Abstract-Progressive edge-growth (PEG) algorithm was proven to be a simple and effective approach to design good LDPC codes. However, the Tanner graph constructed by PEG algorithm is non-structured which leads the positions of &apos;s of the corresponding parity check matrix fully random. In this paper, we propose a general method based on PEG algorithm to construct structured Tanner graphs. These hardware-oriented LDPC codes can reduce the VLSI implementation complexity. Similar to PEG method, our CP-PEG approach can be used to construct both regular and irregular Tanner graphs with flexible parameters. For the consideration of encoding complexity and error floor, the modifications of proposed algorithm are discussed. Simulation results show that our codes, in terms of bit error rate (BER) or packet error rate (PER), outperform other PEG-based LDPC codes and are better than the codes in IEEE 802.16e

Chih-Lung Chen

Hsie-Chia Chang

Yen-Chin Liao

Yi-Kai Lin

CiteSeerX

Structured LDPC Codes with Low Error Floor
Based on PEG Tanner Graphs
Yi-Kai Lin, Chih-Lung Chen, Yen-Chin Liao, and Hsie-Chia Chang
Department of Electronics Engineering
National Chiao Tung University,
1001 Ta-Hsueh Road, Hsinchu, Taiwan, R.O.C.
Tel: +886-3-5131369 email: hcchang@si2lab.org
Abstract— Progressive edge-growth (PEG) algorithm was
proven to be a simple and effective approach to design good
LDPC codes. However, the Tanner graph constructed by PEG
algorithm is non-structured which leads the positions of ’s of
the corresponding parity check matrix fully random. In this
paper, we propose a general method based on PEG algorithm
to construct structured Tanner graphs. These hardware-oriented
LDPC codes can reduce the VLSI implementation complexity.
Similar to PEG method, our CP-PEG approach can be used to
construct both regular and irregular Tanner graphs with flexible
parameters. For the consideration of encoding complexity and
error floor, the modifications of proposed algorithm are discussed.
Simulation results show that our codes, in terms of bit error rate
(BER) or packet error rate (PER), outperform other PEG-based
LDPC codes and are better than the codes in IEEE 802.16e.
I. INTRODUCTION
Low-density parity-check (LDPC) code, a linear block code
defined by a very sparse parity check matrix, was first invented
by Gallager in 1960’s [1]. However, it has been ignored
for about thirty years. The rediscovery of LDPC codes was
done by MacKay and Neal [2]. An LDPC code with long
block length shows good capacity-approached capability under
iterative decoding algorithm, so it attracts much research
interests in recent years. In practical applications, construction
of good LDPC codes at short to intermediate block length is
of great importance.
PEG method was shown to be an effective one to construct
Tanner graphs, or equivalently parity check matrices, in an
edge-by-edge manner [3]. It maximizes the local girth for the
proceeding variable node when adding an edge into the current
graph. Code parameters specified in PEG method are highly
flexible, so they can be chosen for the practical applications.
However, the positions of 1’s of parity check matrix con-
structed by PEG algorithm are fully random and this makes it
incur higher complexity for VLSI design. A structured LDPC
code reduces both encoder and decoder complexity and is
suitable for hardware implementation. The recently proposed
communication standard [4] adopts structured LDPC codes
as error-correcting codes. Z. Li et al. [5] added a constraint
into original PEG algorithm to construct a class of structured
LDPC codes, named PEG-QC LDPC codes. It had shown
performance comparable to several existing methods at high
code rate. However, through simulation, we find that PEG-
QC algorithm is much easier to construct a Tanner graph with
small girth which degrades the error-correcting performance.
This makes it sometimes not suitable for usage.
In this paper, we propose a general method based on PEG
algorithm to construct structured Tanner graphs. Simulation
results show that the proposed algorithm can suppress the
probability to generate a graph with short cycles. Moreover, we
