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Design of Binary LDPC Codes with Parallel Vector Message Passing

December 2017
IEEE Transactions on Communications PP(99):1-1

December 2017
PP(99):1-1

DOI:10.1109/TCOMM.2017.2783624

Authors:

Xingcheng Liu

Sun Yat-Sen University | Guangzhou Xinhua University

Shuo Liang

Sun Yat-Sen University

Many studies were carried out for the construction of low density parity-check (LDPC) codes. They usually focused on introducing the construction methods for good LDPC codes instead of a general method for code optimization. This paper proposes a method with high versatility, called the parallel vector message passing-based edge exchange (PMPE), for optimizing a type of graph-based LDPC codes, without changing the code parameters of mother codes such as the code length, code rate and degree distribution.With the approximately nearest codewords (ANC) searching approach, we find the optimization method can increase the Hamming distance of the LDPC codes. For the quasi-cyclic (QC) LDPC codes, an optimization method, called the parallel vector message passing oriented-to the QC-LDPC codes (QC-PMP), is further suggested, with which the quasi-cyclic characteristics of QC-LDPC codes can remain unchanged in the optimization. To evaluate the performance of the parity-check matrix corresponding to a Tanner graph, a very simple metric, the Cycles Metric (CM), is introduced to work with the proposed PMPE and QC-PMP algorithms. The experimental results show that the performance of the LDPC codes optimized with the proposed PMPE can be improved significantly at low BER range compared to the mother codes of the random codes, including the regular MacKay code of rate 0:5 and the regular PEG code of rate 0:9. For the case of the regular and irregular QC-LDPC codes with different code lengths and code rates, the optimized LDPC codes with the proposed QC-PMP algorithm significantly outperform the mother codes.

A subgraph with cycles.

…

Tanner graph

…

Error correction performance of the PMPE optimized codes and the MacKay codes.

…

Error correction performance of the (440, 396) and (500, 451) PMPE codes and PEG codes.

…

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Transactions on Communications

Design of Binary LDPC Codes with Parallel Vector

Message Passing

Xingcheng Liu, Senior Member, IEEE, Feng Xiong, Zhongfeng Wang, Fellow, IEEE, and Shuo Liang

Abstract—Many studies were carried out for the construction

of low density parity-check (LDPC) codes. They usually focused

on introducing the construction methods for good LDPC codes

instead of a general method for code optimization. This paper

proposes a method with high versatility, called the parallel vector

message passing-based edge exchange (PMPE), for optimizing a

type of graph-based LDPC codes, without changing the code

parameters of mother codes such as the code length, code rate

and degree distribution. With the approximately nearest codewords

(ANC) searching approach, we ﬁnd the optimization method can

increase the Hamming distance of the LDPC codes. For the

quasi-cyclic (QC) LDPC codes, an optimization method, called

the parallel vector message passing oriented-to the QC-LDPC codes

(QC-PMP), is further suggested, with which the quasi-cyclic

characteristics of QC-LDPC codes can remain unchanged in the

optimization. To evaluate the performance of the parity-check

matrix corresponding to a Tanner graph, a very simple metric,

the Cycles Metric (CM), is introduced to work with the proposed

PMPE and QC-PMP algorithms. The experimental results show

that the performance of the LDPC codes optimized with the

proposed PMPE can be improved signiﬁcantly at low BER range

compared to the mother codes of the random codes, including

the regular MacKay code of rate 0.5and the regular PEG code

of rate 0.9. For the case of the regular and irregular QC-LDPC

codes with different code lengths and code rates, the optimized

LDPC codes with the proposed QC-PMP algorithm signiﬁcantly

outperform the mother codes.

Index Terms—Low density parity-check (LDPC) codes; quasi-

cyclic codes; optimization; parallel vector message passing; cycles

metric.

I. INT ROD UC TI ON

LOw density parity-check (LDPC) codes, ﬁrst introduced

by Gallager in 1962 [1], and rediscovered by Mackay et

al. in 1996 [2], are currently ones of the most popular coding

techniques for error control in communication and storage

systems such as the DVB-S2, the IEEE 802.11 (Wi-Fi), the

IEEE 802.16e standard for WiMAX, and the patterned media

storage [3], due to their capacity-approaching performance

and practically implementable decoding algorithms. Numerous

studies have been focused on the construction, encoding,

decoding, and application of LDPC codes. Recently, long

LDPC codes were adopted for one of the standard channel

codes in the 5G mobile communications. The application of

The work was supported by the National Natural Science Foundation of

China (Grant Nos. 61572534 and 61602531).

Xingcheng Liu and Shuo Liang are with the School of Electronics and

Information Technology, Sun Yat-sen University, Guangzhou, 510006, China

(e-mail: isslxc@mail.sysu.edu.cn). Xingcheng Liu is also currently afﬁliated

with Xinhua College of Sun Yat-sen University, Guangzhou, 510520. Feng X-

iong was with the School of Electronics and Information Technology, Sun

Yat-sen University, Guangzhou, 510006. Now, he is with the FiberHome

Technologies Group, Wuhan, 430205, China. Zhongfeng Wang is with the

School of Electronic Science and Engineering, Nanjing University, Nanjing,

210023, China.

LDPC codes in high density storage systems such as NAND

ﬂash memory aroused great interests in academia [4] [5] [6].

An LDPC code whose parity-check matrix has a constant

column weight γand a constant row weight ρis said to be

regular and called a (γ, ρ)-regular LDPC code. An LDPC code

whose parity-check matrix does not have a unique column

weight or row weight is called an irregular LDPC code [7]. The

overwhelming majority of the methods for constructing LDPC

codes can be divided into two types: the algebraic-based and

the graph-based methods, also known as the structured and the

random methods, respectively. Algebraic-based constructions

of LDPC codes are mainly developed based on the ﬁnite ﬁelds

[8] [9] and the ﬁnite geometries [10] [11]. The PEG (progres-

sive edge growth) [7] and the protograph-based methods [12]

[13] are among the most well-known graph-based construction

methods.

Generally, LDPC codes constructed with random methods

have good performance because they increase the girth of the

Tanner graph or avoid short cycles in constructing the Tanner

graph or the parity-check matrix as possible as they can.

Among them, the PEG algorithm is the most effective random

construction method. Many PEG-based algorithms [14] [15]

with extrinsic message degree (EMD) or approximate cycle

EMD (ACE) properties [16] are proposed to further enhance

the performance of the PEG-based LDPC codes. The EMD

and ACE properties emphasize that not only the length of

cycles, but also the vertex connectivity in Tanner graphs play

important roles in error-ﬂoor performance. Some short cycles

with good graph connectivity are allowed, while long cycles

with poor graph connectivity are precluded. Applying EMD or

ACE properties to the design of LDPC codes, the PEG-based

algorithm can reduce the number of small stopping sets [17] or

trapping sets [18], which determine the error ﬂoors of LDPC

codes. Nevertheless, the construction algorithms of the PEG-

based codes and even all of the graph-based codes, are greedy

in maximizing the local girth of each symbol node, because

the Tanner graphs are constructed through establishing edges

or connections between variable nodes and check nodes in

an edge-by-edge manner. Each edge to be established is just

optimal to the subgraph in the current round of construction.

The edges established at early stages have many available

nodes while the edges to be established at later stages of code

construction have few ones, which results in a small local

girth for some symbol nodes, especially in the construction of

high-rate LDPC codes.

The ACE spectrum [19] is a useful tool for design and

evaluation of LDPC codes. It is a good metric to measure

the quality of a Tanner graph [20] as a whole. A generalized

ACE constrained design based on the ACE spectrum is shown

to be more ﬂexible and efﬁcient than the original proposal

http://ieeexplore.ieee.org/document/8207605/

Transactions on Communications

[16]. However, by increasing the ACE spectrum components

(a dmax-tuple, and dmax is the depth of the ACE spectrum),

the generalized ACE constrained design [19] leads to computa-

tional complexity problems since ﬁnding appropriate value for

each entry of the dmax-tuple is an unsolvable task. To tackle

this problem, a method called decoder-optimized progressive

edge growh (DO-PEG) algorithm [21] has been put forward.

