A novel algorithm for removing cycles in quasi-cyclic LDPC codes

Abstract

In this paper, an algorithm for removing cycles in quasi-cyclic(QC) LDPC codes is presented. This algorithm can ensure that the code after cycle removal process preserves the quasi-cyclic structure and significantly improves the flexibility in parameter selection (such as the length of the code) of algebraic constructions of QC-LDPC codes. Besides, it has far lower computational complexity than the existing cycle removal algorithm. Experimental results show that this algorithm is very effective in improving the performance of the QC-LDPC codes and can construct code which has better performance than the corresponding binary LDPC code based on IEEE 802.16e standard.

Full text

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A Novel Algorithm for Removing Cycles in Quasi-
Cyclic LDPC Codes
Keke Liu, Zesong Fei, Jingming Kuang, Xiangming Li
School of Information and Electronics
Beijing Institute of Technology
Beijing, P. R. China
liukeke498@126.com, feizesong@bit.edu.cn


Abstract—In this paper, an algorithm for removing cycles in
quasi-cyclic(QC) LDPC codes is presented. This algorithm can
ensure that the code after cycle removal process preserves the
quasi-cyclic structure and significantly improves the flexibility in
parameter selection (such as the length of the code) of algebraic
constructions of QC-LDPC codes. Besides, it has far lower
computational complexity than the existing cycle removal
algorithm. Experimental results show that this algorithm is very
effective in improving the performance of the QC-LDPC codes
and can construct code which has better performance than the
corresponding binary LDPC code based on IEEE 802.16e
standard.
Keywords-Quasi-Cyclic LDPC codes; cycles; large girth;
combination
I.
INTRODUCTION
LDPC codes, first discovered by Gallager [1] in 1962 and
rediscovered by Mackay in 1990’s [2,3], have caught the
attention of Channel Coding Field due to its near-capacity
performance, and the constructions of LDPC codes has been a
focal point of research in LDPC codes. Recently, constructions
of Quasi-Cyclic(QC) LDPC codes has caused wide concern
because QC-LDPC codes enable linear encoding complexity[4]
as opposed to the quadratic complexity in computer-generated
random LDPC codes [5].
The girth is one of key factors affecting the performance of
an LDPC code. Although recent research indicates that a large
girth might result in a relatively higher error floor, but more
simulation results[6] has demonstrated that a relatively larger
girth helps to improve the bit-error rate(BER) performance
within the range of 10-2 to 10-6, and the block-error rate(BLER)
within the range of 10-1 to 10-4, which is sufficient for practical
applications of an LDPC code in many situations.
Consequently the research into construction of an LDPC codes
with relatively large girth(larger than 6) is still very valuable
for practical application.
There have been several approaches, including direct
construction approaches[7,15] and cycle removal based
approaches[9], to remove short cycles in tanner graph and
construct LDPC codes with large girth. However, these
approaches are either not sufficiently flexible in the code
parameter selection due to the constraint on girth, or unable to
preserve the quasi-cyclic structure thereby increasing the
encoding complexity.
In this paper, we present a novel algorithm aimed at cycle
removal of quasi-cyclic LDPC(QC-LDPC) code. Compared to
the existing methods for cycle removal or constructing LDPC
codes with large girth, our new algorithm has the following
advantages: 1) The proposed algorithm ensures that the code
after cycle removal preserves the quasi-cyclic structure, which
is essential for low-complexity encoding, but not taken into
consideration in some other cycle removal methods, such as the
algorithms proposed in [9]. 2) The proposed algorithm
significantly enhances the flexibility in parameter selection of
algebraic constructions of QC-LDPC codes, especially non-
binary QC-LDPC codes constructions. 3) The proposed
algorithm makes good use of the structure of the parity-check
matrix of QC-LDPC code so that it has far lower computational
complexity than the existing cycle removal method with
respect to QC-LDPC codes.
II.
QUASI-CYCLIC LDPC CODES BASED ON CIRCULANT
PERMUTATION MATRICES
The parity-check matrix of a QC-LDPC code from circulant
permutation matrices[10] can be represented by
()(
()(I p I p
⎢
=⎢
⎢
⎢
⎣
where for 11jJ
≤≤− ,1l
≤ ≤
circulant permutation matrix obtained by shifting the p
identity matrix I to the right by
0
j l
pp
≤<
. If we replace some circulant permutation
matrices by pp
×
zero matrices, we obtain a family of QC-
LDPC codes with various degree distributions. For the sake of
convenience, we define pp
×
obvious that for given , J L and p , the parity-check matrix
H is characterized by the following matrix called base matrix:
pp
pp
⎢
=⎢
⎢
⎢
⎣
1,11,21,
2,12,2
?
2,
,1,2
L
,
represents the
)
)
(
(
)
)
()()
− ,
()
L
L
JJ J L
)
I pI p I p
I p
I pI pI p
⎡
⎢
⎤
⎥
⎥
⎥
⎥
⎥
⎦
H
?
?
?
?
1
??
(1)
,
(
j l
I p
p
×