present that code performance of our LDPC codes outperforms
that of codes based on PEG-QC algorithm. In addition, for
the consideration to encoders, a simple modification of the
proposed algorithm is given. In order to lower the error floor
of irregular codes, an additional criterion is combined into our
proposed algorithm.
The remainder of this paper is organized as follows. Section
II introduces the necessary definitions and notations on graphs
and describes the proposed algorithm. In Section III, we
give the modifications of proposed algorithm. The simulation
results will be presented in Section IV. Finally, Section V
concludes this paper.
II. PROPOSED STRUCTURED LDPC CODES
Using the same notations and definitions as in [3], we
further give several definitions and notations which will be
used, hereafter. Assume there are variable nodes and
check nodes in the Tanner graph. We partition the variable
nodes and check nodes into subgroups where each group
has nodes. After partition it forms variable node subgroups
and check node subgroups. We also partition edges
in term of variable node subgroup as
, with containing all edges emanate
from th variable node subgroup . Check node subgroup
is connected to variable node subgroup if there
is an edge , where ,
, , , and
. Moreover, if check node subgroup
is connected to variable node subgroup under current
graph setting, check nodes contained in form the set
. Its complementary set, , is defined as the check-
node set , i.e. excludes . is the
complementary one of set of all check nodes reached by a
978-1-4244-1684-4/08/$25.00 ©2008 IEEE 1846
tree spreading from variable node within depth . A
complementary set , where
, is a set of candidates. Finally, a variable-subgroup degree
sequence is defined as ,
where . In a variable-subgroup
degree sequence, denotes the degree of , where
and .
Here, we describe how to design a Tanner graph using
proposed algorithm. First, we specify the number of variable
node subgroups and the number of check node subgroups ,
with each subgroup has nodes. For the specified variable-
subgroup degree sequence , construction of the Tanner
graph is described by the following pseudo-code which adds
edges to Tanner graph group by group instead of node by node.
CP-PEG Algorithm:
for to do
for to do
if
, where is a check node with
lowest check-node degree under the current sub-
graph setting. (randomly pick one if such check
nodes are more than one)
else
Expand a tree from variable node up to depth
under the current graph setting such that the cardi-
nality of stops increasing but is less than ,
or but , then ,
where is a check node picked from having
the lowest check-node degree. (randomly pick one
if such check nodes are more than one).
for to do
.
In the above algorithm, is the th edge incident on
and we call this else-statement as tree spreading procedure
for convenience. The corresponding of the generated graph
is a structured matrix as
...
...
. . .
...
(1)
where is a circulant permutation (CP) or all-
zero matrix. A circulant permutation matrix is a square matrix,
where each row vector contains one and is rotated one
element to the right relative to the preceding row vector. Due
to the circulant permutation form of , our proposed LDPC
codes are suitable for layered decoding [6] which speeds up
the convergence rate of iterative decoding about two times.
Table I gives four sets of code parameters whose degree
distribution are optimized from density evolution [7]. For
irregular ones, the maximum variable-node degree are
for both rate and and for rate . We construct
TABLE I
SETS OF CODE PARAMETERS WHICH WILL BE USED HEREAFTER
Index Type Rate Dimension of Size of
I irregular
II irregular
III irregular
IV (3,27)-regular
TABLE II
OCCURRENCE PROBABILITIES OF CODES WITH VARIOUS GIRTH
CONSTRUCTED BY PROPOSED CP-PEG AND PEG-QC ALGORITHMS
Index Algorithm Prob. of girth=4 Prob. of girth=6
I
PEG-QC
CP-PEG
II
PEG-QC
CP-PEG
III
PEG-QC
CP-PEG
IV
PEG-QC
CP-PEG
code ensembles by proposed CP-PEG and previously men-
tioned PEG-QC algorithm. Each code ensemble contains one
