a new algorithm to construct ldpc codes with large stopping sets
DESCRIPTION
A new algorithm to construct good low-density parity-check (LDPC) codes with large stopping sets is presented. Since the minimum stop- ping set characterizes an LDPC code, searching for stopping sets in LDPC codes is an important issue. Large minimum stopping sets avoid the LDPC code to get trapped in cycles specially on the binary erasure channel. Dealing with stopping sets is not an easy task since their discovering is a well known NP hard prob- lem. Conversely, we propose an algorithm in order to construct an LDPC code from a stopping set which is demonstrated to be large. Results of simulations showing the performance of the LDPC code obtained this way are analyzed.TRANSCRIPT
A new Algorithm to construct LDPC codes with large stopping sets
A new Algorithm to construct LDPC codes withlarge stopping sets
Juan Camilo Salazar Ripoll† and Nestor R. Barraza‡
Septiembre - 2013
†Universidad de los Andes.‡Universidad Nacional de Tres de Febrero y Facultad de Ingenierıa, UBA
A new Algorithm to construct LDPC codes with large stopping sets
Indice
1 IntroductionLDPC codesBipartite Tanner graph - Stopping setVertex Edge Incidence MatrixProperties of Graphs - Girth
2 The AlgorithmThe aimThe methodGetting the LDPC code
3 Simulation
4 Conclusions
A new Algorithm to construct LDPC codes with large stopping sets
Indice
1 IntroductionLDPC codesBipartite Tanner graph - Stopping setVertex Edge Incidence MatrixProperties of Graphs - Girth
2 The AlgorithmThe aimThe methodGetting the LDPC code
3 Simulation
4 Conclusions
A new Algorithm to construct LDPC codes with large stopping sets
Indice
1 IntroductionLDPC codesBipartite Tanner graph - Stopping setVertex Edge Incidence MatrixProperties of Graphs - Girth
2 The AlgorithmThe aimThe methodGetting the LDPC code
3 Simulation
4 Conclusions
A new Algorithm to construct LDPC codes with large stopping sets
Indice
1 IntroductionLDPC codesBipartite Tanner graph - Stopping setVertex Edge Incidence MatrixProperties of Graphs - Girth
2 The AlgorithmThe aimThe methodGetting the LDPC code
3 Simulation
4 Conclusions
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
LDPC codes
H =
1 0 1 0 1 0 00 1 0 1 0 1 00 0 0 1 0 1 11 0 1 0 0 1 00 1 0 0 1 0 1
x1 + x3 + x5 = 0
x2 + x4 + x6 = 0
x4 + x6 + x7 = 0
x1 + x3 + x6 = 0
x2 + x5 + x7 = 0 (1)
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Bipartite Tanner graph - Stopping set
1
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6
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9
10
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12
x1 + x3 + x5 = 0
Variable nodes
Check nodes
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Bipartite Tanner graph - Stopping set
1
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3
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9
10
11
12
x1 + x3 + x5 = 0
x2 + x4 + x6 = 0
Variable nodes
Check nodes
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Bipartite Tanner graph - Stopping set
1
2
3
4
5
6
7
8
9
10
11
12
x1 + x3 + x5 = 0
x2 + x4 + x6 = 0
x4 + x6 + x7 = 0
Variable nodes
Check nodes
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Bipartite Tanner graph - Stopping set
1
2
3
4
5
6
7
8
9
10
11
12
x1 + x3 + x5 = 0
x2 + x4 + x6 = 0
x4 + x6 + x7 = 0
x1 + x3 + x6 = 0
Variable nodes
Check nodes
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Bipartite Tanner graph - Stopping set
1
2
3
4
5
6
7
8
9
10
11
12
x1 + x3 + x5 = 0
x2 + x4 + x6 = 0
x4 + x6 + x7 = 0
x1 + x3 + x6 = 0
x2 + x5 + x7 = 0
Variable nodes
Check nodes
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Bipartite Tanner graph - Stopping set
1
2
3
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Stopping Set
Variable nodes
Check nodes
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Bipartite Tanner graph - Stopping set
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Stopping Set
Message Passing
Variable nodes
Check nodes
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Bipartite Tanner graph - Stopping set
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12
Stopping Set
Message Passing
Variable nodes
Check nodes
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Vertex Edge Incidence Matrix
1 2 4
3
6
5a c
b
e
d
VE =
a b c d e
1 1 0 0 0 02 1 1 1 0 03 0 1 0 0 04 0 0 1 1 15 0 0 0 1 06 0 0 0 0 1
H?= VE (T )
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Vertex Edge Incidence Matrix
1 2 4
3
6
5a c
b
e
d
VE =
a b c d e
1 1 0 0 0 02 1 1 1 0 03 0 1 0 0 04 0 0 1 1 15 0 0 0 1 06 0 0 0 0 1
H
?= VE (T )
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Properties of Graphs - Girth
36
2
4
15
9
7
10
8
Petersen Graph. Girth = 5
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Properties of Graphs - Girth
36
2
4
15
9
7
10
8
Petersen Graph. Girth = 5
VE =
a b c d e e f g h i j k l m n o
1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 02 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 03 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 04 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 05 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 06 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 07 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 08 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 09 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1
10 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Properties of Graphs - Girth
36
2
4
15
9
7
10
8
Petersen Graph. Girth = 5
Tanner Graph. H = VET .
