The progressive edge-growth algorithm is a well-known procedure to construct
regular and irregular low-density parity-check codes. In this paper, we propose
a modification of the original algorithm that improves the performance of these
codes in the waterfall region when constructing codes complying with both,
check and symbol node degree distributions. The proposed algorithm is thus
interesting if a family of irregular codes with a complex check node degree
distribution is used.Comment: 3 pages, 3 figure

Elkouss, David

Martin, Vicente

Martinez-Mateo, Jesus

English

arXiv

The progressive edge-growth algorithm is a well-known procedure to construct regular and irregular low-density parity-check codes. In this paper, we propose a modification of the original algorithm that improves the performance of these codes in the waterfall region when constructing codes complying with both, check and symbol node degree distributions. The proposed algorithm is thus interesting if a family of irregular codes with a complex check node degree distribution is used

Martinez Mateo, Jesus

Elkouss Coronas, David

Martín Ayuso, Vicente

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Improved Construction of 
Irregular Progressive Edge-Growth Tanner Graphs 
Jesus Martinez-Mateo, David Elkouss, and Vicente Martin 
Abstract—The progressive edge-growth algorithm is a well-
known procedure to construct regular and irregular low-density 
parity-check codes. In this paper, we propose a modification of 
the original algorithm that improves the performance of these 
codes in the waterfall region when constructing codes complying 
with both, check and symbol node degree distributions. The 
proposed algorithm is thus interesting if a family of irregular 
codes with a complex check node degree distribution is used. 
Index Terms—Irregular low-density parity-check codes, pro-
gressive edge-growth algorithm, waterfall region. 
I. INTRODUCTION 
LOW-DENSITY parity-check (LDPC) codes were intro-duced by Gallager in the early 1960s [1]. These codes 
are capacity achieving for many communication channels, and 
though forgotten for years, are nowadays ubiquitous as they 
have been found useful in many communications problems. 
A family of LDPC codes is usually defined by two gen-
erating polynomials, A(x) and p{x). The coefficients of these 
polynomials define the distribution of incident edges to symbol 
and check nodes respectively. Richardson et al. showed that 
the asymptotic behaviour of a family of codes defined by both 
polynomials can be analysed by using the density evolution 
algorithm [2]. Families of LDPC codes performing close to 
the channel capacity can be then designed by optimising both 
generating polynomials [3], [4]. 
An LDPC code is a linear code identified by an sparse 
parity-check matrix or its equivalent bipartite graph, also 
called Tanner graph. It is known that some iterative algorithms, 
such as belief propagation based algorithms, provide optimum 
decoding over cycle-free Tanner graphs [5]. However, any 
finite-length graph has necessarily cycles, and it has been 
shown that a large girth (length of the shortest cycle) improves 
the performance of LDPC codes using iterative decoding as 
it enforces a reasonable minimum distance [6]. Therefore, 
taking into account that a finite-length graph has cycles, it 
is important to make its girth as large as possible. 
The progressive-edge-growth (PEG) algorithm is an ef-
ficient method for constructing Tanner graphs with large 
girth [6], in most cases with better performance than ran-
domly constructed codes. PEG algorithm's interest lies in its 
simplicity, and its flexibility when constructing codes from a 
complex symbol node degree distribution. However, note that 
a large girth does not automatically imply a a large minimum 
distance. The performance of these codes can be improved, 
for instance, in the error floor region. Significant research in 
PEG-based algorithms has been done to achieve this improved 
performance. 
The rest of this paper has been organised as follows. In 
Section II, a modified PEG algorithm is proposed for the 