In this method, the edge placement is enhanced by use of

the Sum Product algorithm in the design of the parity-check

matrix. The main idea is comparing the performance of the

code under the current graph setting for each of the candidate

CNs with the maximum distance possible from the current

VN. Moreover, the application of the DO-PEG is limited to

be employed in optimizing the LDPC codes generated with

the PEG method. By contrast, the proposed PMPE can be

employed on a given code generated with methods not limited

to the PEG to get improved performance. In this paper, we

introduce a very simple metric called the Cycles Metric

(CM) to measure the performance of the parity-check matrix

corresponding to a Tanner graph. Furthermore, an efﬁcient

method called parallel vector message passing (PMP) for

computing the CM of a Tanner graph is designed based on

the message passing algorithm (MPA) [22] [23]. On this basis,

a method called parallel vector message passing-based edge

exchange (PMPE) is proposed to optimize the global Tanner

graph. Although the algorithm in [22] may be recast in a form

equivalent to the proposed PMP and PMPE, the PMP and

PMPE both target at the optimization from the whole Tanner

graph, and can be employed for a given code to get improved

performance. Since the metric, CM, considers all the cycles-

2lin the Tanner graph as a whole, it gives a weight value to

each cycle-2lto measure the magnitude of the harm on the

error ﬂoor performance. A smaller value of CM means that the

number of cycles is fewer and/or their lengths are longer on the

whole Tanner graph. Consequently, reducing the cycles metric,

CM, intuitively improves the error correction performance of

the codes. The experimental evidence is also provided in the

numerical result section to support this claim. Accordingly,

the code construction is carried out for decreasing the CM of

a Tanner graph.

Compared to the random LDPC codes, the QC-LDPC

codes [24] have advantages in hardware implementations of

encoding and decoding. Encoding of a QC-LDPC code can

be efﬁciently implemented with simple shift registers. In hard-

ware implementation of a QC-LDPC decoder, the quasi-cyclic

structure of the code simpliﬁes the wire routing for message

passing. Furthermore, well designed QC-LDPC codes perform

as well as any other types of random LDPC codes in the

waterfall region. All these advantages will inevitably make the

QC-LDPC codes become the mainstream of study on LDPC

codes. In this paper, we propose a method to optimize binary

QC-LDPC codes, whose parity-check matrices are composed

of arrays of zero matrices (ZMs) and circulant permutation

matrices (CPMs). In code optimization, we present a simple

and very ﬂexible method for optimizing the base matrices

of QC-LDPC codes based on the ﬁnite ﬁelds [8] [9] or the

pseudo-random methods [25] [26]. The QC-LDPC codes can

be uniquely determined by their respective base matrices since

the codes constructed with these methods are just different in

the modes that the elements of base matrix are mapped into the

CPMs. The code optimization is performed through adjusting

the elements of a given base matrix without changing the basic

parameters of codes, such as the code lengths and the code

rates. The total CM of a parity-check matrix to which the

current base matrix is associated is computed with the PMP

algorithm after each adjustment. The principle of optimization

is to decrease the total CM until it can not go down further.

Experimental results show that the codes constructed with the

proposed method perform well and how low the error ﬂoor

occurs.

The rest of the paper is organized as follows. Section

II brieﬂy introduces notations and preliminaries about the

graph-based method. Section III presents the CM to measure

the quality of a Tanner graph, and the PMP algorithm for

computing the CM more efﬁciently. In Section IV, a versatile

algorithm, called PMPE, is introduced for optimizing the

Tanner graphs constructed with the random construction of

LDPC code, and the proposed CM and PMP are further

employed in the optimization of QC-LDPC codes. Section

V presents the simulation results and gives analyses of the

proposed algorithms. Finally, Section VI concludes the paper

with comments.

II. NOTATI ON S AN D PRE LI MI NAR IE S

Denote nas the number of variable nodes (VNs), mas the

number of check nodes (CNs), Neas the number of edges,

vj,j= 0, 1, ..., n−1as the j-th VN, ci,i= 0, 1, ..., m−1

as the i-th CN, dvjas the degree of variable node vj,Ej,i as

the edge connecting vjwith ci, and cycle-tas the cycle with

length tin a Tanner graph.

A. Basic Deﬁnitions

In the construction of LDPC codes, many considerations

such as ACE property [16] [19], EMD property [15], and

trapping set [18] have been taken into account in measuring

the quality of a Tanner graph. Some basic deﬁnitions related

to LDPC code construction are provided as follows.

Deﬁnition 1: Let Vbe a set of VNs, and Γ(V)be the set

of adjacent CNs of V. A subgraph induced by Vis the graph

containing nodes in V∪Γ(V)and the edges between Vand

Γ(V).

Deﬁnition 2: Astopping set (SS) is a set of VNs to which

every adjacent CN of this set is connected at least twice [17].

It is worth noting that the empty set is a stopping set, and the

stopping set connected to only one CN does not form a cycle.

If the degree of each VN is at least 2in a subgraph induced

by a stopping set, the subgraph always contains cycles. Large

stopping sets can be recursively derived by small ones, and

there is no efﬁcient algorithm for explicitly identifying and

removing them [27].

Deﬁnition 3: Atrapping set (TS) [18] is a non-empty set

of VNs in a Tanner graph with parameters aand b, where a

is the number of VNs in this set and bis the number of CNs

with odd degrees in the subgraph induced by the aVNs and

their adjacent CNs. We denote this set as TS (a, b).

Transactions on Communications

Fig. 1. A subgraph with cycles.

A TS is said to be elementary if all check nodes connected

to this TS have degree 1and/or 2. If the degree of each VN is

at least 2in a subgraph induced by a trapping set, the subgraph

always contains cycles. It is known that ﬁnding all trapping

sets in a Tanner graph is also an NP-hard problem [28].

Deﬁnition 4: The extrinsic message degree (EMD) of a cycle

in an LDPC code Tanner graph is the number of check nodes

connected only once to the set of variable nodes in this cycle

[19].

If a cycle in the Tanner graph of an LDPC code has a low

EMD, its corresponding connectivity with the rest of the graph

vertices is poor, which results in the phenomenon that the set

of variable nodes in this cycle receives a limited number of

extrinsic messages from the rest of the graph. In the general

case of the SS and TS, the EMD of an SS is zero while the

EMD of an elementary TS (a, b)is b.

Deﬁnition 5: The approximated cycle EMD (ACE) of a

cycle is the sum of components, which are the degree of each

variable node within the cycle minus two [19].

It is hard to ﬁnd the EMDs of all short cycles since it takes

additional steps to identify whether an edge is extrinsic or not

for a cycle. The ACE metric is a simpliﬁed version of the

EMD. For their taken values, the EMD of a cycle is less than

or equal to the ACE of it.

Example 1: As is shown in Fig. 1, {v0,c0,v1,c2} is a cycle

with length = 4, its EMD = 1 since only c1connects the VN

set once, and its ACE = 1 since (dv0−2) + (dv1−2) = 1.

{v0,c1,v2,c2} is a cycle with length = 4, its EMD =ACE

= 3 after similar computations. {v0,c0,v1,c2,v2,c1} is a

cycle with length = 6, EMD = 2 and ACE = 3.v2,c3,v3,

c4is a cycle with length = 4, its EMD =ACE = 2. {v0,v1,

v2,v3} is an SS related to these cycles. In Fig. 1, a TS (3,3)

is formed since there are 3VNs and 3CNs with odd degrees,

i.e., dc2= 3 and dc3=dc4= 1, in the subgraph induced by

{v0,v1,v2}.

The performance of LDPC codes with iterative decoding

algorithms is closely related to the ACE and EMD of cycles

or the number of small SS and TS involved in cycles. Thus,

the problem of error ﬂoors actually depends on the cycles.

B. MacKay and PEG-based LDPC Codes

MacKay introduced four methods for the construction of

LDPC codes [29] [30], which are called Constructions 1A,

2A, 1B and 2B, respectively. The codes generated with these

methods are collectively known as the MacKay codes. In

the parity-check matrix designed with Constructions 1A or

2A, the overlap between any two columns is no greater than

1, which results in the Tanner graph corresponding to the

parity-check matrix free of cycle-4. Constructions 1B and 2B

are generalized based on Constructions 1A and 2A. A small

number of columns are deleted from the matrix produced with

Constructions 1A and 2A, respectively. In this way, the Tanner

graph corresponding to the matrix has no short cycles of length

less than some value l. With this method, the elements in the

parity-check matrix are generated randomly. The principle is

that the cycle-4and cycle-6in the current matrix are forbidden

as possible as it could. Because of generating the elements

of the matrix one by one, the MacKay methods do not take

the cycles of the whole Tanner graph corresponding to the

matrix into consideration. Consequently, the performance of

the MacKay code can be improved by further optimizing

the cycle structure. Later in this paper, we introduce our

optimization method and apply it to the MacKay code. The

improvement of error-correcting performance is demonstrated.

The progressive edge growth (PEG) algorithm is also one

of the most promising graph-based methods for constructing

LDPC codes. The original PEG algorithm [7] constructs

Tanner graphs by progressively establishing edges between

VNs and CNs. Only one edge is put during each step in order

to achieve the largest local girth or form as few short cycles

as possible in the current subgraph. The variants of the PEG-

based algorithms [31] [14] [32] are all similar to some extent.

They do some modiﬁcations or add some constraints, such

as applying the ACE [16] property in each step to achieve

better cycle-connectivity. The PEG-based method only makes

the optimization in establishing edges among the current VNs.