thousand LDPC codes. Table II shows the occurrence prob-
abilities of codes with various girth. We observe that Tanner
graphs constructed by proposed CP-PEG algorithm have lower
probabilities to contain short cycles than those by PEG-QC
algorithm. Explanation for this phenomenon, submatrices of
constructed by PEG-QC algorithm are of circulant form. A
matrix with circulant submatrices sometimes induces short
cycles for its corresponding Tanner graph. Here, we give an
example in equation (2).
(2)
where the positions of ’s in entries , , , and
form a -cycle. However, in our CP-PEG algorithm, we
construct with circulant permutation submatrices.
III. MODIFICATIONS OF CP-PEG ALGORITHM
A. Approximate Lower Triangular (ALT) Form
Assume that the parity check matrix is in approximate lower
triangular (ALT) form [8], [9] as shown in Fig. 1. Let
is a codeword of where denotes the systematic
part, and combined denote the parity part, has length
, and has length . Given a message ,
the systematic encoding can be achieved by
(3)
and
(4)
where is assumed to be nonsingular.
1847
 (m-)p
I
I
I
M= m p
N= n p
(n-m) p
g = p
p  (m- ) p
 (m- ) p
Fig. 1. The parity check matrix in approximate lower triangular form.
For the encoding consideration, we can restrict the parity
check matrix generated by proposed CP-PEG algorithm in
ALT form and assign the variable nodes with higher degree to
be the systematic part to get better protection from noises.
This can be accomplished by the following pseudo-code.
ALT-CP-PEG Algorithm:
for to do
for to do
if
if
.
else
, where is a check node with
lowest check-node degree under the current sub-
graph setting.
else
if
Setup that are con-
nected to .
Expand a tree from variable node up to depth
under the current graph setting such that the cardi-
nality of stops increasing but is less than ,
or but , then ,
where is a check node picked from having
the lowest check-node degree.
for to do
.
In the encoding procedure for codes with ALT form, the
overall complexity to determine and are O( ) and
O( ), respectively. To reduce the complexity, we choose (or
) as small as possible under degree distribution setting .
Then, after constructing a parity check matrix , we perform
block-column permutation to make nonsingular and keep
the resulting structured. This is usually possible when
is not rank deficient. By the way, when operating the block-
column permutation, is made to be an identity matrix if
possible. This can further reduce the complexity because of
.
B. EMD Criterion
Comparing with regular codes, irregular codes with op-
timized degree distribution often suffer higher error floor.
Stopping sets in a Tanner graph are defined below. It was
proven that preventing small stopping sets can avoid small
minimum distance [10]. Moreover, small minimum distance
degrades performance in error-floor region.
Definition 1: ( Stopping set) A variable node set is called
an set if it has elements and all its neighbors are
connected to it at least twice.
In order to avoid small stopping sets, we have to make the
subsets of variable nodes have as many extrinsic edges as pos-
sible. There are two quantities to define the number of extrinsic
edges, extrinsic message degree (EMD) and approximate cycle
EMD (ACE).
Definition 2: The EMD of a variable node set is the number
of check nodes that singly connected to this variable node set.
Definition 3: The ACE of a length cycle is ,
where is the degree of the th variable in this cycle.
In the tree spreading procedure of proposed CP-PEG al-
gorithm, among all check-node candidates with the same
check-node degrees, we choose one at random. For irregular
PEG Tanner graph, similar to above situation, [11] suggests
us to choose the check node that maximizes the minimum
ACE for the new cycles. This causes the improvement in
error floor region. However, it may be still more than one
candidate after adopting this criterion. In [12], it further gives
an EMD criterion which improving the performance in error
floor region further. We adopt the EMD criterion into our code
construction procedure to choose a proper check node from
, and confirm that it also lowers the error floor of our