a b c d e f g h i j k l m n o
1 2 3 4 5 6 7 8 9 10
A new Algorithm to construct LDPC codes with large stopping sets
Introduction
Properties of Graphs - Girth
36
2
4
15
9
7
10
8
Petersen Graph. Girth = 5
Tanner Graph. H = VET . Cycles in graph → Stopping sets.
a b c d e f g h i j k l m n o
1 2 3 4 5 6 7 8 9 10
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The aim
Construct a big graph with a big girth
Generate the LDPC code from the transpose of the vertex-edgeincidence matrix
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
The aim is to get a graph which determines the minimumstopping set of the obtained code.
The parity check matrix of the code is obtained as thetranspose of the vertex-edge incidence matrix of the graph.
This method allows to construct LDPC codes up to astopping set size of 12, and with a slight variation the girthcan be increased to 14.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
The aim is to get a graph which determines the minimumstopping set of the obtained code.
The parity check matrix of the code is obtained as thetranspose of the vertex-edge incidence matrix of the graph.
This method allows to construct LDPC codes up to astopping set size of 12, and with a slight variation the girthcan be increased to 14.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
The aim is to get a graph which determines the minimumstopping set of the obtained code.
The parity check matrix of the code is obtained as thetranspose of the vertex-edge incidence matrix of the graph.
This method allows to construct LDPC codes up to astopping set size of 12, and with a slight variation the girthcan be increased to 14.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
Take a core C which is a simple graph, its girth determinesthe stopping set size of the LDPC code.
Make 2|C |+ 1 copies of the core obtaining 2|C |+ 2 subgraphs.
Divide the subgraphs into two sets: a left set and a right set,each one of |C |+ 1 subgraphs. Lets name the subgraphs inthe left set 0, 1, · · · , |C | and the subgraphs in the right set0′, 1′, · · · , |C |′.Connecting the nodes
Take the node i from the graph j and connect it to the node jof the graph i ′ for i 6= j with 1 ≤ i , j ≤ |C |.Connect the node i from the graph i to the node i of thegraph 0′, in a similar way connect the node i from the graph i ′
to the node i of the graph 0.
A graph with 2|C |(|C |+ 1) nodes and girth(graph) =mın(girth(core),12) is obtained. The degree of each node isincreased by one.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
Take a core C which is a simple graph, its girth determinesthe stopping set size of the LDPC code.
Make 2|C |+ 1 copies of the core obtaining 2|C |+ 2 subgraphs.
Divide the subgraphs into two sets: a left set and a right set,each one of |C |+ 1 subgraphs. Lets name the subgraphs inthe left set 0, 1, · · · , |C | and the subgraphs in the right set0′, 1′, · · · , |C |′.Connecting the nodes
Take the node i from the graph j and connect it to the node jof the graph i ′ for i 6= j with 1 ≤ i , j ≤ |C |.Connect the node i from the graph i to the node i of thegraph 0′, in a similar way connect the node i from the graph i ′
to the node i of the graph 0.
A graph with 2|C |(|C |+ 1) nodes and girth(graph) =mın(girth(core),12) is obtained. The degree of each node isincreased by one.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
Take a core C which is a simple graph, its girth determinesthe stopping set size of the LDPC code.
Make 2|C |+ 1 copies of the core obtaining 2|C |+ 2 subgraphs.