Because of its greediness, the VNs to be treated later in

code construction are hard to achieve a large local girth than

the former ones. In addition, the established edges of the

later VNs may decrease the local girths of the former VNs.

Especially, in the situation of high code-rate, the number

of available CNs is limited at each step, resulting in the

consequence that each edge to be established will be involved

in many cycles. Although the PEG-based codes may achieve

better performance than the original PEG code, the code-rates

of them are all around 0.5. To overcome the difﬁculty, we

proposed an optimization method that is devised based on the

established Tanner graph of an original PEG code, focusing

on the situation of high code-rate up to 0.9. The experimental

results will demonstrate the validity of the proposed method.

III. PAR AL LE L VEC TO R MES SAG E PASSING AND CYC LE S

MET RI C

The ACE spectrum [19] and the EMD spectrum [15] can be

used to evaluate the quality of a Tanner graph corresponding

to a parity-check matrix for an LDPC code. But both of them

need to know the speciﬁc VNs involved in each cycle. More-

over, the EMD spectrum needs to know the speciﬁc extrinsic

CNs of each cycle. To tackle this computational problem, we

introduce a very simple metric called the Cycles Metric (CM),

which only needs to compute the number of cycles, but without

computing the speciﬁc nodes involved in them. Furthermore,

an efﬁcient method called parallel vector message passing

Transactions on Communications

(PMP) for computing the number of short cycles in a Tanner

graph is designed based on the message passing algorithm

(MPA) [22] [23]. In this section, we ﬁrst give an introduction

to the PMP, then to the CM.

A. Parallel Vector Message Passing

The message passing algorithm (MPA) [22] [23] is a rep-

resentative for counting the number of short cycles. Just like

the sum-product algorithm (SPA) [33] in LDPC decoding, the

message that a CN cpasses to a VN vequals the product of

the messages that ccollects from all the VNs connected to the

CN cexcept the VN v.

The MPA is also an iterative method with three procedures,

initialization, iterative message passing, and stopping. After

several iterations, the original MPA [22] can count the number

of cycles with length 2l(2 ≤l≤(g−1)), where lis the

number of iterations and gis the girth of the Tanner graph. The

improvement [23] to MPA can count the number of cycles with

length greater than (2g−2). Both of them are implemented

sequentially, i.e., calculating short cycles of all the nodes in

a Tanner graph one by one. At the beginning, MPA uses an

integer to initialize the edge’s message in order to distinguish

different initial messages corresponding to different edges.

During iterative message passing, the product of integers from

different edges is used to update the edges’ messages. In the

calculation of iterative message passing procedure, the division

operations are performed to ﬁnd the number of different

integers corresponding to the initial messages of different

edges.

In the following sections, we need to compute the Cycles

Metric of each edge or node in a Tanner graph with MPA

efﬁciently. Thus the parallel vector message passing (PMP)

is devised with some modiﬁcations to MPA [22]. In the

initialization procedure, we set the initial message of each

edge to a unique vector Xk(J, I)based on the Tanner graph.

Xk(J, I)can be represented by the index row vector [J I],

which denotes the k-th edge EJ,I in the Tanner graph with

the J-th variable node vJconnected with the I-th check node,

cI. The indices fall in the range of [0, n −1] for J,[0, m −1]

for I, and [0, Ne−1] for k. In the iterative message passing

procedure, we just record the number of different vectors when

updating the edge messages from CNs to VNs (C2V) or from

VNs to CNs (V2C). Then we compute the number of cycles

involving each edge according to this edge’s last C2V message

in the stopping procedure. For the sake of clarity, the notations

in the proposed PMP algorithm are shown in TABLE I. The

detailed procedures of the PMP are described as follows.

1) Initialization:

Initialize all vector messages from all the edges of a

Tanner graph. For example, Xk(J, I) = [J I ]represents

the initial vector message to which EJ,I is associated.

That is, there is a one-to-one mapping between Xk(J, I),

[J I], and EJ,I. It is clear that there are Neinitial

vectors. Then, the initial message from vjto cican be

initialized as follows:

mvj→ci=Xk(j, i)1=Xk(J, I)1= [J I ]1,(1)

where, j=Jand i=Iat the present stage and

Xk(J, I)1shows that the initial V2C message of Ej,i

contains only 1vector Xk(J, I)and no other ones.

2) Iterative Message Passing:

a) Update all the messages from CNs to VNs (C2V)

by computing

mci→vj={Xk(J, I)Nci→vj

k|k∈[0, Ne−1]},(2)

where, mci→vjrepresents the message from check

node cito variable node vj.mci→vjis a set that

consists of Nevectors with different counts since

vjmay receive any one of the Neinitial vectors

during the iterative message passing procedure.

The number of Xk(J, I)in mci→vj, i.e., Nci→vj

equals the sum of the counts of Xk(J, I)from all

the variable nodes in V(ci)\vjto ci. We can get

Nci→vj

kby computing

Nci→vj

k=∑

vj′∈V(ci)\vj

Nvj′→ci

k,(3)

where, Nvj′→ci

kdenotes the number of Xk(J, I)

in mvj′→ci.

b) Update all the messages from VNs to CNs (V2C)

by computing

mvj→ci={Xk(J, I)Nvj→ci

k|k∈[0, Ne−1]},(4)

where, the computational principle of mvj→ciis

the same as that of mci→vj, and Nvj→ci

kcan be

computed by

Nvj→ci

k=∑

ci′∈C(vj)\ci

Nci′→vj

k.(5)

c) Calculate the number of cycles-2linvolving edge

Ej,i by

Nj,i

2l=∑

ci′∈C(vj)\ci

Nci′→vj

k,(6)

where, lis the current number of iterations. If

two edges, Ej,i1and Ej,i2, are involved in one

(or more) cycle(s)-2l, the initial vector message

of edge Ej,i1, i.e., Xk(j, i1), will appear in the

C2V message of edge Ej,i2, i.e., mci2→vj, after

literations of message passing. The number of

times that Xk(j, i1)appears in Ej,i2equals the

number of cycles-2lwhich Ej,i1and Ej,i2are

both involved in. It is noted that the girth of the

Tanner graph, g, is obtained when Nj,i

2ldoes not

equal 0for the ﬁrst time. If only we continuously

increase the number of iterations, the girth gcan

be computed by g= 2l′, where l′is the minimum

number of iterations that holds Nj,i

2l̸= 0.

3) Stopping Procedure:

If the algorithm runs till the speciﬁed number of itera-

tions, it ends, else it goes back to Step 2) and continues

the message passing.

The PMP algorithm has a computational complexity O(Ne)

in the initialization procedure. Here, we consider a regular

Transactions on Communications

Tanner graph as an example. In the iterative message passing

procedure, each node , say vj, has a computation complexity

O((dvj−1)Ne)when updating the C2V message. Therefore,

if the number of check nodes in this Tanner graph is n, the av-

erage computation complexity of Step 2a) is O(n(dvj−1)Ne),

i.e., O(Ne2). By the same token, the computation complexity

for Step 2b) is O(Ne2). In the meantime, Step 2c) has a com-

putation complexity O(dvj−1). According to the above, the

total computation complexity of PMP algorithm is O(lNe2)

with literations, which is equivalent to the complexity of

MPA. However, the PMP is easy to be implemented in parallel

with a little cost in memory requirements. It is easy to see that

the same computation complexity also applies to the irregular

Tanner graph. With the PMP algorithm, we can efﬁciently

obtain the number of short cycles in the Tanner graph, which

is beneﬁcial to computing Cycles Metric of a Tanner graph.

B. Cycles Metric

The SPA assumes that all nodes are independent in the

decoding of LDPC codes of ﬁnite lengths. However, the cycles

in a Tanner graph contradict with this assumption. Short cycles

are considered to be harmful to the decoding performance of

LDPC codes. Therefore, the PEG-based algorithms all focus

on maximizing the local girth or reducing the number of short

cycles in each stage of edge establishment. Some of them add

the ACE constraint to increase the connectivity of cycles with

the rest of a Tanner graph while some short cycles with a good

graph connectivity are allowed. However, all of these algo-

rithms just take the local optimum into consideration, where,

the selected CN is just optimal to the current subgraph. The

result is that the previously established edges will inﬂuence

the available CNs in the subsequent construction and selection

procedures. Thus, the ﬁnally established Tanner graph is not

necessarily optimal when taking it as a whole. To cope with

this problem, we propose a new metric to evaluate the code

parameters that affect the overall performance of a Tanner

graph as follows.

