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1288 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 8, AUGUST 2012
A PEG Construction of Finite-Length LDPC Codes with Low Error Floor
Sina Khazraie, Reza Asvadi, Member, IEEE, and Amir H. Banihashemi, Senior Member, IEEE
Abstract—The progressive-edge-growth (PEG) algorithm of
Hu et al. is modified to improve the error floor performance of the
constructed low-density parity-check (LDPC) codes. To improve
the error floor, the original PEG algorithm is equipped with
an efficient algorithm to find the dominant elementary trapping
sets (ETS’s) that are added to the Tanner graph of the under-
construction code by the addition of each variable node and
its adjacent edges. The aim is to select the edges, among the
candidates available at each step of the original PEG algorithm,
that prevent the creation of dominant ETS’s. The proposed
method is applicable to both regular and irregular variable
node degree distributions. Simulation results are presented to
demonstrate the superior ETS distribution and error floor
performance of the constructed codes compared to similar codes
constructed by the original and other modifications of the PEG
algorithm.
Index Terms—Low density parity-check (LDPC) codes, trap-
ping sets, elementary trapping sets, error floor, progressive edge
growth (PEG) algorithm, code construction.
I. INTRODUCTION AND PRELIMINARIES
THE Progressive-edge-growth (PEG) algorithm of [2] is
well-known for the construction of good low-density
parity-check codes with short to medium block length. Since
the introduction of the PEG algorithm, a number of modi-
fications have appeared in the literature, including [8], [10],
[11]. In [10], the error floor performance of the PEG con-
struction was enhanced for irregular codes using the ap-
proximate cycle extrinsic message degree (ACE) metric [7].
In [8], PEG construction and generalized ACE constrained
design of [9] were combined for even superior error floor
performance. In [11], Zheng et al. devised a modified
PEG algorithm, called PEG Approximate-minimum-Cycle-
Set-Extrinsic-message-degree (PEG-ACSE), to design short
irregular LDPC codes. To lower the error floor of PEG
construction, all the existing literature [8], [10], [11], use an
indirect measure of error floor performance. This measure is
ACE in [8], [10], and ACSE in [11]. In addition, both [10],
[11] are limited to the construction of codes with irregular
variable node degree distributions.
In this work, unlike the existing PEG-based constructions,
we use a direct measure of error floor performance, i.e., the
spectrum of elementary trapping sets (ETS’s). For a trapping
Manuscript received April 17, 2012. The associate editor coordinating the
review of this letter and approving it for publication was M. Flanagan.
S. Khazraie is with the Department of Electrical Engineering, Sharif
University of Technology, Tehran, Iran (e-mail: khazraie@alum.sharif.edu).
R. Asvadi is with the Department of Electrical and Computer Engi-
neering, K. N. Toosi University of Technology, Tehran, Iran (e-mail: as-
vadi@ee.kntu.ac.ir).
A. H. Banihashemi is with the Broadband Communications and Wire-
less Systems (BCWS) Center, Department of Systems and Computer En-
gineering, Carleton University, Ottawa, ON K1S 5B6, Canada (e-mail:
ahashemi@sce.carleton.ca).
Digital Object Identifier 10.1109/LCOMM.2012.060112.120844
set S,weusetheterm“(a, b)trapping set”, if Shas size a
and if the induced subgraph G(S)of Sin the code’s Tanner
graph Gcontains bunsatisfied check nodes. An (a, b)trapping
set is called elementary [6] if every check node in G(S)
has either degree 1 or degree 2. Following the general belief
and supported by much experimental evidence, we assume
that the most harmful trapping sets are elementary and that
among ETS’s those with smaller values of aand bare more
harmful (see, e.g., [6], [11]). These ETS’s are categorically
called dominant.Wedefinethedepth-dETS spectrum of a
Tanner graph Gas the d-tuple (ξ1,...,ξ
d),whereξa,for
1≤a≤d, is the minimum value of bfor all (a, b)ETS’s
in G. To improve the error floor, the goal is thus to achieve
ETS spectrums with components as large as possible. This is
particulary important for the components with smaller indices
(corresponding to smaller ETS’s).