construction of LDPC codes following both A(x) and p(x) 
degree distributions for symbol and check nodes, respectively. 
In Section III, simulation results are shown. These results 
are compared with the original PEG algorithm and similar 
alternatives. In Section IV, the modified PEG algorithm and 
its applications in the construction of new codes are discussed. 
II. MODIFIED PEG ALGORITHM 
A PEG-based algorithm consists of two basic procedures: a 
local graph expansion and a check node selection procedure. 
Both procedures are executed sequentially in order to construct 
a Tanner graph connecting symbol and check nodes in an 
edge-by-edge manner. In the first procedure it is performed 
the expansion of the local graph from a symbol node, this 
expansion is used to detect and avoid short cycles when adding 
a new edge. The result is that check nodes that will produce a 
cycle are pruned, or if it is not possible to avoid a cycle, there 
only remains a set of candidate check nodes producing the 
largest cycle. The selection procedure is used to reduce this 
list of candidate nodes according to the current graph setting. 
In typical PEG algorithms, this procedure attempts to balance 
the degree of any check node selecting those candidates with 
the lowest check node degree. 
Recent research has been focused on the case when there are 
several candidate nodes after both procedures. At this point, 
it is possible to improve the performance of a PEG-based 
algorithm, for instance avoiding small stopping sets for the 
binary erasure channel (BEC) [7]-[9] or trapping sets in the 
binary symmetric channel (BSC). New definitions have also 
been introduced, such as the extrinsic message degree (EMD) 
or the approximate cycle EMD (ACE) [10], [11], which are 
two common measures used to calculate the connectivity of 
symbol nodes. 
In the original PEG algorithm [6], a code is constructed 
according to a symbol node degree sequence. This sequence 
is previously calculated with the number of symbol nodes n 
and the edge degree distribution established by A(x). Note that 
the original proposal does not take into account the second 
polynomial, p{x), for the check node degree distribution. The 
algorithm proposed in this paper follows both degree distri-
butions during the code construction procedures, changing the 
edge selection criterion, thus obtaining a better approach to 
codes with an irregular degree distribution. 
Lowest check Highest free 
degree check node degree 
initial I r d ( c i ) = 3 f d f e ) = 5 
zig-zag pattern | \ f^) =
 2 \ f(Cj) = 3 
Si s 2 S3 i 1 i 
(a) (b) (c) 
Fig. 1. Tanner graphs illustrating some characteristics of the algorithm. In 
(a) the zig-zag pattern used for 2-degree symbol nodes is shown, (b) and 
(c) show check nodes with different degrees, <¿(c¿) and d(cj), and different 
number of incident edges (partially assigned), d (c¿) = 1 and d (CJ) = 2. 
First check node, c¿, is selected if a lowest check node degree criterion is 
used, while the second node, Cj, is selected if the criterion used is the FCD 
(see Section II-A). 
A. Free Check-node Degree (FCD) Criterion 
The edge selection procedure used in this proposal differs 
from the selection procedure proposed in the original PEG 
algorithm. The graph is analysed to avoid local short cycles, 
however check nodes are not chosen according to its number 
of assigned edges, dfc(c¿), i.e. its current (or partial) degree. 
Instead of this, the check node with the highest differ-
ence between its partial and final-defined degree is chosen, 
/ (CJ) = d(ci) — dk(ci), i.e. the difference between the number 
of currently assigned edges and the total number of edges 
to be assigned. The lowest check node degree procedure is 
replaced by a highest free check node degree (FCD) procedure 
(see Fig. 1). The FCD concept, comes from the concept of 
"sockets" previously described in [2], [12]. 
We introduce the concept of compliance of a constructed 
code as the distance between the distribution of nodes (symbol 
or check nodes) in the code and the pre-established node 
