraic-based QC-LDPC code constructions with girth 6 in the literature are special cases of our construction. Our experimental results show that removing ETSs with small size contribute to have better performance curves in the error floor region.

Advances in Mathematics of Communications•Jan 1, 2020•10 citations

On characterization of elementary trapping sets of variable-regular LDPC codes

Mehdi Karimi, Amir H. Banihashemi

In this paper, we study the graphical structure of elementary trapping sets (ETS) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree dland girth g, we identify all the non-isomorphic structures of an arbitrary class of (a, b) ETSs, where a is the number of variable nodes and b is the number of odd-degree check nodes in the induced subgraph of the ETS. Our study leads to a simple characterization of dominant classes of ETSs (those with relatively small values of a and b) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set S in the class is a layered superset (LSS) of a short cycle, where the term “layered” is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate S. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for dl= 3, 4, 5, 6, g = 6, 8 and a, b ≤ 10. The results of this work can be used for the error floor analysis and for the design of LDPC codes with low error floors.

International Symposium on Turbo Codes and Iterative Information Processing•Aug 1, 2014•8 citations

On Characterization of Elementary Trapping Sets of Variable-Regular LDPC Codes

Mehdi Karimi, Amir H. Banihashemi

In this paper, we study the graphical structure of elementary trapping sets (ETSs) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree dl and girth g, we identify all the nonisomorphic structures of an arbitrary class of (a, b) ETSs, where a is the number of variable nodes and b is the number of odd-degree check nodes in the induced subgraph of the ETS. This paper leads to a simple characterization of dominant classes of ETSs (those with relatively small values of a and b) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set S in the class is a layered superset (LSS) of a short cycle, where the term layered is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate S. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for dl= 3, 4, 5, 6, g = 6, 8, and a, b ≤ 10 in this paper. The results of this paper can be used for the error floor analysis and for the design of LDPC codes with low error floors.