To avoid the creation of dominant ETS’s, we use the original
PEG algorithm of [2] in conjunction with an efficient recursive
search algorithm that can find the dominant ETS’s of the
under-construction Tanner graph at each step of the PEG
algorithm. Among the candidates available for the edges of
a given variable node, we then select those that prevent the
creation of dominant ETS’s (those that satisfy a target ETS
spectrum). Our search algorithm for finding dominant ETS’s
is a modification of the recursive algorithms presented in [3]
and [4], for regular and irregular LDPC codes, respectively.
The direct application of the algorithms of [3], [4] to our
search problem appears to be too complex to implement.
The modifications are made to simplify the original search
algorithms, without sacrificing the accuracy, by tailoring them
to the problem of finding those dominant ETS’s that contain
a specific variable node of the graph (not all the dominant
ETS’s in the graph). In the context of this work, this variable
node is the one that is being added to the Tanner graph at
each step of the PEG algorithm.
The proposed PEG algorithm, referred to as ETS-
constrained PEG, is applicable to codes with both regular and
irregular variable node degree distributions. In Section III, we
demonstrate that the proposed construction provides signifi-
cant improvement over the existing PEG-based constructions
in terms of both the ETS spectrum and the error floor
performance. One should also note that instead of ETS’s,
the proposed algorithm can target other graphical objects of
interest such as absorbing sets [1]. This is due to the fact that
the search algorithms of [3] and [4] can be easily tailored to
find graphical objects other than ETS’s.
II. PROPOSED PEG ALGORITHM
We use the notation vi,1≤i≤n, to denote the variable
node that is being added to the under-construction Tanner
1089-7798/12$31.00 c
2012 IEEE
KHAZRAIE et al.: A PEG CONSTRUCTION OF FINITE-LENGTH LDPC CODES WITH LOW ERROR FLOOR 1289
graph at the current step of the PEG algorithm, where nis the
block length of the code. Corresponding to vi,wedefinethe
graph Giand the integer gias the constructed Tanner graph
at step iof the PEG algorithm (the subgraph of Ginduced
by {v1,...,v
i}) and the size of the shortest cycle(s) in Gi
containing vi, respectively. Similar to [11], we consider ETS’s
Swhose induced subgraphs G(S)are connected, and each of
whose variable nodes is connected to at least two satisfied
check nodes in G(S). Such ETS’s are denoted by Tand are
the only ETS’s that are checked for satisfying the target ETS
spectrum. As the size of the ETS’s in Tis at least g,where
gis the girth of the Tanner graph, we are only interested in
the components of ETS spectra with indices at least equal to
g/2. In the rest of this paper, we thus consider ETS spectra
of the general form ξ=(ξg/2,...,ξ
d).
Starting from a target ETS spectrum ξ,1the general frame-
work of the proposed construction is as follows:
1) Apply the PEG construction of [2] to establish the edges
of the Tanner graph one variable node at a time (select
all the edges connected to the variable node viunder
consideration based on the original PEG algorithm).
2) Using Algorithm A, find the set Siof dominant ETS’s
of Giin Tthat contain vi, and check to see if all the
elements of Sisatisfy the target ETS spectrum, i.e., if
every (a, b)ETS in Siwith gi/2≤a≤dsatisfies
b≥ξa. If they do, then accept the addition of the edges
to the Tanner graph and move to the next variable node
vi+1. If they do not, then select a new set of edges for vi
by randomly choosing another set within the framework
of the PEG algorithm. In this case, continue the process
of selecting new set of edges for viuntil the target ETS
spectrum is satisfied or a maximum number of trials is
reached. In the latter case, declare a construction failure.
3) Continue the construction until all the edges of the
Tanner graph are established or a construction failure
is declared.
Algorithm A, used in Step 2 of the proposed construction,
is responsible for finding those dominant ETS’s of Giof size
up to dthat contain vi. This algorithm is described below.
Algorithm A: Finding the dominant (a, b)ETS’s of Giin T
with a≤dthat contain vi.