Divide the subgraphs into two sets: a left set and a right set,each one of |C |+ 1 subgraphs. Lets name the subgraphs inthe left set 0, 1, · · · , |C | and the subgraphs in the right set0′, 1′, · · · , |C |′.
Connecting the nodes
Take the node i from the graph j and connect it to the node jof the graph i ′ for i 6= j with 1 ≤ i , j ≤ |C |.Connect the node i from the graph i to the node i of thegraph 0′, in a similar way connect the node i from the graph i ′
to the node i of the graph 0.
A graph with 2|C |(|C |+ 1) nodes and girth(graph) =mın(girth(core),12) is obtained. The degree of each node isincreased by one.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
Take a core C which is a simple graph, its girth determinesthe stopping set size of the LDPC code.
Make 2|C |+ 1 copies of the core obtaining 2|C |+ 2 subgraphs.
Divide the subgraphs into two sets: a left set and a right set,each one of |C |+ 1 subgraphs. Lets name the subgraphs inthe left set 0, 1, · · · , |C | and the subgraphs in the right set0′, 1′, · · · , |C |′.Connecting the nodes
Take the node i from the graph j and connect it to the node jof the graph i ′ for i 6= j with 1 ≤ i , j ≤ |C |.Connect the node i from the graph i to the node i of thegraph 0′, in a similar way connect the node i from the graph i ′
to the node i of the graph 0.
A graph with 2|C |(|C |+ 1) nodes and girth(graph) =mın(girth(core),12) is obtained. The degree of each node isincreased by one.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
Take a core C which is a simple graph, its girth determinesthe stopping set size of the LDPC code.
Make 2|C |+ 1 copies of the core obtaining 2|C |+ 2 subgraphs.
Divide the subgraphs into two sets: a left set and a right set,each one of |C |+ 1 subgraphs. Lets name the subgraphs inthe left set 0, 1, · · · , |C | and the subgraphs in the right set0′, 1′, · · · , |C |′.Connecting the nodes
Take the node i from the graph j and connect it to the node jof the graph i ′ for i 6= j with 1 ≤ i , j ≤ |C |.
Connect the node i from the graph i to the node i of thegraph 0′, in a similar way connect the node i from the graph i ′
to the node i of the graph 0.
A graph with 2|C |(|C |+ 1) nodes and girth(graph) =mın(girth(core),12) is obtained. The degree of each node isincreased by one.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
Take a core C which is a simple graph, its girth determinesthe stopping set size of the LDPC code.
Make 2|C |+ 1 copies of the core obtaining 2|C |+ 2 subgraphs.
Divide the subgraphs into two sets: a left set and a right set,each one of |C |+ 1 subgraphs. Lets name the subgraphs inthe left set 0, 1, · · · , |C | and the subgraphs in the right set0′, 1′, · · · , |C |′.Connecting the nodes
Take the node i from the graph j and connect it to the node jof the graph i ′ for i 6= j with 1 ≤ i , j ≤ |C |.Connect the node i from the graph i to the node i of thegraph 0′, in a similar way connect the node i from the graph i ′
to the node i of the graph 0.
A graph with 2|C |(|C |+ 1) nodes and girth(graph) =mın(girth(core),12) is obtained. The degree of each node isincreased by one.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
Take a core C which is a simple graph, its girth determinesthe stopping set size of the LDPC code.
Make 2|C |+ 1 copies of the core obtaining 2|C |+ 2 subgraphs.
Divide the subgraphs into two sets: a left set and a right set,each one of |C |+ 1 subgraphs. Lets name the subgraphs inthe left set 0, 1, · · · , |C | and the subgraphs in the right set0′, 1′, · · · , |C |′.Connecting the nodes
Take the node i from the graph j and connect it to the node jof the graph i ′ for i 6= j with 1 ≤ i , j ≤ |C |.Connect the node i from the graph i to the node i of thegraph 0′, in a similar way connect the node i from the graph i ′
to the node i of the graph 0.
A graph with 2|C |(|C |+ 1) nodes and girth(graph) =mın(girth(core),12) is obtained. The degree of each node isincreased by one.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
Take a core C which is a simple graph, its girth determinesthe stopping set size of the LDPC code.
Make 2|C |+ 1 copies of the core obtaining 2|C |+ 2 subgraphs.