, j l
p
places where
,
zero matrix as ()I −∞ . It is
1,11,2 1,
2,1
?
2,2
?
2,
?
,1,2,
L
L
JJ J L
p
p
ppp
⎡
⎢
⎤
⎥
⎥
⎥
⎥
⎥
⎦
P
?
?
?
?
(2)

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For a given nonzero element in a circulant permutation
matrices, we define its adjacent nonzero element as the nonzero
element in the row below it(If the given nonzero element is in
the last row of the circulant permutation matrix, its adjacent
nonzero element is the one in the first row). For QC-LDPC
codes, if several nonzero elements in the parity-check matrix
constitute a cycle of length 2l (called cycle 1), we readily
know that their respective adjacent nonzero element also
constitute a cycle of length 2l (called cycle 2). We called cycle
2 the adjacent cycle of cycle 1 and define the mapping from a
cycle to its adjacent cycle as adjacent cycle mapping. Based on
the fact above, we conclude that in QC-LDPC code, the cycles
exist in the form of group, called cycle-group. Given a cycle in
a cycle-group, we can obtain any other cycle in this cycle-
group by exerting adjacent cycle mapping on this cycle several
times. That is to say, in a cycle-group, one cycle can
completely characterize all the cycles in this cycle-group. This
conclusion is peculiar to QC-LDPC code and lays the
theoretical foundation for our new low-complexity cycle
removal algorithm. Additionally, if one or more elements in a
given cycle is included in a sub-matrix A , we say that this
cycle passes sub-matrix A .
III.
THE LOW-COMPLEXITY CYCLE REMOVAL ALGORITHM
FOR QC-LDPC CODES
A. The Low-Complexity Cycle Removal Algorithm
Our proposed cycle removal algorithm can be divided into
two basic steps: the step of searching cycles and the step of
removing cycles. We can divide the parity-check matrix H
into several sections and each section is a row of circulant
permutation matrices in H . For 1 i
cycle-set ( )C i formed by all the cycles that pass the first row
of the i th section of H but don’t pass any row above the i th
section of H , in other words, ( ) C i is formed by cycles which
pass the first row of the i th section and are included in the last
1Ji
− − sections of H . Then we get a series of cycle-sets,
denoted by
(1), (2),..., ( )CC C J , which have the following
properties: 1) any two cycle-sets don’t have any cycles in
common; 2) any two cycles, whether belong to the same cycle-
set or not, belong to the different cycle-groups; 3) For any
given cycle-group, there exists a cycle-set in which we can find
a cycle in the given cycle-group. These three properties
indicate that each cycle in a cycle-set corresponds to a cycle-
group. Let C be the union of
elements in C are all the cycles we need to search.
J
≤ ≤
, we construct a
(1), (2),..., ( )CCC J
. The
For a given check node r in the tanner graph, we construct
a tree which has the following properties: 1) r is the root node
of the tree; 2) For an arbitrary node of the tree, its child nodes
are all the nodes connected with it in the tanner graph except its
parent node. We call such a tree the check node r ’s
neighborhood tree, as shown in Fig.1(b). If the check node r
is included in a cycle of length 2l (the cycle consists of the red
edges in Fig.1(a) and we take
nodes in this cycle connected with r in the tanner graph can be
both regarded as the node at the 1st level and the node at the
2l =
for example), the two
(2
appear in the 2l th level of its neighborhood tree for two times,
and there must exists two path of length 2l in the
neighborhood tree of r corresponding to the cycle, which start
from the root node (the two paths are shown in Fig.1(b) by red
edges) . These two paths represent the same cycle, so we only
need to select one path between them, called cycle-path, such
that in this path the variable node at the 1st level has a smaller
label number than the node at the ( 2
neighborhood tree(the thicker one in the two paths). Now we
propose the algorithm for searching cycles in the set C .
1)l −
th level of r ’s neighborhood tree, hence r must
1l − )th level of r ’s
1234
123
Variable Nodes
Check Nodes