1: if the variable node degree distribution is regular, then
2: Inputs: vi,Gi,gi,lmax,β
3: Recursively apply Algorithm 1 of [3] by initializing G
with Gi,kwith d,andthesetLin with all the cycles
of length gi,...,g
i+(2lmax −2) in Githat contain vi.
At each recursion, for each value of a, only the (a,b
)
ETS’s with the βsmallest values of bare considered,
and further expanded.
4: else
5: Inputs: vi,Gi,gi,lmax,TAC E and β
6: Recursively apply Algorithm 3 of [4] (except for the
1To achieve a desirable ETS spectrum in our simulations, we aim at
maximizing the components of the target ETS spectrum in a greedy fashion
starting from components with smaller indices.
command on Line 4) by initializing Gwith Gi,k
with d,andthesetLin with all the cycles of length
gi,...,g
i+(2lmax −2) in Githat contain viplus all
the cycles of larger size up to size 2dcontaining viwith
ACE value up to TACE . At each recursion, for each
value of a, only the (a,b
)ETS’s with the βsmallest
values of bare considered, and further expanded.
7: if at any stage of running Algorithm 1 of [3] (within
the recursions of Algorithm 3 of [4]), an input ETS
contains a 2-chain, then
8: step 8 of Algorithm 1 is only executed for the check
node cthat is connected to a degree-2 variable node
at the tail of the ETS.
9: end if
10: end if
The main idea behind the Algorithms of [3], [4] is to
start from an initial set of ETS’s and then expand them
recursively to larger ETS’s. The initial set for regular codes
contains short cycles of the Tanner graph up to a certain
length, and for irregular codes, in addition to short cycles,
it contains larger cycles with small ACE values and low-
degree variable nodes.2In the proposed PEG construction,
Algorithm A is executed numerous times. It is thus important
to reduce the complexity of the search algorithms of [3], [4].
To achieve this in Algorithm A, we have made the following
two modifications: (i)Instead of initiating the search with all
the short cycles of up to a certain length, we only use the short
cycles that contain vi. This makes the results less accurate if
the maximum cycle length in the initial set is not changed. To
ensure high accuracy, we include longer cycles by choosing a
large enough value for lmax;(ii)Step 4 of Algorithm 3 (the
addition of low-degree variable nodes as part of the initial
ETS’s) is removed. Instead, and to maintain the accuracy of
the algorithm, Steps 7 and 8 are added to Algorithm A. Our
experiments show that in all cases considered in this paper
and a few others, the ETS’s obtained by Algorithm A are
identical to those obtained by the Algorithms of [3], [4]. In
such experiments, the algorithms of [3], [4] are applied to Gi,
and then among all the obtained ETS’s, those that contain vi
are compared to the ones obtained by Algorithm A.
Increasing the values of lmax,βand TAC E, in general,
increases the accuracy of Algorithm A, but at the cost of
higher complexity. In Section III, proper values for these
parameters that strike the right balance between the accuracy
and complexity are given.
III. SIMULATION RESULTS AND DISCUSSIONS
In this section, we report a number of code construction
examples using the proposed algorithm, for both irregular and
regular variable node degree distributions. All the simulation
results presented in this section are for BPSK modulation over
the AWGN channel. The iterative decoding algorithm is belief
propagation in log-likelihood ratio domain with maximum
2Under certain conditions [5], algorithms of [3] and [4] are guaranteed to
find all the dominant ETS’s (up to a certain size and with a certain number
of unsatisfied check nodes). Extensive simulation results are also provided
in [5] to demonstrate the effectiveness of the algorithms.
1290 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 8, AUGUST 2012
number of iterations equal to 50. To obtain each point on the
error rate curves, simulations are run until 100 block (frame)
errors occur.
Example 1: For this example, we design a rate-1/2
(1008,504) code with the variable node degree distribution
λ1(x)=0.25105x+0.30938x2+0.00104x3+0.43853x9.
This distribution is optimized by density evolution and has
also been used in Example C of [11].
