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Limited-Search Chase Decoding Algorithm for LDPC

Coded Underwater Acoustic Multiuser Channels

Xingming Li1, Zhiliang Qin2*, Yu Qin2, Yuanhao Sun1, Qidong Lu2, and Xiaowei Liu2 Weihai Cloud Computing Center, China

2 Weihai Beiyang Electrical Group Co., Ltd, Weihai, Shandong, China

Email: {lixingming; qinzhiliang; qinyu; sunyuanhao; luqidong; liuxiaowei}@beiyang.com

Abstract—In this paper, we propose a low-complexity soft-

input/soft-output (SISO) Chase multiuser detector that has a

polynomial computational complexity in terms of the number of

the least reliable bit positions for low-density parity-check

(LDPC) coded code-division multiple-access (CDMA) systems,

which is a potentially competitive technology for underwater

acoustic networks (UWAN). Simulation results over highly

correlated channels show that the proposed detector can afford

searching over a larger number of the least reliable bit positions

and achieve better bit-error-rate (BER) performance as

compared with the Chase-II detector at much lower complexity.

Index Terms—Chase decoding, coded CDMA, local

neighborhood, multiuser detection, soft-input/soft-output

I. INTRODUCTION

The turbo principle, which consists of iterative

processing, random interleaving, and soft-input/soft-

output (SISO) decoding, has significantly stimulated the

research on multiuser detection for coded code-division

multiple-access (CDMA) systems, which is considered as

a candidate for underwater acoustic networks (UWAN)

[1]. In [2], it is observed that a synchronous CDMA

channel can be viewed as a block code; while an

asynchronous channel is equivalent to a convolutional

code. This observation has led to the natural format of a

serially concatenated system that consists of the CDMA

channel as an inner code and the single-user channel code

as an outer code. In [3], an SISO multiuser detector based

on the A Posteriori Probability (APP) algorithm [4] is

developed to generate reliability information for single-

user channel decoders and is shown to provide near-

single-user Bit-Error-Rate (BER) performance. The

computational complexity of the APP multiuser detector,

however, is exponential in terms of the number of users K

in the system, i.e., O(2K) per iteration.

In [5], a low-complexity iterative receiver based on the

Chase-II decoding algorithm [6] was proposed for turbo-

coded CDMA systems. The proposed Chase-II multiuser

detector first constructs 2q candidate vectors by

identifying and perturbing the q (0<q<<K) least reliable

bit positions and then produces the a posteriori log-

likelihood ratios (LLR) for single-user turbo decoders.

Simulation results in [5] have shown that the Chase-II

Manuscript received August 24, 2020; revised January 15, 2021.

Corresponding author email: [email protected]

doi: 10.12720/jcm.16.2.76-81

multiuser detector can significantly reduce the

computational complexity while with only a small

performance loss as compared with the APP algorithm

for moderate-to-high signal-to-noise ratios (SNR). A

limitation of the Chase-II detector, however, is that its

computational complexity is exponential in terms of q,

which is still intensive for large values of q or K. In this

paper, we propose an improved Chase multiuser detector

based on the concept of the local neighborhood of the q

least reliable bit estimates, which has a polynomial

complexity of O(q2/2-q/2+K+1) per iteration. Simulation

results over highly correlated low-density parity-check

(LDPC) coded channels show that the proposed detector

can afford searching over a larger number of the least

reliable bit positions and performs better than Chase-II

detector [5] while with much lower computational

complexity. Moreover, compared with other well-known

SISO schemes such as the soft interference cancellation

and minimum-mean-square-error filtering (SICMMSE)

multiuser detector [7], the proposed scheme is shown to

achieve better BER performance and converge more

quickly for the considered systems.

This paper is organized as follows. In Section II, the

system model is described. In Section III, the proposed

SISO Chase multiuser detector is developed. In Section

IV, simulation results for highly correlated LDPC coded

systems are presented. Finally, the conclusion is drawn in

Section V.

II. SYSTEM MODEL

We consider a synchronous system with K users. For

the kth user, k=0,,K-1, a frame of binary data bits dk is

encoded by a channel encoder with code rate Rk and

passed into a random interleaver (Intl). We assume the

same LDPC code is used by all users. The interleaved

code bit stream is binary-phase-shift-keying (BPSK)

modulated, multiplied by a spreading waveform skt with

duration N chips. At the receiver, the sampled signal at

the ith bit interval can be expressed as,

iiii zWbRy (1)

where TK ibibi 10 ,, b is a K1 vector of K users’

LDPC code bits, W is a KK diagonal amplitude matrix,

i.e., 10 ,,diag Kww W , and TK izizi 10 ,, z

is a colored Gaussian noise vector with zero mean and

Journal of Communications Vol. 16, No. 2, February 2021

76©2021 Journal of Communications

1

iiiET

Rzz 2 , R(i)=S(i)S(i)T is a K×K correlation

matrix, TKi 10 ,, ssS , 1,0, ,, Nkkk ss s is the

spreading sequence assigned to the kth user with uniform

probability over NN 1,1 . We assume that the

channel is K-symmetrical [8], i.e., the correlation matrix

R is characterized by Ri,i=1, Ri,j=, ji , i,j=0,,K-1. For

a synchronous system, it is well known that y(i)

constitutes a set of sufficient statistics for detecting all K

bits at the ith interval [4]. Hence, we drop the time index i

in (1) to simplify notations in the following sections.