Figure 1. (a) Tanner Graph with cycles of length 4
1
1
3
23
34
223
12
122
132
4th level
3rd level
2nd level
1st level
0th level
cycle-path

Figure 1. (b) The neighborhood tree of No.3 Check node
The Proposed Algorithm for Searching Cycles
Step 1. Set a counter t and let
0t =
.
Step 2. Construct a tanner graph corresponding to sub-matrix
consisting of the last J
( )G t
).
t
−
sections of H (call it
Step 3. Select the check node corresponds to the first row of
the last Jt
−
sections of H , and construct its
neighborhood tree corresponding to
( )T t
).
( )G t
(call it
Step 4. Identify all the cycle-paths in
corresponding to the cycle-paths form the set (
( )T t
, and the cycles
C t +
1)
.
Step 5. If
STOP.
1tJ
≠−
,
1tt
=+
and go to Step 2; else

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In the above algorithm, Step 2 and 3 ensures that any two
sets of (1), (2),..., ( )CC C J don’t have any cycles in common
and that there is a one-to-one correspondence between all the
found cycle-paths and the cycles in C .
Before we propose our algorithm for removing cycles, we
first label all the cycles C and all the s
permutation matrices(including all-zero matrices) in H , then
we introduce an auxiliary binary matrix M with size of c s
whose rows correspond to all the c cycles in C and whose
columns correspond to all the s circulant permutation matrices
in H . The element at the intersection of the i th row and the
j th column of L is equal to ‘1’ if and only if the i th cycle in
C passes the j th circulant permutation matrix in H . Now we
discuss the algorithm for removing cycles in QC-LDPC codes.
J L
⋅=
circulant
× ,
The Proposed Algorithm for Removing Cycles
Step 1. Construct the auxiliary binary matrix M.
Step 2. Find the number i , such that i th column of M is
the column with the largest column weight of M.
Step 3. Find the circulant permutation matrix in H
corresponding to i th column of M , and replace it by
zero matrix with the same size.(We call this step ‘delete
the circulant permutation matrix’)
Step 4. For each j such that
elements ‘1’ in j th column of M by zeros.
,
1
j i=
M
, replace all the
Step 5. If M is not an all-zero matrix, go to Step 2; else
STOP.
Now we explain this cycle removal algorithm in details.
Step 1 is the initialization of our algorithm and the binary
matrix M characterize the cycle distribution of the quasi-
cyclic parity-check matrix H . The purpose of Step 2 and Step
3 is to find out and delete the circulant permutation matrix that
is passed by the most cycles in C . In fact, cycle removal
process will inevitably reduce the column weight of the parity-
check matrix, thereby reducing the minimum distance of the
code. If the code’s minimum distance is too small, it might
have a relatively higher error floor. Consequently, in order to
prevent higher error floor, we need to achieve the girth we need
by deleting as few circulant permutation matrices as possible.
While step 2 can ensure that the number of deleted circulant
permutation matrices is minimized. Since the circulant
permutation matrix selected in Step 3 is deleted, the cycles in
C passing it are broken. Step 4 is to modify the auxiliary
binary matrix M in order to make it correspond to the parity-
check matrix H after Step 3. If all the cycles in C are