We use the proposed PEG algorithm with parameters
lmax =4,β=4,TAC E =4,andd=14to construct a code
with variable node degree distribution λ1(x). The constructed
code has the following ETS spectrum for 6≤a≤14:
ξa=3if 6≤a≤10
2if 11 ≤a≤14
For comparison, we also construct two codes with the same
rate, block length and variable node degree distribution using
the original PEG algorithm of [2] and the generalized-ACE-
constrained PEG algorithm of [8].
Although all the three constructed codes have the same girth
6, they differ greatly in their ETS distribution. The distribution
of (a, b)ETS’s for 6≤a≤14 and b≤2, are reported in
Table I for the three codes. (For 3≤a≤5, none of the codes
has any (a, b)ETS with b≤2.) These results are obtained by
Algorithm 3 of [4]. A careful inspection of the table shows that
although the generalized-ACE-constrained PEG construction
eliminates some of the smaller ETS’s with small values of
bthat exist in the original PEG construction, it still has a
great number of dominant (a, b)ETS’s which are absent in
the ETS-constrained PEG construction proposed in this work.
Monte Carlo simulation results of the three codes are
presented in Fig. 1. This figure demonstrates that the ETS-
constrained PEG code outperforms the other PEG codes,
particularly in the error floor region. This is consistent with
the superior ETS distribution of the former in comparison
with the latter. The ETS-constrained code also outperforms the
similar PEG-ACSE code constructed in Example C of [11].
For example, the code of [11] achieves bit error rate (BER)
values of 1.19 ×10−7and 8.79 ×10−8,atEb/N0values
2.8dB and 3dB, respectively [11], where Ebis the energy
per information bit and N0is the one-sided power spectral
density of the AWGN. The corresponding values for the ETS-
constrained code are 4.76×10−8and 1.30×10−8, respectively.
Example 2: In this example, we design a rate-1/2
(504,252) code with variable node degree 3. With design
parameters lmax =4,β=2,andd= 13, we obtain a code
with the following ETS spectrum for 4≤a≤13:
ξa=⎧
⎨
⎩
5if a=5,7,9
4if a=4,6,8,10,12
3if a= 11,13
Similar codes are also constructed using the PEG algorithms
of [2] and [8]. The ETS distributions and the error rate
curves for all three codes are given in Table II and Fig.2,
respectively. The results in Table II are obtained by using
Algorithm 1 of [3]. (For a=4, none of the codes has
any (a, b)ETS with b≤3.) Note that although all three
codes have the same girth of 8, the ETS-constrained code
has a significantly superior ETS distribution compared to both
1.6 1.8 2 2.2 2.4 2.6 2.8 3
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
Eb/N0 (dB)
FER/BER
FER: Solid lines, BER: Dashed lines
Original PEG [2]
Generalized−ACE−Constrained PEG [8]
ETS−Constrained PEG
Fig. 1. Bit error rate (BER) and frame error rate (FER) performance of the
LDPC codes designed in Example 1.
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
Eb/N0 (dB)
FER/BER
FER: Solid lines, BER: Dashed lines
Original PEG [2]
Generalized−ACE−Constrained PEG [8]
ETS−Constrained PEG
Fig. 2. BER/FER performance of the LDPC codes designed in Example 2.
the original PEG and the generalized-ACE-constrained PEG
codes. Correspondingly, the former code outperforms the latter
in the error floor region.
Example 3: In this example, we construct two rate-1/2
irregular codes with lengths n= 1008 and 2016, both with a
density evolution optimized variable node degree distribution
given by: λ2(x)=0.23802x+0.20997x2+0.03492x3+
0.12015x4+0.01587x6+0.00480x13 +0.37627x14 .This
degree distribution has also been used to design codes of the
same block lengths in Examples A and B of [11], respectively.
For both block lengths, we use the parameters lmax =3,
β=4,TACE =4and d=16in the construction. The
ETS distribution of the ETS-constrained code, in both cases,
is superior to those of the PEG constructions of [2] and [8].