degree distribution. Let pj be the pre-established probability 
distribution for a degree j check node, and p* the actual 
probability of a degree j check node in the constructed graph, 
we calculate the p-compliance of a code as: 
rl=Y^ ¡Pi -Pj\ c1) 
d™ a x 
where p*(x) = J2j=2 PjxJ > anc^ ^™ax = max{d(c¿)}. 
B. Modified Progressive Edge-Growth (PEG) Algorithm 
A modified PEG algorithm is described below. 
Require: d(s¿) < d(sj) Vi < j and /(c¿) = <á(c¿) Vi 
for j = 1 to n do 
for k = 1 to d(sj) do 
if k = 1 then 
if d(sj) = 2 then 
Ej. <— (CÍ,SJ), where Ej. is the first edge 
incident to SJ and c¿ is a check node such that 
it has the lowest check-node degree under the 
current graph setting ESl U ES2 U • • • U ESj_1. 
else 
Ej. <— (CÍ,SJ), where Ej. is the first edge 
incident to SJ and c¿ is a check node such that 
it has the highest free check-node degree. 
end if 
else 
Expand a subgraph from symbol node SJ up to depth 
/ under the current graph setting, such that Afls. = 
Ml+\ or77l.^0but77l+1 = 0. 
Sj Sj Sj 
Eg. <— (CJ, SJ), where E%. is the kth edge incident 
to SJ and c¿ is a check node picked from the set 
J\fs. having the highest free check-node degree. 
end if 
f(Ci) = f(Ci) ~ 1 
end for 
end for 
The same notation as in [6] is used, d(sj) is the SJ symbol 
node degree (The number of incident edges. It corresponds to 
the cardinality of the ensemble ESj after the code construc-
tion), d(ci) is the Cj check node degree, /(c¿) is the number 
of edges that can be added to the check node c¿ under the 
current graph setting, such that dfc(c¿) = /(c¿) — d(c¿), ESj is 
the ensemble of edges connected to the symbol node SJ, Ek. 
the edge added in the step k of the progressive construction, 
and Aí¡. is the ensemble of nodes reached after the graph 
expansion from the symbol node SJ up to depth /. 
A zig-zag construction for 2-degree symbol nodes (see 
Fig. 1) is forced by using a special criterion when adding 
the first edge to a symbol node. In this particular selection, a 
list of eligible check nodes is limited to those check nodes 
already connected under the current graph setting, Es> = 
ESl U ES2 U • • • U ESj_1, i.e. to the list of check nodes that 
have been chosen at least once from the first to j-th step. 
This construction is used to avoid cycles with 2-degree symbol 
nodes, thus obtaining a good performance in the error floor 
region as the results show (see Section III below). 
Relaxed edge selection: The proposed PEG algorithm can 
be modified to work with a relaxed edge selection. In this 
case, if there are not check nodes with free edges in the final 
ensemble of candidate check nodes, 77' , check nodes with 
free edges are searched in the previous candidate ensemble, 
7VS. . This procedure improves the p-compliance, rj, with the 
target check node degree distribution, p(x), at the expense of 
the current local cycle length (see Table I). 
III. SIMULATION RESULTS 
Simulations results have been done for different LDPC 
code construction methods. All constructed codes have a 
codeword length n = 105 and rate one half, R = 0.5. 
Performance has been measured under iterative decoding by 
using belief propagation. The maximum number of iterations 
for the decoder was set to 2000. Five different PEG-based 
construction methods are compared: (1) the original PEG 
algorithm as proposed in [6]; (2) the modified PEG algorithm 
10" 
1(T 
10 
"•-••lx> 
m Í 
-3 
n-4 
¥ N X 
10" 
10-5 r 
^ \ 
10 
10-' 
-6 
-*— Original PEG (1) 
•^— PEG prop, by Richter (2) 
•©•- Prev. 
•••••• PEG proposed here (4) 
~0~" PEG mixed criteria (5) 
% -
- ^ 
« a » — A 
o 
$ 
0.4 0.6 0.8 1 1.2 
SNR E¡/N0 (dB) 
1.4 1.6 
Fig. 2. Frame error rate (FER) as a function of the signal-to-noise ratio 
(SNR) for five construction methods, as explained in the text. Results have 
been obtained with the simulation of transmissions over the AWGN channel. 
Generating polynomials used in the code construction have been extracted 
from Table II (with a maximum symbol node degree of 50) in [2]. 
10"' 
1 0 ' 
g i o - 2 
10"" 
io-: 
10 c 
o 
A — 
*s£^£. 
J Original PEG (1) PEG prop, by Richter (2) Prev. PEG modified (3) PEG proposed here (4) 