removed, we terminate the cycle removal process, otherwise
we continue the cycle removal process by going back to step 2
until all the cycles are removed.
B. The Analysis of Comlexity of Our Cycle Removal
Algorithm
From the detail of our algorithm, we know that there are
two reasons why our algorithm has low complexity compared
to the conventional cycle removal algorithm. 1)our algorithm
don’t need to identify all the cycles in H , instead we only
need to identify one cycle in each cycle-group, which can
totally characterize all the cycles. 2) The construction of the
series of sets
(1), (2),..., ( )
CC
repetitive search of one cycle.
C J
successfully avoid
The computational complexity is mainly decided by the
process of searching cycles because in this process we have to
traverse almost all the path of length 2l in the neighborhood
tree. According to the process of searching paths in the tree, we
know that the computational complexity of searching paths in
the neighborhood tree is linearly proportional to the number of
branches of the neighborhood tree from the root to the 2l th
level. We assume that the column and row weight of H are J
and L , respectively. In other words, none of the circulant
permutation matrices H is all-zero matrix. For a regular tanner
graph with check node degree ρ and variable node degree γ ,
although there areρ branches connecting the root check node
to the variable nodes of the 1st level, we don’t need to search in
the paths including the last branch connecting to the root node
because in the cycle-path the 1st variable node can’t have the
largest label number in the path. Hence the total number of
branches of the neighborhood tree from the root to the 2l th
level can be calculated as below:
1
11
γ
1
1
1
( , ,2 )
ρ γ
( 1) (1)(1) (1)
(1)
(
ρ
(
1)(
−
1)
−
( 1)
( 1) (1)
1) 1
−
ll
iiii
ii
ll
l
ii
i
Nl
ργργ
ρρ
γργγ
γ
−
==
+
−
=
=−−+−−
−−−−
=−−=
∑∑
∑
(3)
In (3), the first term corresponds to the branches from 2i th
level to (21)i +
th level(
0,1,...,i
=
tree, and the second term corresponds to the branches from
(21)i −
th level to 2i th level(
neighborhood tree. In our algorithm for searching cycles, the
degree of both the check node and variable node in the tanner
graph
( )G t
decrease as t increases. In fact, for the tanner
graph
( ) G t
, the total number of branches of the
corresponding neighborhood tree from the root to the 2l th
level is
(, ,2 )N Lt J t l
−−
for the reason that as the counter t
increases by 1, both the degree of check node and variable node
decrease by 1. Hence in the whole process of searching cycles
in our algorithm, the number of the branches in all the t tanner
graphs
(0),...,(1)GG J −
calculated as below:
1l
− ) of the neighborhood
1,2,...,il
=
) of the
, denoted by
(2 )Ql
can be
11
1
001
(2)Q l(,,2)()(1) (1)
J
∑
J
∑
l
ii
tti
N L t J t l
−
J t
−
L t
− −
J t
− −
−−
−
===
=−=
∑
(4)
For comparison, we also provide complexity analysis of the
cycle removal approach proposed in [10]. In this approach, the
process of identifying all the ‘elementary paths’ is equivalent to
the process of constructing the neighborhood tree of each check
node in the tanner graph corresponding to H and identifying
all the cycle-paths. For the reason that this approach need to