For example, for the (1008,504) code, the ETS spectra of the
ETS-constrained code and the generalized-ACE-constrained
code for 6≤a≤16 are respectively (for a≤5, none of
KHAZRAIE et al.: A PEG CONSTRUCTION OF FINITE-LENGTH LDPC CODES WITH LOW ERROR FLOOR 1291
TAB L E I
DOMINANT ETS DISTRIBUTIONS OF THE (1008,504) CODES CONS TRUCTED BY THE PROPOSED ETS-CONSTRAINED PEG ALGORITHM,THE
ORIGINAL PEG ALGORITHM OF [2] AND THE GENERALIZED-ACE-CONSTRAINED PEG ALGORITHM OF [8]. THE THREE NUMBERS OF EAC H
ELEMENT OF THE TABLE FROM LEFT TO RIGHT ARE THE MULTIPLICITY OF (a, b)ETS’SWITHb=0,1,AND 2,RESPECTIVELY (EXAMPLE 1).
ETS size aOriginal PEG [2] Generalized-ACE-Constrained PEG [8] ETS-Constrained PEG
(6,b< 3) 0; 0; 27 0; 0; 0 0; 0; 0
(7,b< 3) 0; 1; 49 0; 0; 43 0; 0; 0
(8,b< 3) 0; 1; 60 0; 2; 64 0; 0; 0
(9,b< 3) 0; 0; 74 0; 1; 59 0; 0; 0
(10,b<3) 0; 8; 130 0; 0; 90 0; 0; 0
(11,b<3) 0; 13; 254 0; 19; 163 0; 0; 133
(12,b<3) 0; 18; 431 0; 10; 346 0; 0; 411
(13,b<3) 1; 31; 666 1; 23; 597 0; 0; 582
(14,b<3) 1; 49; 1012 0; 35; 887 0; 0; 711
TAB L E I I
DOMINANT ETS DISTRIBUTIONS OF THE (504,252) CODES CONSTRUCTED BY THE PROPOSED ETS-CONSTRAINED PEG ALGORITHM,THE ORIGINAL
PEG ALGORITHM OF [2] AND THE GENERALIZED-ACE-CONSTRAINED PEG ALGORITHM OF [8]. THE FOUR NUMBERS OF EAC H ELEMENT OF THE
TABLE FROM LEFT TO RIGHT ARE THE MULTIPLICITY OF (a, b)ETS’SWITHb=0,1,2AND 3,RESPECTIVELY (EXAMPLE 2).
ETS size aOriginal PEG [2] Generalized-ACE-Constrained PEG [8] ETS-Constrained PEG
(5,b< 4) 0; 0; 0; 14 0; 0; 0; 1 0; 0; 0; 0
(6,b< 4) 0; 0; 0; 0 0; 0; 0; 0 0; 0; 0; 0
(7,b< 4) 0; 0; 0; 57 0; 0; 0; 18 0; 0; 0; 0
(8,b< 4) 0; 0; 5; 0 0; 0; 1; 0 0; 0; 0; 0
(9,b< 4) 0; 1; 0; 156 0; 0; 0; 71 0; 0; 0; 0
(10,b<4) 0; 0; 6; 0 0; 0; 3; 0 0; 0; 0; 0
(11,b<4) 0; 0; 0; 605 0; 0; 0; 366 0; 0; 0; 155
(12,b<4) 0; 0; 25; 0 0; 0; 17; 0 0; 0; 0; 0
(13,b<4) 0; 1; 0; 2824 0; 1; 0; 1523 0; 0; 0; 941
the two codes has any (a, b)ETS with b≤3):
ξa=3if 6≤a≤11
2if 12 ≤a≤16
ξPEG−AC E
a=⎧
⎪
⎪
⎨
⎪
⎪
⎩
3if a=6
2if a=7
1if a=8,9,10,11,13,14
0if a= 12,15,16
As it can be seen, the latter spectrum is much inferior to the
former one. Correspondingly, the error floor performance of
the ETS-constrained code is superior to those of the other
two PEG constructions. Moreover, the ETS-constrained codes
outperform the similar codes constructed in Examples A and
B of [11], in the error floor region, respectively. For example,
for n= 2016 and Eb/N0=2.4dB, the BER of the two codes
are about 6×10−9and 1.46 ×10−8, respectively.
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