Deﬁnition 6: The cycles with length 2l(2 ≤l≤g−1) that

contains Ej,i can be quantiﬁed as a whole by a parameter,

which is called the Cycles Metric (CM). The CM of Ej,i can

be computed by

Mvj,ci=

g−1

∑

l=2

βl−2·Nj,i

2l,(7)

where, Nj,i

2lis the number of cycles-2linvolving edge Ej,i

and it can be computed by (6) in the PMP algorithm. gis

the girth of a Tanner graph and also can be obtained with

the PMP algorithm. It is noted that we only take cycles with

length from 4to 2g−2into consideration. β(0 < β < 1) is

a constant and βl−2represents the weight of inﬂuence that a

cycle with length 2lhas to the overall Tanner graph. Then the

CM of vjequals the sum of the CMs of its adjacent edges,

computed by

Mvj=∑

ci∈C(vj)

Mvj,ci.(8)

Finally, the total CM of a Tanner graph (or a check-parity

matrix) can be calculated by

MT=

n−1

∑

j=0

Mvj.(9)

Example 2: The shortest cycles are the most harmful to the

decoding performance of an LDPC code. Hence, a weight val-

ue is simply given to each cycle-2lto measure the magnitude

of this harm. The weight of inﬂuence for a cycle-2lon the

decoding performance is set to be βl−2. Based on this weight

set, the weight of inﬂuence for a cycle of cycle-4will be 1,

the one for a cycle of cycle-6will be 0.1if βis set to 0.1, and

that one for a cycle of cycle-8will be 0.01. Accordingly, it is

easily understood that 10 cycles of cycle-6are as equivalently

harmful as 1cycle of cycle-4to the decoding performance,

and that 10 cycles of cycle-8are also as harmful as 1cycle

of cycle-6.

The number of edges in the Tanner graph is constant for the

proposed algorithm. Hence, when the number of cycles with

length 2ldecreases, the number of cycles with other lengths

may increase. Then, all the cycles of cycle-2lin the Tanner

graph are considered as a whole and measured with MT. The

MTtakes the number of cycles and their lengths into account

in order to ﬁnd a balance between the cycles and lengths.

That is, the performance of LDPC codes is evaluated using

the MT. A smaller value of MTmeans that the number of

cycles is fewer and/or their lengths are longer on the entire

Tanner graph.

IV. OPT IM IZ ATION OF LDPC CODES

In this section, two optimization algorithms for construction

of LDPC codes, the PMPE for random LDPC codes and the

QC-PMP for quasi-cyclic LDPC codes, are proposed based

on the parallel vector message passing (PMP). In view of

the global optimization, the two proposed algorithms aim to

minimize MT. For ease of exposition, the PMPE and the

QC-PMP are to be introduced in two subsections. Moreover,

to provide proofs for the improvement of the optimization

method on LDPC codes, the distributions of codewords with

low Hamming weights are estimated with the bit-reversing

method [34].

A. Parallel Vector Message Passing-Based Edge Exchange

The MacKay codes and PEG codes establish a Tanner graph

getting trapped in the local optimum. To solve or alleviate the

problem, we can try to rearrange some of the established edges

in the whole graph. That is, we can change some edges to

make MTas small as possible. Here we give some deﬁnitions

related to edges and nodes as follows.

Deﬁnition 7: An edge exchange is said to appear in a

Tanner graph if two edges in the Tanner graph exchange their

connected CNs.

Deﬁnition 8: In a Tanner graph, a direct connection between

two edges exists when the edges connect a common VN or

CN, while an indirect connection between two edges occurs if

two edges are not connected directly but there exists another

Transactions on Communications

TABLE I

NOTATIO NS EM PL OYED I N PMP

Xk(J, I)the vector message corresponding to edge EJ,I (there are Nevectors)

Xk(J, I)bthe number of vector Xk(J, I)is b

mvj→cithe message from vjto ci(it is a set that contains Nevectors with different counts)

mci→vjthe message from cito vj(it is a set that contains Nevectors with different counts)

Nvj→ci

kthe number of vector Xk(J, I)in mvj→ci

Nci→vj

kthe number of vector Xk(J, I)in mci→vj

C(vj)the set of CNs connected with vj

V(ci)the set of VNs connected with ci

C(vj)\cithe set of CNs connected with vjexcept ci

V(ci)\vjthe set of VNs connected with ciexcept vj

Algorithm 1 Computation of MTwith the PMP

1: for k←0to (Ne−1) do

2: // Ej,i is the k-th edge in the Tanner graph

3: Initialize the V2C message of edge Ej,i;

4: end for

5: // lis the current number of iterations, and gis the girth

of Tanner graph and temporarily set to +∞

6: Initialize l←0,g←+∞;

7: // the maximum number of iterations is g−1

8: while l≤(g−1) do

9: Perform iterative message passing once, l←(l+ 1);

10: //compute the number of cycles-2lfor each edge

11: for k←0to (Ne−1) do

12: Compute Nj,i

2lwith Eq.(6);

13: // obtain the actual girth

14: if g= +∞and Nj,i

2l>0then

15: g←2l;

16: end if

17: end for

18: end while

19: //compute the number of cycles for each edge

20: for k←0to (Ne−1) do

21: Compute Mvj,ciwith Eq.(7);

22: end for

23: //compute the number of cycles for each variable node

24: for j←0to (n−1) do

25: Compute Mvjwith Eq.(8);

26: end for

27: Compute MTwith Eq.(9);

28: End;

edge connected directly to both of them. If two edges are

directly connected or indirectly connected, we call them

an unexchangeable edges pair (UEP).

Example 3: A Tanner subgraph is shown in Fig. 2. There

are 3edges represented in bold lines, i.e., E0,0,E1,0and

E1,1. Obviously, E0,0and E1,0are directly connected with

a common CN c0, while E1,0and E1,1are also directly

connected with a common VN v1. For E0,0and E1,1, there

exists an edge E1,0that is connected directly to both of them,

so they are indirectly connected. Thus, the three edge pairs,

(E0,0, E1,0),(E1,0, E1,1)and (E0,0, E1,1), are all UEPs.

It is noted that an edge exchange between two directly con-

nected edges will not change the structure of a Tanner graph,

while an edge exchange between two indirectly connected

Fig. 2. Tanner graph

edges will decrease the number of edges in a Tanner graph.

The above two situations are not desired in code construction.

Deﬁnition 9: In a Tanner graph, if two edges do not form a

UEP, and an edge exchange between them does not decrease

the MT, we call them an invalid edge pair (IEP).

A Tanner graph with a smaller MTis intuitively expected

to have a better error correction performance. Hence, an

edge exchange without decreasing the MTis also not desired.

An edge exchange between two edges that do not form

a UEP never changes the original degree distribution of the

Tanner graph, but may have an effect on the cycles in which

the two edges are involved. Since it is impossible that the

degrees of VNs all equal 1in the Tanner graph corresponding

to a practical LDPC code , the cycles existed in the Tanner

graph can not be eliminated completely. Moreover, an edge is

deﬁnitely not involved in one cycle. A slight change after an

edge exchange may affect the cycles as a whole. Therefore,

MTis employed to measure the performance of the Tanner

graph. With the PMP algorithm, we can compute MTfast.

After an edge exchange, the performance of a Tanner graph is

improved as a whole if MTis decreased.

To move forward, we ﬁrstly compute the CM of each VN

and MTwith Algorithm 1. Then, two sets Sand S′are

initialized respectively. For each VN, we select the edge with

largest CM from the edges connected to this VN. There will

be nselected edges and we put them into S. And S′is used

to keep UEPs and IEPs and is temporarily set to an empty set

at the beginning. We randomly select two edges, say Ej1,i1

and Ej2,i2, from Saccording to the rule that (Ej1,i1, Ej2,i2)

is not in S′. Then we determine whether they form a new UEP

according to Deﬁnition 8. If so, we put this UEP into S′and re-

select another two edges different from the former. Otherwise,

we try to make an edge exchange on them and compute MT

for the Tanner graph with Algorithm 1, determining whether

Transactions on Communications

they form an IEP based on Deﬁnition 9. If not, a successful

edge exchange is performed, so we accept this edge exchange

and remove Ej1,i1and Ej2,i2from S. Otherwise, we undo

this edge exchange and put the IEP into S′. Especially, in the

selection of two edges, if an edge forms a UEP or IEP with

every one of other edges in S, it should be removed directly

from S. We keep the above procedure until Sbecomes an

empty set. At that time, a round of edge exchanges (multiple

edge exchanges) is completed and a new Tanner graph is

established with several edges changed. Then, we re-initialize

the sets Sand S′from the new Tanner graph and begin the

next round of edge exchanges as before. Finally, if no more

edge exchange is accepted at the current round, it suggests

that MTwill not be changed, indicating the termination of

the algorithm.

The algorithm is performed with edge exchange procedures

for optimization of random LDPC codes, which is illustrated

in Algorithm 2. We call it the parallel vector message passing-

based edge exchange, in short, PMPE, where Skand S′

represent the sets at the k-th round.