Divide the subgraphs into two sets: a left set and a right set,each one of |C |+ 1 subgraphs. Lets name the subgraphs inthe left set 0, 1, · · · , |C | and the subgraphs in the right set0′, 1′, · · · , |C |′.Connecting the nodes
Take the node i from the graph j and connect it to the node jof the graph i ′ for i 6= j with 1 ≤ i , j ≤ |C |.Connect the node i from the graph i to the node i of thegraph 0′, in a similar way connect the node i from the graph i ′
to the node i of the graph 0.
A graph with 2|C |(|C |+ 1) nodes and girth(graph) =mın(girth(core),12) is obtained. The degree of each node isincreased by one.
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
The method
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A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
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A new Algorithm to construct LDPC codes with large stopping sets
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A new Algorithm to construct LDPC codes with large stopping sets
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A new Algorithm to construct LDPC codes with large stopping sets
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A new Algorithm to construct LDPC codes with large stopping sets
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A new Algorithm to construct LDPC codes with large stopping sets
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A new Algorithm to construct LDPC codes with large stopping sets
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A new Algorithm to construct LDPC codes with large stopping sets
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A new Algorithm to construct LDPC codes with large stopping sets
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A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
Getting the LDPC code
The parity check matrix H is obtained as the transpose of thevertex-edge incidence matrix of the graph.
The nodes of the graph are the check nodes of the code andthe edges of the graph are the variable nodes.
Cycles of length k give cycles of length 2k in the Tannergraph. Then, the size of the stopping set in the LDPC codewill not be less than the girth of the graph.
If a regular graph is chosen as the core, being dv the degree ofeach node, the number of nodes in the generated graph is(|C |) (2 |C |+ 2) and the number of edges is(dv + 1)(|C |)(|C |+ 1).
As a consequence, we get an LDPC code withn = (dv + 1)(|C |)(|C |+ 1) and rate R = dv−1
dv+1 .
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
Getting the LDPC code
The parity check matrix H is obtained as the transpose of thevertex-edge incidence matrix of the graph.
The nodes of the graph are the check nodes of the code andthe edges of the graph are the variable nodes.
Cycles of length k give cycles of length 2k in the Tannergraph. Then, the size of the stopping set in the LDPC codewill not be less than the girth of the graph.
If a regular graph is chosen as the core, being dv the degree ofeach node, the number of nodes in the generated graph is(|C |) (2 |C |+ 2) and the number of edges is(dv + 1)(|C |)(|C |+ 1).
As a consequence, we get an LDPC code withn = (dv + 1)(|C |)(|C |+ 1) and rate R = dv−1
dv+1 .
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
Getting the LDPC code
The parity check matrix H is obtained as the transpose of thevertex-edge incidence matrix of the graph.
The nodes of the graph are the check nodes of the code andthe edges of the graph are the variable nodes.
Cycles of length k give cycles of length 2k in the Tannergraph. Then, the size of the stopping set in the LDPC codewill not be less than the girth of the graph.
If a regular graph is chosen as the core, being dv the degree ofeach node, the number of nodes in the generated graph is(|C |) (2 |C |+ 2) and the number of edges is(dv + 1)(|C |)(|C |+ 1).
As a consequence, we get an LDPC code withn = (dv + 1)(|C |)(|C |+ 1) and rate R = dv−1
dv+1 .
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
Getting the LDPC code
The parity check matrix H is obtained as the transpose of thevertex-edge incidence matrix of the graph.
The nodes of the graph are the check nodes of the code andthe edges of the graph are the variable nodes.
Cycles of length k give cycles of length 2k in the Tannergraph. Then, the size of the stopping set in the LDPC codewill not be less than the girth of the graph.
If a regular graph is chosen as the core, being dv the degree ofeach node, the number of nodes in the generated graph is(|C |) (2 |C |+ 2) and the number of edges is(dv + 1)(|C |)(|C |+ 1).
As a consequence, we get an LDPC code withn = (dv + 1)(|C |)(|C |+ 1) and rate R = dv−1
dv+1 .
A new Algorithm to construct LDPC codes with large stopping sets
The Algorithm
Getting the LDPC code
The parity check matrix H is obtained as the transpose of thevertex-edge incidence matrix of the graph.
The nodes of the graph are the check nodes of the code andthe edges of the graph are the variable nodes.