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identify all the cycles in H , the total number of the branches
in the process of searching cycles denoted by
to:
'(2 )Ql amounts
1
1
'(2 )( 1) ( 1)
l
ii
i
Ql pJ JLJ
−
=
=⋅−−
∑
(5)
where pJ is the number of rows of the parity-check matrix H .
Through comparison between (4) and (5) we know that
'(2 )(2 )QlQ l
>>
as long as p is not too small. For example,
let
63, 4,8pJL
=== and
l =
while
'(2 ) 155232Ql =
, almost 165 time as large as (2 )
Consequently, we see that our algorithm can greatly reduce the
complexity of cycle removal. Table I shows the complexity
comparison between our algorithm and the algorithm proposed
in [9] with some specific pairs of column and row weights of
H when removing cycles of length 6.
2
, then we get (2 )914Q l =
,
Q l .
TABLE I.
THE COMPUTATIONAL COMPLEXITY COMPARISON BETWEEN
OUR ALGORITHM AND THE ALGORITHM PROPOSED IN [9]
Size of
circulant
matrices p
Column
weight
J
Row
weight
L
Complexity
of our
algorithm
Complexity
of
algorithms
in [9]
127 4 12 63688 25101296
63 3 6 1836 314685
31 4 16 163176 15408240
C. Simulation Results
In all the examples to be given in this paper, we assume
BPSK transmission over AWGN channel. We adopt SPA
decoding for binary LDPC codes and FFT-QSPA decoding for
non-binary LDPC codes with the maximum number of
iterations set to 50.
Example 1: We choose two QC-LDPC codes, one
code(called Code C1) is a (4113, 2288) code with rate 0.556,
constructed in [12], the other code(called code C2) is a 64-ary
(756,519) QC-LDPC codes with rate 0.687 constructed in [13].
By our algorithm for searching cycles, we find that C1 has a
girth 8 and C2 has a girth 6. Applying our cycle removal
algorithm respectively in these two codes, we obtain a new
binary (4113, 2285) QC-LDPC code with rate 0.555 and girth
at least 10 (called code C1’), and a new (756, 504) 64-ary QC-
LDPC code with rate 0.667 and girth at least 8(called code
C2’). The performances all the codes are shown in Fig.2, from
which we see that at BER of 10-6, C1’ with girth 10
outperforms C1 with girth-8 by around 0.4dB, and that taking
off the 0.1dB Shannon limit gap between C2 and C2’, C2’
with girth 8 outperforms C2 with girth 6 by around 0.4dB.
IV. CONSTRUCTION OF NON-BINARY QUASI-CYCLIC LDPC
CODES BY COMBINATION OF PARITY-CHECK MATRICES USING
CYCLE REMOVAL ALGORITHM
Algebraic tools, such as finite fields and combinatorial
design have shown their effectiveness in constructing Quasi-
Cyclic LDPC Codes. Taking finite fields for example, many
classes of finite-field based QC-LDPC codes have been
proposed, such as codes based on additive subgroups,
multiplicative subgroups, primitive elements, affine mappings
[11-13], multiplicative inverses[14] and so forth.
0.511.522.5
EbN0dB
33.54 4.55
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Error Rate


C2(girth6)BER
C2(girth6)FER
C2'(girth8)BER
C2'(girth8)FER
Shannon Limit(C2')
Shannon Limit(C2)
Uncoded BPSK
C1'(girth10)BER
C1'(girth10)FER
C1(girth8)BER
C1(girth8)FER
Shannon limit(C1&C1')