To compare the performance of the MacKay codes and the

optimized ones, the Hamming weight distribution is chosen

and shown in Table II for the LDPC codes of rate 0.5, where

we denote was the Hamming weight of the searched code-

words and Nwas the number of codewords with weight w. It

should be noted that the values in Table II are estimates. From

the table, we can see that the MacKay codes optimized with

PMPE have a larger Hamming distance among codewords or

fewer codewords with low weights than the original MacKay

codes. This just explains the validity of the proposed PMPE

algorithm.

B. Parallel Vector Message Passing Oriented-to the QC-

LDPC Codes

There are two general categories for the construction of

QC-LDPC codes, the algebraic constructions such as those

based on the ﬁnite ﬁelds [35] [36] and the pseudo-random

constructions such as those based on the PEG algorithm [25]

[26] [37]. A QC-LDPC code can be represented by an R×C

matrix, W, called the base matrix with the form







w0,0w0,1· · · w0,C−1

w1,0w1,1· · · w1,C−1

.....

wR−1,0wR−1,1· · · wR−1,C−1







where, each entry is represented by wr,c (0 ≤r≤R−1,

0≤c≤C−1). By replacing each entry wr,c of Wwith an

L×Lmatrix A(wr,c), we obtain an R×Carray







A(w0,0)A(w0,1)· · · A(w0,C−1)

A(w1,0)A(w1,1)· · · A(w1,C−1)

.....

A(wR−1,0)A(wR−1,1)· · · A(wR−1,C−1)







For the algebraic construction based on the ﬁnite ﬁelds [35]

[36], wr,c is an element taken over GF(q). A(wr,c ) is an L×

Lzero matrix (ZM) if wr,c = 0; otherwise, it is an L×L

circulant permutation matrix (CPM), where, L=q−1and

Algorithm 2 PMPE

1: Construct an LDPC code with a random method and

conﬁgure the Tanner graph;

2: // kis the number of rounds for edge exchange

3: Initialize k←0;

4: Compute MTwith Algorithm 1;

5: // tis the time of accepted edge exchanges at the current

round

6: Initialize Skand S′

k,t←0;

7: while Sk̸=∅do

8: // denote Ej1,i1and Ej2,i2as the current selected two

edges

9: Randomly select two edges from Skthat satisfy

(Ej1,i1, Ej2,i2)/∈S′

10: if (Ej1,i1, Ej2,i2)is a UEP then

11: Put (Ej1,i1, Ej2,i2)into S′

k, goto Line 9;

12: end if

13: Try an edge exchange between Ej1,i1and Ej2,i2;

14: // denote MTtmp as the CM of Tanner graph after an

edge exchange

15: if MTtmp <MTthen

16: Accept this edge exchange, remove Ej1,i1and Ej2,i2

form Sk;

17: MT←MTtmp,t←(t+ 1);

18: else

19: Undo this edge exchange, put (Ej1,i1, Ej2,i2)into S′

20: end if

21: Remove the edge from Skthat forms a UEP or IEP

with every one of the other edges in Sk;

22: end while

23: // t= 0 shows that MTis unchanged at the current round

of edge exchanges

24: if t= 0 then

25: Exit;

26: else

27: k←(k+ 1), go to Line 6;

28: end if

1≤wr,c ≤L. Let αbe a primitive element of GF(q). Then,

each nonzero wr,c can be denoted with the exponential form

of α, i.e., α0=αq−1= 1,α1,α2,. . .,αq−2. The power of α

represents the position of "1" at the top row of A(wr,c). For

the pseudo-random construction [26] [38], A(wr,c) is an L×L

ZM if wr,c is "-" or a negative value (e.g., −1); otherwise, it

is an L×LCPM, where, wr,c represents the position of "1" at

the top row of A(wr,c), and 0≤wr,c ≤L−1. That is, there is

a one-to-one mapping between wr,c and a ZM (or CPM) for

both of the two forms of base matrix. It is noted that qshould

be a prime or a power of a prime, so the size of submatrix,

L=q−1, with algebraic construction is limited, while Lcan

be arbitrary in the pseudo-random construction.

With such dispersion of Wto H, the null space of Hgives a

QC-LDPC code of length CL. Based on the special structure

of the QC-LDPC codes, a simple method like the PMPE for

random codes is proposed to change the graph of the QC-

LDPC codes. For the convenience of discussion, A(wr,c ) is an

L×LZM in the proposed algorithm if wr,c = 0; otherwise,

Transactions on Communications

TABLE II

HAM MIN G WE IGH T DI STR IBU TI ON (ES TI MATED VAL UE S)OF T HE MAC KAY AND PMPE C OD ES

w6 8 10 12 14 16 18 20 22

Nw- MacKay(96,48) 1 4 53 509 4845

Nw- PMPE(96,48) 1 2 44 442 4377

Nw- MacKay(204,102) 1 1 5 43 304

Nw- PMPE(204,102) 0 0 2 12 154

Nw- MacKay(408,204) 1 1 12 26 162

Nw- PMPE(408,204) 0 1 0 11 52

it is an L×LCPM, where, wr,c represents the position of

"1" at the top row of A(wr,c), the index of the position begins

from 1, and 1≤wr,c ≤L.

We still measure the performance of QC-LDPC code with

the total CM of the Tanner graph (or parity-check matrix).

Here we denote Hbqc (i.e., H, previously) of dimension m×n,

as the ﬁnal binary matrix of QC-LDPC code, and Hnb (i.e.,

W), as the base matrix of Hbqc. If Hnb is provided, Hbqc can

be obtained by the dispersion of Hnb in order not to destroy the

quasi-cyclic characteristic. The method of constructing Hnb is

divided into two steps, the base matrix construction (BMC)

and the element adjustment with message passing (MPEA).

As the name suggests, the BMC procedure is to construct

a base matrix, [Hnb]R×C, whose elements are "1" or "0"

with a random method, such as the PEG algorithm and the

proposed PMPE algorithm for optimization. The principle is

still to minimize MTof the Tanner graph for Hnb . Then,

we replace all the "1"’s in Hnb with nonzero elements taken

values between 1to Lrandomly. Through the BMC procedure,

we obtain an Hnb whose elements are between 0and L. It

is noted that the BMC procedure is alternative. That is, we

can start our algorithm directly from an existed base matrix

(mother code) Hnb with any other methods. This suggests

our method be highly versatile. We denote MT(Hbqc )as the

MTof the Tanner graph for Hbqc corresponding to Hnb . The

purpose of the MPEA procedure is to adjust the elements of

the Hnb between 1and Lin order to decrease the MT(Hbqc).

It should be noted that we do not adjust a nonzero element

wr,c to zero in order not to change the degree distribution

of the mother code. The algorithm is shown in Algorithm 3

for the details performing the the core MPEA procedure. It

begins from the parity-check matrix, Hbqc, from the existing

construction methods of QC-LDPC codes [35] [36] [25] [26]

[37]. We call it the parallel vector message passing oriented-

to the QC-LDPC codes, in short QC-PMP.

For the case of QC-LDPC codes, we compute the weight

distributions for two 802.11ad standard codes [38] and their

optimized versions with the QC-PMP algorithm. We use

the same computational method as stated in the previous

subsection. The Hamming weight distributions of these codes

are shown in Table III. We can see that the optimized codes

have less number of low weight codewords than the original

IEEE802.11ad ones. In other words, the proposed LDPC codes

optimized with QC-PMP are supposed to bring fewer errors

with large noises.

Algorithm 3 QC-PMP

1: Initialize the Hbqc and the corresponding Hnb;

2: Compute the initial MT(Hbqc)with Algorithm 1, denoted

with Minit;

3: for r←0to (R−1) do

4: for c←0to (C−1) do

5: if wr,c ̸= 0 then

6: for k= 1 to Ldo

7: wr,c ←(wr,c + 1) mod (L+ 1); // adjust wr,c

between 1and L

8: if wr,c = 0 then

9: wr,c ←1; // skip "0" in order not to change

the predeﬁned degree distribution

10: end if

11: Obtain new Hbqc and compute MT(Hbqc)de-

noted with Mtmp;

12: if Mtmp <MT(Hbqc)then

13: MT(Hbqc)←Mtmp ;

14: else

15: if wr,c = 1 then

16: wr,c ←1;

17: else

18: wr,c ←(wr,c −1) mod (L+ 1);

19: end if

20: end if

21: end for

22: end if

23: end for

24: end for

25: if MT(Hbqc) = Minit then

26: Exit;

27: else

28: Goto Line 2;

29: end if

V. SIMULATIO N RES ULTS

In this section, we investigate the performance of the

proposed PMPE and QC-PMP algorithms through simulations.

Random and QC-LDPC codes with different code parameters

are employed to evaluate the validity of the proposed algo-

rithm. In simulations, the value of βis investigated for LDPC

codes of different lengths and code rates over the AWGN

channel. It is found that the decoding performance of the

LDPC codes is optimal when the value of βis taken at 0.01.