Cycles of length k give cycles of length 2k in the Tannergraph. Then, the size of the stopping set in the LDPC codewill not be less than the girth of the graph.
If a regular graph is chosen as the core, being dv the degree ofeach node, the number of nodes in the generated graph is(|C |) (2 |C |+ 2) and the number of edges is(dv + 1)(|C |)(|C |+ 1).
As a consequence, we get an LDPC code withn = (dv + 1)(|C |)(|C |+ 1) and rate R = dv−1
dv+1 .
A new Algorithm to construct LDPC codes with large stopping sets
Simulation
22
21
20
19
181716
15
14
13
12
11
10
9
8
76 5
4
3
2
1
Regular core |C | = 22, dv = 2
0 0
*
1 1
1− ε
ε
ε
1− ε
Binary erasure channel (BEC).
A new Algorithm to construct LDPC codes with large stopping sets
Simulation
Core Generated graph LDPC codeRegular Regular Regular, variable node
degree = 2
|C | = 22 |C |(2|C |+ 2) = 1012 no-des
1012 check nodes
dv = 2 node degree = dv +1 = 3 check nodes degree = 3dv+1
2 |C |(2|C | + 2) =1518 edges
1518 variable nodes
girth = 22 girth = 14 stopping set size = 14
R = dv−1dv+1 = 1
3
A new Algorithm to construct LDPC codes with large stopping sets
Simulation
0,10,20,30,40,50,6
10−5
10−4
10−3
10−2
10−1
100
ε
BE
R
Performance of the regular LDPC code in a BEC (R = 1/3, n =1518, girth = 28) with error probability ε.
A new Algorithm to construct LDPC codes with large stopping sets
Simulation
0,10,20,30,40,50,6
10−5
10−4
10−3
10−2
10−1
100
ε
BE
R
Performance of the regular LDPC code in a BEC (R = 1/3, n =1518, girth = 28) with error probability ε.
A new Algorithm to construct LDPC codes with large stopping sets
Conclusions
A new algorithm to construct an LDPC code from agenerated graph was presented
This graph is generated by making some connections betweenseveral copies of a given core
Since the stopping set of the LDPC code is related to thegirth of the graph, a large stopping set size is obtained
The parity check matrix is quite sparse, then, the generatedLDPC code converges in just a few iterations
It is possible to generate bigger codes using the obtainedgraph as the core. We are working now on this issue and itwill be shown in a future work
A new Algorithm to construct LDPC codes with large stopping sets
Conclusions
A new algorithm to construct an LDPC code from agenerated graph was presented
This graph is generated by making some connections betweenseveral copies of a given core
Since the stopping set of the LDPC code is related to thegirth of the graph, a large stopping set size is obtained
The parity check matrix is quite sparse, then, the generatedLDPC code converges in just a few iterations
It is possible to generate bigger codes using the obtainedgraph as the core. We are working now on this issue and itwill be shown in a future work
A new Algorithm to construct LDPC codes with large stopping sets
Conclusions
A new algorithm to construct an LDPC code from agenerated graph was presented
This graph is generated by making some connections betweenseveral copies of a given core
Since the stopping set of the LDPC code is related to thegirth of the graph, a large stopping set size is obtained
The parity check matrix is quite sparse, then, the generatedLDPC code converges in just a few iterations
It is possible to generate bigger codes using the obtainedgraph as the core. We are working now on this issue and itwill be shown in a future work
A new Algorithm to construct LDPC codes with large stopping sets
Conclusions
A new algorithm to construct an LDPC code from agenerated graph was presented
This graph is generated by making some connections betweenseveral copies of a given core
Since the stopping set of the LDPC code is related to thegirth of the graph, a large stopping set size is obtained
The parity check matrix is quite sparse, then, the generatedLDPC code converges in just a few iterations
It is possible to generate bigger codes using the obtainedgraph as the core. We are working now on this issue and itwill be shown in a future work
A new Algorithm to construct LDPC codes with large stopping sets
Conclusions
A new algorithm to construct an LDPC code from agenerated graph was presented
This graph is generated by making some connections betweenseveral copies of a given core
Since the stopping set of the LDPC code is related to thegirth of the graph, a large stopping set size is obtained
The parity check matrix is quite sparse, then, the generatedLDPC code converges in just a few iterations
It is possible to generate bigger codes using the obtainedgraph as the core. We are working now on this issue and itwill be shown in a future work