Figure 2. The performances of codes in Example 1
However, there exists a main problem with all of these
classes of codes: the parameter selection, especially the code
length is not sufficiently flexible, due to the constraint on the
size of base matrix. For binary LDPC codes, the problem can
be solved by using the finite fields of larger size for code
construction (the maximum size of the base matrix is directly
decided by the size of finite field we use). However, for non-
binary codes, this solution doesn’t make sense. For example, if
we construct a 16-ary QC-LDPC codes based on finite fields,
any of the proposed classes of finite field based codes can only
provide the base matrix with size no larger than 16 16
thereby limiting the maximum code length to 16 15
If we choose a field of larger size, such as
increase the maximum code length but the corresponding code
we construct is a 32-ary code rather than a 16-ary code. In
other words, if we want to keep the size of the field which the
code is over not increasing, choosing the fields of larger sizes is
not effective in increasing the maximum code length.
×
240
,
.
×=
5
(2 )GF
, we can
In this section, we present an approach for constructing
non-binary QC-LDPC code applying our cycle removal
algorithm. First we construct a series of quasi-cyclic parity-
check matrices with girth at least 6, denoted by
all of which have the same number of rows and are arrays of
circulant permutation matrices with the same size. Arranging
H ,H ,...,H in a row, we obtain a new parity-check matrix
H =[HHH ]
?
. The properties of
mentioned above ensure that H is still a quasi-cyclic parity-
check matrix. However, H usually has short cycles, such as
cycles of length 4, which could cause serious performance
degradation. Consequently, we use our cycle removal
12n
H ,H ,...,H ,
12n
12n12n
H ,H ,...,H

Page 5

algorithm to remove the short cycles to obtain desirable
performance.
The combination of several quasi-cyclic parity-check
matrices enables constructing longer non-binary QC-LDPC
codes over fields of small size..
Example 2: In this example we construct a QC-LDPC code
over
(2 ) GF
. First we construct two parity-check matrix
H ,H , respectively by multiplicative subgroups[13] and
GF
15 15
×
arrays of 15 15
×
circulant permutation matrices. We
choose the 8 12
×
sub-array at the upper right corner of
denoted by
1
H , and the 8 12
×
corner of
2
H , denoted by
2
H . Arranging
we obtain an 8 24
×
array of 15 15
matrices denoted by H(0) containing cycles of length 4.
Applying our cycle removal algorithms in H(0)for two times,
we obtain two new parity-check matrix H(1) and H(2). The
null space of H(1) gives a 16-ary (360,240) QC-LDPC code
with girth 6, and the null space of H(2) gives a 16-ary
(360,240) QC-LDPC code with girth 8. The performance of the
two (360,240) QC-LDPC codes are shown in Fig.4. We see
that at BER of 10-6, the girth-8 (360,240) code obviously
outperforms girth-6 (360,240) code by 1.25dB. To prove the
excellent performance of the constructed girth-8 code, we also
include the performance of a binary (1440, 960) LDPC code
constructed based on IEEE 802.16e standard, which has the
same rate and equivalent binary code length as our girth-8 code.
We see that at BER of 10-6, our girth-8 (360,240) 16-ary code
outperforms the binary (1440, 960) LDPC code based on IEEE
802.16e standard by 0.15dB, and that at FER of 10-4, our girth-
8 code outperforms the corresponding IEEE 802.16e code by
0.2dB.
4
12
multiplicative inverses[14] in
4
(2 )
. Both
1
H and
2
H are
1
H ,
'
sub-array at the upper left
H ,
×
circulant permutation
''
1
'
2
H in a row,
11.522.533.54 4.555.56
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Eb/N0(dB)
Error Rate


Uncoded BPSK
QC(360,240)girth8BER
QC(360,240)girth8FER
802.16e(1440,960)BER
802.16e(1440,960)FER
Shannon limit
QC(360,240)girth6BER
QC(360,240)girth6FER

Figure 3. The performances of codes in Example 2
V.
CONCLUSIONS
In this paper, we have presented an algorithm for removing
cycles in QC-LDPC codes. Taking full advantage of the
structure of the parity-check matrix of QC-LDPC code, our
algorithm has far lower computational complexity than the
existing algorithm and preserves the quasi-cyclic structure after
cycle removal process, which is very important for low-
complexity encoding. Simulation results shows that this
algorithm is very effective in improving the performance of the
QC-LDPC codes and allows for constructing longer QC-LDPC
codes with large girth and far more flexible parameter selection.
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