Therefore, the βin Eq. (7) is set to 0.01 in the following sim-

ulations. The error correction performance is simulated with

the SPA decoding algorithm over the AWGN channel with the

Transactions on Communications

TABLE III

HAMMING WEIGHT DISTRIBUTION (EST IM ATED VALU ES )OF TH E 802.11AD A ND QC-PMP CODES

w8 9 10 11 12 13 14

Nw- 802.11ad(648,486) 27 0 81 135 654 1683 4706

Nw- QC-PMP(648,486) 0 27 54 81 339 1626 4061

Nw- 802.11ad(1944,1620) 162 405 1117 4058 12195 28985

Nw- QC-PMP(1944,1620) 0 162 1038 2498 8111 20659

2 3 4 5 6 7

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Eb/N0(dB)

BER

MacKay (408,204)

PMPE (408,204)

MacKay (204,102)

PMPE (204,102)

MacKay (96,48)

PMPE (96,48)

Fig. 3. Error correction performance of the PMPE optimized codes and the

MacKay codes.

BPSK modulation. The maximum number of iterations for the

two long codes, (3960,2640) and (4032,3264), is set to 20

while the maximum for the other codes is set to 50.

A. Performance of PMPE Codes

In this subsection, we optimize some relatively short random

codes to show the validity of the PMPE algorithm. Focusing

on the code rate of 0.5, we simulate three (3,6)-regular

MacKay codes [30], (96,48),(204,102) and (408,204), and

the optimized codes with the proposed PMPE algorithm.

As is shown in Fig. 3, the three MacKay codes optimized

with the PMPE algorithm all outperform the original ones to

some extent.

For the high code rate around 0.9, we simulate two codes,

(500,451) and (440,396), which are constructed with the PEG

algorithm and then optimized with the PMPE algorithm. They

are both regular codes. The former has the column weight of

ρ= 2, which is often used in non-binary LDPC codes [3].

The column weight of the latter is ρ= 3, which is among the

most common column weights of regular LDPC codes.

As is shown in Fig. 4, the two codes optimized with the

PMPE algorithm both outperform the counterparts constructed

with the PEG algorithm. For code (500,451), the code opti-

mized with PMPE gains about 0.25 dB compared with the

PEG code at BER = 9×10−6. For code (440,396), the code

generated from the PMPE outperforms that from the PEG by

about 0.5dB at BER = 1 ×10−7.

In addition, the PMPE code is compared with the OptPEG

code [39], which optimizes the ACE property of the designed

code graph. The latter code is generated based on the original

PEG algorithm.

4 4.5 5 5.5 6 6.5 7 7.5

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/N0(dB)

BER/FER

BER, PEG(500,451)

BER, PMPE(500,451)

BER, PEG(440,396)

BER, PMPE(440,396)

FER, PEG(500,451)

FER, PMPE(500,451)

FER, PEG(440,396)

Fig. 4. Error correction performance of the (440,396) and (500,451) PMPE

codes and PEG codes.

5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Eb/N0(dB)

BER/FER

BER, PEG

BER, OptPEG

BER, PMPE

FER, PEG

FER, OptPEG

FER, PMPE

Fig. 5. Error correction performance of the (504, 448) code generated

respectively with the PMPE, the ACE (OptPEG), and the original PEG

algorithms.

As shown in Fig. 5, the performance of the code (504,448)

with three methods is shown. The PMPE code and the OptPEG

code perform very closely while the proposed PMPE outper-

forms the OptPEG at Eb/No>6.4dB, and they both surpass

the original PEG code when Eb/No>5.2dB.

B. Performance of QC-PMP Codes

The QC-LDPC code is a mainstream of the research in

recent studies. Hence, we pay more attention to the QC-LDPC

codes optimized with the proposed QC-PMP algorithm. With-

out loss of generality, different QC-LDPC codes are chosen

in simulations with the QC-PMP algorithm. The chosen codes

Transactions on Communications

1.5 2 2.5 3

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/N0(dB)

BER/FER

BER, QC−PMP

FER, QC−PMP

BER, CPM−QC

FER, CPM−QC

Fig. 6. Error correction performance of the (3960,2640) regular QC-PMP

and CPM-QC LDPC codes with rate 2/3.

2.8 3 3.2 3.4 3.6 3.8

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Eb/N0(dB)

BER/FER

BER, C2

FER, C2

BER, C3

FER, C3

BER, C2’

FER, C2’

BER, C3’

FER, C3’

Fig. 7. Error correction performance of the (4032,3264) irregular QC-LDPC

codes with rate 17/21.

contain those used in academia [36] [37] [40] and in industry

[38].

Firstly, we validate the proposed algorithm on the regular

QC-LDPC code. We can construct a 4×12 base matrix

with L= 330, [W1]4×12, of a CPM-QC LDPC code [36].

Through mapping each entry of W1to a 330×330 submatrix,

a1320 ×3960 parity-check matrix, H1, can be obtained.

The null space of H1gives a (3,9)-regular CPM-QC LDPC

code, C1(3960,2640). Then, we apply the proposed QC-

PMP algorithm to W1, resulting in an optimized base matrix,

W1′, and a parity-check matrix, H1′. Consequently, the null

space of the matrix, H1′, gives a (3,9)-regular QC-PMP code,

C1′(3960,2640).

From Fig. 6, we can see that the two codes behave the same

in the range of BER ≥2×10−6. However, in the range of

the BER < 2×10−6, the QC-PMP code outperforms the

CPM-QC code, exhibiting a lower error ﬂoor.

With the column weight distribution λ(x) = 0.4052x3+

0.3927x4+ 0.1466x7+ 0.0555x8and the row weight distri-

bution ξ(x)=0.1667x22 + 0.8333x23 [40], a 12 ×63 base

matrix with L= 64, [W2]12×63, can be obtained by using the

2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/N0(dB)

BER/FER

BER, 802.16e

FER, 802.16e

BER, SQC−LDPC

FER, SQC−LDPC

BER, QC−PMP

FER, QC−PMP

0.2 dB

Fig. 8. Error correction performance of the (2304,1920) irregular QC-PMP,

802.16e [37] and SQC-LDPC codes with rate 5/6.

geometry decomposition approach and ACE algorithm with

parameters (δ2d= 4, δAC E = 3). Through mapping each

entry of W2to a 64 ×64 submatrix, a 768 ×4032 parity-

check matrix, H2, can be built. The null space of H2gives

an irregular QC-LDPC code, C2(4032,3264). Then, we apply

the proposed QC-PMP algorithm to W2. Consequently, we can

obtain the optimized base matrix, W2′, and the parity-check

matrix, H2′. The null space of H2′gives an irregular QC-

PMP code, C2′(4032,3264). Performing the same steps with

parameters (δ2d= 4, δAC E = 4) [40] for construction and the

proposed QC-PMP for optimization, we can obtain one more

irregular QC-LDPC code, C3(4032,3264), and its optimized

version, C3′(4032,3264).

As shown in Fig. 7, the performance of C2′based on C2

is improved greatly. It gains about 0.15 dB compared with

C2at BER = 8 ×10−8. At the beginning of the point,

Eb/No=3.5dB, the slope of curve C2begins to reduce signiﬁ-

cantly. Hence, the error ﬂoor appears for code C2. Meanwhile,

C3′outperforms C3in the range of BER < 4×10−8.

In addition, with a different column weight distribution

λ(x) = 0.125x2+ 0.4167x3+ 0.4583x4and the uniform row

weight of 20, a 4×24 base matrix with L= 96,W4, can

be generated [37]. Through mapping each entry of W4to a

96 ×96 submatrix, a 384 ×2304 parity-check matrix, H4, can

be achieved. The null space of H4gives an irregular QC-LDPC

code, C4(2304,1920), called the SQC-LDPC code (QC-LDPC

code with single-weight circulants) [37]. The SQC-LDPC code

shows a lower error ﬂoor than the 802.16e standard code with

the same basic code parameters as those in [37]. Thus, we

apply the proposed QC-PMP algorithm to W4. The optimized

base matrix, W4′, and the parity-check matrix, H4′, can be

formed. The null space of H4′gives an irregular QC-PMP

code, C4′(2304,1920).

Fig. 8 shows the error correction performance of the irreg-

ular QC-PMP, SQC-LDPC and 802.16e standard codes. It is

noted that the experimental result of the 802.16e standard code

is extracted from [37]. From the curve of BER vs. Eb/N0,

we can see that the QC-PMP code based on the SQC-LDPC

code gains about 0.1dB compared with the SQC-LDPC code

at BER = 3 ×10−8and about 0.2dB compared with the

Transactions on Communications

3 3.5 4 4.5 5

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Eb/N0(dB)

BER/FER

BER, 802.11ad

FER, 802.11ad

BER, QC−PMP

FER, QC−PMP

Fig. 9. Error correction performance of the (648,486) QC-PMP and 802.11ad

codes with rate 3/4.