PEG mixed criteria (5) 
0.075 0.08 0.085 0.09 0.095 0.1 0.105 
Fig. 3. FER as a function of the crossover probability, e, over the BSC 
for five construction methods (see text). Codes have been constructed using 
the optimised generating polynomials from [4]. Since the p(x) distribution 
is more complex in this set of codes, it is possible to better appreciate the 
differences among the various algorithms used for constructing the codes. 
proposed by Richter in [12]; (3) a modification of (2) by 
inforcing the selection of a check node in the current graph 
when the first edge is added to a symbol node; (4) the modified 
PEG algorithm proposed here; and (5) a mixed version, the 
lowest check node degree criterion is used to connect the first 
edge to a symbol node (not only to 2-degree symbol nodes as 
proposed here) and the FCD criterion is used for the remaining 
edges. 
Figures 2 and 3 show the performance of the codes over the 
additive white Gaussian noise (AWGN) channel and the BSC. 
The error floor is improved using the zig-zag construction for 
2-degree symbol nodes. On the other hand, within a given 
graph setting, when the first check node connected to a symbol 
node rule is used, there is no relevant improvement, as this 
allows for different structures to the zig-zag. 
TABLE i 
p-COMPLIANCE CALCULATED AS DEFINED IN EQ. (1) 
Algorithm 
AWGN 
BSC 
AWGNfl 
BSC 
(1) 
1.495936 
1.938722 
— 
(2) 
0.022604 
0.050631 
0.001482 
0.001623 
(3) 
0.016141 
0.051067 
0.001403 
0.000459 
(4) 
0.020870 
0.057456 
0.001673 
0.001634 
(5) 
0.022998 
0.050050 
0.000806 
0.000510 
"Using relaxed edge-selection. 
irregular symbol degree distribution. Simulation results show 
that this method behaves very well in the waterfall region 
while also maintaining a low error floor. Good performance 
in both regions is relevant as it allows to use the same code 
independently of the working point, e.g. this construction 
might find application in rate compatible solutions. 
ACKNOWLEDGMENT 
Partially supported by project Quantum Information Tech-
nologies Madrid1, P2009/ESP-1594, Comunidad Autónoma de 
Madrid. The authors acknowledge the resources and assistance 
provided by the Centro de Supercomputación y Visualización 
de Madrid2 and the Spanish Supercomputing Network. 
R E F E R E N C E S 
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vol. 8, no. 1, pp. 21-28, Jan. 1962. 
[2] T. Richardson and R. Urbanke, ' T h e capacity of low-density parity-
check codes under message-passing decoding," IEEE Trans. Inf. Theory, 
vol. 47, no. 2, pp. 599-618, Feb. 2001. 
A. Shokrollahi and R. Storn, "Design of efficient erasure codes with 
differential evolution," in Proc. IEEE Int. Sympos. on Inf. Theory, p. 5, 
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no. 1, pp. 386-398, Jan. 2005. 
S.-H. Kim, J.-S. Kim, and D.-S. Kim, "LDPC code construction with 
low error floor based on the IPEG algorithm," IEEE Commun. Lett., 
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Y.-K. Lin, C.-L. Chen, Y.-C. Liao, and H.-C Chang, "Structured LDPC 
codes with low error floor based on PEG tanner graphs," in Proc. IEEE 
Int. Sympos. on Circuits and Systems, May 2008, pp. 1846-1849. 
X. Jiao, J. Mu, J. Song, and L. Zhou, "Eliminating small stopping sets in 
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no. 6, pp. 435^137, June 2009. 
T. Tian, C. Jones, J. D. Villasenor, and R. D. Wesel, "Construction of 
irregular LDPC codes with low error floors," in Proc. IEEE Intl. Conf. 
Comm, vol. 5, May 2003, pp. 3125-3129. 
H. Xiao and A. Banihashemi, "Improved progressive-edge-growth 
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[12] G. Richter, "An improvement of the PEG algorithm for LDPC codes in 
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[3] 
[4] 
[5] 
[6] 
[7] 
[8] 
[9] 
[10] 
[111 
IV. CONCLUSIONS 
In this paper, we proposed an improved PEG algorithm. The 
constructed codes comply with both an irregular check and an 
1
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2http://www.cesvima.upm.es 


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