codes.
IV. CODE PERFORMANCE
In this section, we show simulation result of the proposed
algorithm and give some comparisons. The log-BP algorithm
is used with maximum number of iteration 80 in binary-input
AWGN channel.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
10-6
10-5
10-4
10-3
10-2
10-1
100
Eb/N0 (dB)
E
rro
r r
at
e
80 iterations 
BER(2560,1280) PEG-QC
BER(2560,1280) ALT-CP-PEG
PER(2560,1280) PEG-QC
PER(2560,1280) ALT-CP-PEG
Fig. 2. Performance comparison of the irregular codes with rate
constructed by proposed and PEG-QC algorithm.
1848
In Fig. 2 with block length 2560 and rate , the code
constructed by proposed ALT-CP-PEG algorithm shows better
performance, in terms of bit error rate (BER) or packet error
rate (PER), than that based on PEG-QC algorithm. Girths of
these two codes are both .
0 0.5 1 1.5 2 2.5 3 3.5
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Eb/N0 (dB)
E
rro
r r
at
e
80 iterations 
BER(2560,1920) PEG-QC
BER(2560,1920) ALT-CP-PEG  w/o_EMD
BER(2560,1920) ALT-CP-PEG  w_EMD
PER(2560,1920) PEG-QC
PER(2560,1920) ALT-CP-PEG  w/o_EMD
PER(2560,1920) ALT-CP-PEG  w_EMD
Fig. 3. Performance comparison of the irregular codes with rate
constructed by proposed and PEG-QC algorithm.
Fig. 3 consists of three codes, all of them are of length
, rate , and girth . As the figure shown, our codes
outperform that based on PEG-QC algorithm. Especially it
shows a PER improvement of one order in magnitude in high-
SNR region. Moreover, we compare the ALT-CP-PEG LDPC
code with EMD criterion to that without EMD criterion. The
former is low-error-floor and shows better performance in
high-SNR region. It confirms that the EMD criterion is also
suitable for our proposed algorithm.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10-6
10-5
10-4
10-3
10-2
10-1
100
Eb/N0 (dB)
E
rro
r r
at
e
80 iterations
BER of proposed (2304,1152), girth = 8
BER of IEEE 802.16e (2304,1152), girth = 6
PER of proposed (2304,1152), girth = 8
PER of IEEE 802.16e (2304,1152), girth = 6
Fig. 4. Performance comparison of the irregular codes with rate
constructed by proposed algorithm and of IEEE 802.16e.
In the end of this section, we compare performance of
the code based on proposed method to irregular LDPC code
adopted in IEEE 802.16e [4]. Here, these two codes have the
same degree distribution pairs and sizes of submatrices of .
Both of them are of length with rate and in ALT
form. Girth of the code constructed by our algorithm is ,
however, that of the code in IEEE 802.16e is . As shown
in Fig. 4, we can see that code performance of our code is
slightly better than that of IEEE 802.16e. Moreover, because
of the ALT form, the complexity of our encoder is similar to
that of IEEE 802.16e.
V. CONCLUSION
In this paper, we propose a general method, called CP-PEG
algorithm, to construct hardware-oriented LDPC codes which
reduce complexity of VLSI design. Moreover, code parame-
ters, including rate, block length, and degree distribution, are
more flexible in proposed algorithm than in other algebraic
methods, which makes proposed algorithm practical. Modified
algorithms, ALT-CP-PEG algorithm and CP-PEG algorithm
with EMD criterion, are also presented to reduce encoding
complexity and lower error floor, respectively.
Simulation results confirm that proposed structured LDPC
codes are better than codes based on PEG-QC algorithm
in terms of BER or PER. Comparing with the LDPC code
adopted in recently proposed communication standard, code
performance of ours outperforms that of IEEE 802.16e. Fi-
nally, for the convergence rate consideration, our structured
codes are much suitable to decode using layered decoding
to achieve around two times faster decoding convergence.
Because of the abovementioned advantages, proposed CP-PEG
algorithm can be a good candidate for designing practical
LDPC codes.
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Structured LDPC codes with low error floor based on PEG Tanner graphs

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Structured LDPC codes with low error floor based on PEG Tanner graphs

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