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Eb/N0(dB)

BER/FER

BER, 802.11ad

FER, 802.11ad

BER, QC−PMP

FER, QC−PMP

0.15 dB

Fig. 10. Error correction performance of the (1944,1620) QC-PMP and

802.11ad codes with rate 5/6.

802.16e code at BER = 2 ×10−7.

Finally, we demonstrate the validity of the proposed algo-

rithm on the 802.11ad standard code [38]. With the column

weight distribution λ(x) = 0.2083x2+ 0.5x3+ 0.2917x6and

the row weight distribution ξ(x) = 0.3333x14 + 0.6667x15,

a6×24 base matrix with L= 27,W5, can be obtained.

Through mapping each entry of W5to a 27 ×27 submatrix,

a162 ×648 parity-check matrix, H5, can be built. The null

space of H5gives an irregular 802.11ad code, C5(648,486).

Then, we apply the proposed QC-PMP algorithm to W5,

resulting in the optimized base matrix, W5′, and the parity-

check matrix, H5′. The null space of H5′gives an irregular

QC-PMP code, C5′(648,486). Meanwhile, with the column

weight distribution λ(x) = 0.125x2+ 0.4583x3+ 0.4167x4

and the row weight distribution ξ(x)=0.25x19 + 0.75x20,

we can obtain a 4×24 base matrix with L= 81,W6.

Through mapping each entry of W6to an 81×81 submatrix, a

324×1944 parity-check matrix, H6, can be achieved. The null

space of H6gives an irregular 802.11ad code, C6(1944,1620).

Then, we apply the proposed QC-PMP algorithm to W6,

resulting in the optimized base matrix, W6′, and the parity-

check matrix, H6′. The null space of H6′gives an irregular

QC-PMP code, C6′(1944,1620). It is noted that the elements

from column-(n−m+ 1) to column-nof W5and W6are not

adjusted during the optimization process, in order to maintain

the block dual-diagonal (BDD) structure of the 802.11ad code.

Figures 9 and 10 show the performance of QC-PMP code

(C5′and C6′) vs. 802.11ad code (C5and C6). From the

curves, we can see the QC-PMP code, C5′, gains about 0.2dB

compared with the 802.11ad code, C5, at BER = 3 ×10−8.

Finally, the QC-PMP code, C6′, outperforms the 802.11ad

code, C6, in the range of the BER < 8×10−6. At

BER = 1 ×10−8, the optimized code, C6′, gains about 0.15

dB compared with its corresponding code, C6.

Regardless of the 802.16e code or the 802.11ad code, the

QC-PMP can optimize the cycles of parity-check matrix as a

whole. It makes the number of cycles become fewer and the

length of cycles larger by continuously reducing the CM of

a parity-check matrix. The number and the length of cycles

within a Tanner graph have a better balance, making the

performance improved to certain degree in the error ﬂoor

range. As shown in all of the above simulation results,

reducing the metric, CM, leads to an exceptional effect of

signiﬁcantly improving the error correction performance of

the LDPC codes.

VI. CO NC LU SI ON

In this paper, we have proposed two optimization methods,

the PMPE algorithm for random codes and the QC-PMP

algorithm for QC-LDPC codes. The proposed algorithms can

decrease the number of short cycles in the Tanner graphs

without changing the code parameters, such as the code length,

code rate and degree distribution after optimization. The

approximately nearest codewords (ANC) searching approach

shows that the optimized codes have fewer codewords of low

weights, compared with the original ones, which means that

the optimized codes have a lower probability to become the

other permissible codes than the original ones when large

noises occur over the channel. Experimental results show

that the PMPE algorithm can improve the error correction

performance of the MacKay codes with code rate around

0.5and the PEG code with code rate around 0.9. For the

QC-LDPC codes, experimental results have shown that the

performance of codes with different code rates and lengths,

including the regular codes and irregular ones, are signiﬁcantly

improved with the proposed QC-PMP algorithm.

VII. AC KN OWLEDGMENT

The authors wish to thank the editor, Enrico Paolini, and

the anonymous reviewers for their comments on this paper

which signiﬁcantly improved the presentation and the quality

of this work. In particular, the simulation example of the

unstructured LDPC code optimized with different algorithms

was added to the paper as a result of such comments. The

authors are very thankful to the fellow researchers, Prof.

Vojin Senk, Dr. Dejan Vukobratovic, Prof. Bane Vasic and Dr.

Milos Ivkovic, for their support in providing their algorithm

details of constructing unstructured LDPC codes. Especially,

we wish to thank Dr. Dejan Vukobratovic for providing us

Transactions on Communications

their parity check matrices and explaining how to obtain the

excellent performance of the optimized algorithm. The above

mentioned researchers greatly speeded up the modiﬁcation and

signiﬁcantly improved our revised version of this work.

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Transactions on Communications

Xingcheng Liu (SM’12) was born in Anfu County,

Jiangxi, China. He received the B.E. and M.E.

degrees in electrical engineering from Huazhong U-

niversity of Science and Technology, Wuhan, China,

and the Ph.D. degree from Sun Yat-sen University

(SYSU), Guangzhou, China.

He received the Royal Society KC Wong Fel-

lowship of the U.K. to do post-doctoral research at

the University of Southampton, Southampton, U.K.,

during May 2002 - May 2003. He was a visiting

scientist at Oregon State University, Corvallis, USA,

from 2004 to 2005. He is a full professor with the School of Electronics and

Information Technology, SYSU. He is also currently afﬁliated with Xinhua

College of SYSU as a leader in the ﬁeld of electrical engineering and

automation. His main research interests include channel coding theory and

applications, mobile communications, wireless sensor networks, and Internet

of Things. He has authored over 100 peer-reviewed papers in journals and

conferences. He is now the primary investigator of several projects on wireless

communications and networking.

Dr. Liu is a Senior Member of the IEEE and the China Institute of

Communications.

Feng Xiong was born in Tianmen City, Hubei,

China. He received the B.E. degree from Yangtze

University, Jinmen City, Hubei in 2014, and the

M.E. degree from Sun Yat-sen University (SYSU),

Guangzhou, Guangdong in June 2017, both in elec-

trical engineering.

He is currently an engineer with the FiberHome

Technologies Group, Wuhan, China. His main re-

search interests include channel coding theory and

applications and mobile communications.

Zhongfeng Wang received both B.E. and M.S.

degrees from Tsinghua University, Beijing, China.

He obtained the Ph.D. degree from the Department

of Electrical and Computer Engineering at the U-

niversity of Minnesota, Minneapolis in 2000. He

joined Nanjing University in 2016 as a Distinguished

Professor through the State ˛a´

rs 1000-Talent Plan after

serving Broadcom Corporation as a leading VLSI

architect for nearly nine years. Prior to that, he was

an Assistant Professor in the School of EECS at

Oregon State University, Corvallis. Even earlier, he

worked for National Semiconductor Corporation, Santa Clara, USA.

Dr. Wang is a world-recognized expert on VLSI for Signal Processing

Systems. He has published over one hundred and ﬁfty technical papers,

edited one book (“VLSI”) and ﬁled tens of U.S. patent applications and

disclosures. He was the recipient of the IEEE Circuits and Systems Society

VLSI Transactions Best Paper Award in 2007. During his tenure at Broadcom,

he has contributed signiﬁcantly on 10Gbps and beyond high-speed networking

products. Additionally, he has made critical contributions in designing FEC

coding schemes for 100Gbps and 400Gbps Ethernet standards. So far, his

technical proposals have been adopted by many international networking

standards.

Since 2004 Dr. Wang has served as Associate Editor for the IEEE Trans.

on Circuits and Systems-I (TCAS-I), TCAS-II, and IEEE Trans. on VLSI

Systems for multiple terms. He is currently a Guest Editor for a special issue

of IEEE Journal on Emerging and Selected Topics in Circuits and Systems.

Furthermore, he has served as Technical Program Committee Member (or Co-

Chair), Session (or Track) Chair, and Review Committee Member for tens of

international conferences. In 2013, he served in the Best Paper Award selection

committee for the IEEE Circuits and System Society. His current research

interests are in the area of Digital Communications, Machine Learning, and

Efﬁcient VLSI Implementation. He is a Fellow of IEEE since 2016.

Shuo Liang was born in Yinchuan, Ningxia, China.

He received the B.E. degree from Sichuan University

in 2012, and the M.E. degree from Shefﬁeld Univer-

sity in October 2014, both in electrical engineering.

He is a PhD candidate in Sun Yat-sen Universi-

ty, Guangzhou, China. His main research interests

include channel coding theory and mobile commu-

nications.

Mr. Liang is a student member of the China

Institute of Electronics.