IEEE Transactions on Information Theory•Jul 2, 2014•68 citations
One of the phenomena that causes high decoding failure rates is trapping sets. Characterization of (a,b)(a,b)(a,b) elementary trapping sets (ETSs), their graphical properties and the lower bounds on their size in variable regular LDPC codes with column weights 3, 4, 5 and 6, where aaa is the size of the ETS and bbb is the number of degree-one check nodes, have been an interesting subject among researchers. Although progressive-edge-growth method (PEG) to construct LDPC codes free of an specific ETS has been proposed in the literature, it is mostly applied to LDPC codes with column weight 3. In this paper, we focus on constructing QC-LDPC codes with column weight 4 whose Tanner graphs are free of small ETSs. Using coloring the edges of the variable node (VN) graph corresponding to an ETS, we provide the sufficient conditions to obtain QC-LDPC codes with column weight 4, girth 6 and free of (5,b)(5,b)(5,b) ETSs, where b≤4b\leq4b≤4, and (6,b)(6,b)(6,b) ETs, where b≤2b\leq2b≤2. Moreover, for (4,n)(4,n)(4,n)-regular QC-LDPC codes with girth 8, we present a method to remove (7,4)(7,4)(7,4) ETSs from Tanner graphs.
One of the phenomena that influences significantly the performance of low-density parity-check codes is known as trapping sets. An (a,b)(a,b)(a,b) elementary trapping set, or simply an ETS where aaa is the size and bbb is the number of degree-one check nodes and ba<1\frac{b}{a}<1ab​<1, causes high decoding failure rate and exert a strong influence on the error floor. In this paper, we provide sufficient conditions for exponent matrices to have fully connected (3,n)(3,n)(3,n)-regular QC-LDPC codes with girths 6 and 8 whose Tanner graphs are free of small ETSs. Applying sufficient conditions on the exponent matrix to remove some 8-cycles results in removing all 4-cycles, 6-cycles as well as some small elementary trapping sets. For each girth we obtain a lower bound on the lifting degree and present exponent matrices with column weight three whose corresponding Tanner graph is free of certain ETSs.
In this paper, we study the graphical structure of elementary trapping sets (ETS) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree dld_ldl​ and girth ggg, we identify all the non-isomorphic structures of an arbitrary class of (a,b)(a,b)(a,b) ETSs, where aaa is the number of variable nodes and bbb is the number of odd-degree check nodes in the induced subgraph of the ETS. Our study leads to a simple characterization of dominant classes of ETSs (those with relatively small values of aaa and bbb) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set S{\cal S}S in the class is a layered superset (LSS) of a short cycle, where the term "layered" is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate S{\cal S}S. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for dl=3,4,5,6d_l = 3, 4, 5, 6dl​=3,4,5,6, g=6,8g = 6, 8g=6,8 and a,b≤10a, b \leq 10a,b≤10 in this paper. The results of this paper can be used for the error floor analysis and for the design of LDPC codes with low error floors.
An approach to improve the performance of QC-LDPC codes is the removal of harmful trapping sets by increasing the girth. However, constructing these LDPC codes with large column weights and girth more than 8 is not easy. We are concerned with protograph-based LDPC codes with large column weights and free of small size trapping sets. We use the edge-coloring technique and some concepts from graph theory such as rainbow cycles to show that for large column weights the removal of all 8-cycles but the ones we callrainbow 8-cycle\textit {rainbow 8-cycle}rainbow 8-cyclecauses the elimination of several small size trapping sets. We provide a detailed theoretical analysis of these harmful trapping sets. Then, we apply them to array-based LDPC codes to significantly simplify and optimize the necessary and sufficient conditions to eliminate those 8-cycles from the Tanner graph. The given exponent matrices and simulation results show the impact of this simplification and the removal of the above mentioned 8-cycles.
Trapping sets significantly influence the performance of low-density parity-check codes. An $ (a, b) $ elementary trapping set (ETS) causes high decoding failure rate and exert a strong influence on the error floor of the code, where $ a $ and $ b $ denote the size and the number of unsatisfied check-nodes in the ETS, respectively. The smallest size of an ETS in $ (3, n) $-regular LDPC codes with girth 6 is 4. In this paper, we provide sufficient conditions to construct fully connected $ (3, n) $-regular algebraic-based QC-LDPC codes with girth 6 whose Tanner graphs are free of $ (a, b) $ ETSs with $ a\leq5 $ and $ b\leq2 $. We apply these sufficient conditions to the exponent matrix of a new algebraic-based QC-LDPC code with girth at least 6. As a result, we obtain the maximum size of a submatrix of the exponent matrix which satisfies the sufficient conditions and yields a Tanner graph free of those ETSs with small size. Some algebraic-based QC-LDPC code constructions with girth 6 in the literature are special cases of our construction. Our experimental results show that removing ETSs with small size contribute to have better performance curves in the error floor region.
In this paper, we study the graphical structure of elementary trapping sets (ETS) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree dland girth g, we identify all the non-isomorphic structures of an arbitrary class of (a, b) ETSs, where a is the number of variable nodes and b is the number of odd-degree check nodes in the induced subgraph of the ETS. Our study leads to a simple characterization of dominant classes of ETSs (those with relatively small values of a and b) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set S in the class is a layered superset (LSS) of a short cycle, where the term “layered” is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate S. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for dl= 3, 4, 5, 6, g = 6, 8 and a, b ≤ 10. The results of this work can be used for the error floor analysis and for the design of LDPC codes with low error floors.
In this paper, we study the graphical structure of elementary trapping sets (ETSs) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree dl and girth g, we identify all the nonisomorphic structures of an arbitrary class of (a, b) ETSs, where a is the number of variable nodes and b is the number of odd-degree check nodes in the induced subgraph of the ETS. This paper leads to a simple characterization of dominant classes of ETSs (those with relatively small values of a and b) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set S in the class is a layered superset (LSS) of a short cycle, where the term layered is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate S. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for dl= 3, 4, 5, 6, g = 6, 8, and a, b ≤ 10 in this paper. The results of this paper can be used for the error floor analysis and for the design of LDPC codes with low error floors.