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Long quasi-cyclic low-density parity check (QC-LDPC) codes have been chosen as one of the forward error correction (FEC) schemes in passive optical network (PON) systems benefiting from the excellent performance. About how to get the appropriate LDPC codes, some traditional methods which focus on obtaining large girth or flat approximated cycle extrinsic message degree (ACE) spectrums are proposed for the design of LDPC codes. These methods which may still make the existence of small stopping sets containing short cycles probable are not so useful for long QC-LDPC codes. Recently, the parallel vector message passing (PMP) algorithm oriented-to decreasing the number of short cycles is proved to be an effective method to optimize LDPC codes. Since ignoring short cycles with poor connectivity which can induce error floor, this method can not guarantee its effectiveness for long QC-LDPC codes. In this letter, an edge exchange optimization method based on the global properties of ACE and girth called EEA method is proposed, which tries to reduce cycles with low ACE in the order of cycle length. The numerical results prove that our EEA method can bring the improvements of error floor up to two orders of magnitude to long QC-LDPC codes and obviously outperforms PMP algorithm.

A PEG-like LDPC code design avoiding short trapping sets

Conference Paper

Full-text available

Jun 2015

Design of LDPC Codes Based on Multipath EMD Strategies for Progressive Edge Growth

Article

Full-text available

Jul 2016

Low-density parity-check (LDPC) codes are capable of achieving excellent performance and provide a useful alternative for high performance applications. However, at medium to high signal-to-noise ratios (SNR), an observable error floor arises from the loss of independence of messages passed under iterative graph-based decoding. In this paper, the error floor performance of short block length codes is improved by use of a novel candidate selection metric in code graph construction. The proposed Multipath EMD approach avoids harmful structures in the graph by evaluating certain properties of the cycles which may be introduced in each edge placement. We present Multipath EMD based designs for several structured LDPC codes including quasi-cyclic and irregular repeat accumulate codes. In addition, an extended class of diversity-achieving codes on the challenging block fading channel is proposed and considered with the Multipath EMD design. This combined approach is demonstrated to provide gains in decoder convergence and error rate performance. A simulation study evaluates the performance of the proposed and existing state-of-the-art methods.

Algebraic Quasi-Cyclic LDPC Codes: Construction, Low Error-Floor, Large Girth and a Reduced-Complexity Decoding Scheme

Article

Full-text available

Aug 2014

This paper presents a simple and very flexible method for constructing quasi-cyclic (QC) low density parity-check (LDPC) codes based on finite fields. The code construction is based on two arbitrary subsets of elements from a given field. Some well known constructions of QC-LDPC codes based on finite fields and combinatorial designs are special cases of the proposed construction. The proposed construction in conjunction with a technique, known as masking, results in codes whose Tanner graphs have girth 8 or larger. Experimental results show that codes constructed using the proposed construction perform well and have low error-floors. Also presented in the paper is a reduced-complexity iterative decoding scheme for QC-LDPC codes based on the section-wise cyclic structure of their parity-check matrices. The proposed decoding scheme is an improvement of an earlier proposed reduced-complexity iterative decoding scheme.

Factor Graphs and the Sum-Product Algorithm

Article

Mar 2001

Algorithms that must deal with complicated global functions of many variables often exploit the manner in which the given functions factor as a product of “local” functions, each of which depends on a subset of the variables. Such a factorization can be visualized with a bipartite graph that we call a factor graph, In this tutorial paper, we present a generic message-passing algorithm, the sum-product algorithm, that operates in a factor graph. Following a single, simple computational rule, the sum-product algorithm computes-either exactly or approximately-various marginal functions derived from the global function. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward/backward algorithm, the Viterbi algorithm, the iterative “turbo” decoding algorithm, Pearl's (1988) belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform (FFT) algorithms

Construction of high-rate QC-LDPC codes with multi-weight circulants

Conference Paper

Sep 2016

In this paper, we propose a construction method of quasi-cyclic low-density parity-check (QC-LDPC) codes with multi-weight circulants (MQC-LDPC codes) which is suitable for high-rate LDPC codes with moderate length. MQC-LDPC codes can achieve better degree distributions, Hamming weight distributions, and stopping set weight distributions than conventional QC-LDPC codes with single-weight circulants (SQC-LDPC codes). Within the same framework, we generalize and simplify the error minimization progressive edge growth (EMPEG) algorithm for supporting the structure with multi-weight circulants. Simulation results show that the constructed MQC-LDPC codes offer better performance at both high and low signal to noise ratio (SNR) regions in additive white Gaussian noise (AWGN) channels than the conventional SQC-LDPC codes defined in 802.11ad and 802.16e.

Improved message-passing algorithm for counting short cycles in bipartite graphs

Conference Paper

Jun 2015

The multi-step PEG and ACE constrained PEG algorithms can design the LDPC codes with better cycle-connectivity

Conference Paper

Jun 2015

A 520k (18900, 17010) Array Dispersion LDPC Decoder Architectures for NAND Flash Memory

Article

Aug 2015

Although Latin square is a well-known algorithm to construct low-density parity-check (LDPC) codes for satisfying long code length, high code-rate, good correcting capability, and low error floor, it has a drawback of large submatrix that the hardware implementation will be suffered from large barrel shifter and worse routing congestion in fitting NAND flash applications. In this paper, a top-down design methodology, which not only goes through code construction and optimization, but also hardware implementation to meet all the critical requirements, is presented. A two-step array dispersion algorithm is proposed to construct long LDPC codes with a small submatrix size. Then, the constructed LDPC code is optimized by masking matrix to obtain better bit-error rate (BER) performance and lower error-floor. In addition, our LDPC codes have a diagonal-like structure in the parity-check matrix leading to a proposed hybrid storage architecture, which has the advantages of better area efficiency and large enough data bandwidth for high decoding throughput. To be adopted for NAND flash applications, an (18900, 17010) LDPC code with a code-rate of 0.9 and submatrix size of 63 is constructed and the field-programmable gate array simulations show that the error floor is successfully suppressed down to BER of 10⁻¹². An LDPC decoder using normalized min-sum variable-node-centric sequential scheduling decoding algorithm is implemented in UMC 90-nm CMOS process. The postlayout result shows that the proposed LDPC decoder can achieve a throughput of 1.58 Gb/s at six iterations with a gate count of 520k under a clock frequency of 166.6 MHz. It meets the throughput requirement of both NAND flash memories with Toggle double data rate 1.0 and open NAND flash interface 2.3 NAND interfaces.

Byte-reconfigurable LDPC codec design with application to high-performance ECC of NAND flash memory systems

Article

Jul 2015

The reliability of NAND Flash memory deteriorates due to multi-level cell technique and advanced manufacturing technology. To deal with more errors, LDPC codes show superior performance to conventional BCH codes as ECC of NAND Flash memory systems. However, LDPC codec for NAND Flash memory systems faces problems of high redesign effort, high on-chip memory cost and high-throughput demand. This paper presents a byte-reconfigurable cost-effective high-throughput QC-LDPC codec design for NAND Flash memory systems. Reconfigurable codec design is proposed to support various QC-LDPC codes for different Flash memories. To save on-chip memory cost, shared-memory architecture and rescheduling architecture are presented for encoder and decoder, respectively. The shared-memory architecture can save 23% area cost of the encoder and the rescheduling architecture reduces 15% area cost of decoder. In addition, the proposed sub-iteration based early termination (SIB-ET) scheme reduces 29.6% decoding iteration counts compare with the state-of-the-art early termination scheme when raw BER of Flash memory is $3times 10^{-3}$. Finally, the QC-LDPC codec for NAND Flash memory systems is implemented in TSMC 90 nm technology. The post-layout result shows that the core size is only 6.72 ${rm mm}^{2}$ at 222 MHz operating frequency.

Construction of Quasi-Cyclic LDPC Cycle Codes over Galois Field GF(q) Based on Cycle Entropy and Applications on Patterned Media Storage

Article

Nov 2015

In this paper, we focus on the construction of a type of quasi-cyclic low-density-parity-check (QC LDPC) codes called cycle codes. Based on our previous work, the maximum cycle entropy (MCE) algorithm for constructing nonbinary LDPC codes can be extended to its QC form (QC-MCE), which maintains the QC structure of the parity-check matrix. With this method employed, an elegant distribution of nonzero entries over the Galois field GF(q) can be obtained among the cycles whose length is related to the girth. Thus, the independence of probabilistic information transferred during decoding is increased, leading to a better performance. Extensive simulation results show that the proposed QC-MCE algorithm behaves much better than the conventional random one and performs as well as the existing method over a pattern media channel with both additive white Gaussian noise (AWGN) and transition jitter noise (TJN). The decoding complexity of our proposed codes is reasonably low due to the QC structure of the codes. The codes constructed with the proposed method can be well applied over the patterned media storage.

Design of Binary LDPC Codes with Parallel Vector Message Passing

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