III. ITERATIVE CHANNEL DETECTION

A. Full-Complexity APP Multiuser Detector

An iterative multiuser receiver consists of two parts: an

SISO multiuser detector and a bank of K single-user

LDPC decoders. At each iteration, the multiuser detector

takes as input the a priori information λ2 delivered by

LDPC decoders and produces the a posteriori LLR of bit

bk as [5]

2

2

,1,1

,1log

λy

λy

k

k

kbP

bP

1:1,1

1:1,1

exp

exp

log

kK

kK

b

b

b

b

b

b (2)

where the metric of a K-tuple candidate vector b is

defined as

bλWRWbbWbybTTT

22 2

12

2

1

(3)

The computational complexity of the APP multiuser

detector is given by O(2K) per iteration, where the

complexity here refers to the average number of times

that the metric (3) is evaluated for detecting all K

transmitted bits in one interval [4].

B. Chase Decoding Algorithm

The Chase multiuser detector in [5] is developed based

on the Chase-II algorithm for decoding linear block codes

[6] and can be approached by viewing tentative hard

estimates fed back from single-user LDPC decoders in

the previous multiuser iteration as an initial solution,

which are given,

0,1

0,1ˆ

,2

,2

k

k

kb

(4)

Based on kb̂ˆ b , 1,,0 Kk , we can restrict

the LLR calculation (2) to a candidate list ε as,

1:

1:

,1exp

exp

log

k

k

b

b

k

εb

εb

b

b

(5)

where ε is defined as a subset of {-1,+1}K associated with

the initial solution b̂ .

1) Chase-II algorithm

To apply the Chase-II decoding algorithm to SISO

multiuser detection, we first define the reliability of bit

estimates kb̂ as absolute value of its a priori LLR |λ2,k |.

By ordering the reliabilities of K bit estimates in a

descending order, we refer to bits corresponding to the

smallest q values as the least reliable bits, where q is an

arbitrary integer with 0<q<<K. That is, we assume that

these q bits are most likely to be in error. Identifying and

forming all possible binary combinations over these q bit

positions, we can construct a subset ε1 that consists of 2q

K-tuple candidate vectors. After forming ε1, the next step

is to select a vector b in ε1 that corresponds to the largest

metric as an updated hard decision of the transmitted bit

vector and then form another subset ε2 that consists of

qK neighboring vectors each differing from b over

exactly one reliable bit position. The LLR calculation (5),

which is based on the union 21 Uεεε , thus has a

computational complexity of O(2q+K-q) per iteration.

2) Improved chase decoding algorithm

In [5], it has been shown that BER performance of the

Chase-II multiuser detector can be effectively improved

by using a larger value of q. The computational

complexity of the Chase-II algorithm, however, is

growing exponentially in terms of q. We propose an

improved Chase decoding algorithm that forms candidate

vectors based on the concept of the local neighborhood of

the q least reliable bits 10

ˆ,,ˆˆ

qjj bb v in the initial

solution b̂ , which is defined as [9], [10]

H

qN vvvv ˆ1,1ˆ (6)

From a geometrical perspective, v̂N represents a

Hamming sphere with radius κ that consists of all

possible binary vectors with Hamming distance not more

than κ from the central vector v̂ , and H denotes the

Hamming weight of its vector argument. For all vv ˆN ,

v differs from v̂ by at most κ elements. For example,

the 1-opt neighborhood N1 of {1, 1, 1} consists of three

vectors {-1, 1, 1}, {1, -1, 1}, and {1, 1, -1}. Similarly, we

can form a larger κ-opt neighborhood v̂N by flipping

one up to κ ( q1 ) bits in v̂ . Clearly, if κ=1, the

proposed detector based on the 1-opt local neighborhood

generates 1+q candidate vectors and hence is equivalent

to the Chase-III decoding algorithm [6]. If κ=q and with

qq

i i

q2

0

, the algorithm is equivalent to an exhaustive

search over the q least reliable bit positions as required by

Chase-II algorithm.

The size of a κ-opt local neighborhood, however, is

0i i

qN , which may be prohibitive for large values of

κ or q. Hence, the complexity is still high to search a

complete κ-opt local neighborhood. To efficiently search

for a subset of candidate vectors for LLR calculation, the



principle of the Lin-Kernighan algorithm [11] for solving

the traveling salesperson problem (TSP) [10] can be

applied to deliver high-quality approximate solutions by

restricting the search to the q least reliable bit positions.

The basic idea is that we can partition a κ-opt local

neighborhood into several 1-opt local neighborhoods. At

each step, a variable number of elements in the initial

solution are flipped to arrive at a better neighboring

solution. To find the most profitable move, a sequence of

q(q+1)/2 solutions is produced at each step. The solution

in the sequence with the largest metric can be accepted as

the input for the next step, which may differ in one up to

q elements from the initial solution. For the sake of

achieving low computational complexities, we focus on

the 1-step Lin-Kernighan algorithm that forms a subset of

1+q(q+1)/2 candidate vectors for LLR calculation per

iteration. For clarity purposes, the pseudocode of the

proposed algorithm is given as follows,

1. Initialization: Obtain 10

ˆ,,ˆˆ

qjj bb v as the set of the

q least reliable bits in the K-tuple tentative estimate

b̂ as formed in (4).

2. Generate a set T={0,...,q_1} to record bit-flipping

positions. Let a q-tuple vector v denote the current

trial solution and set vv ˆ .

a. Find the best neighboring solution vi by flipping

only elements recorded in T, such as Ω(vi)≥

Ω(vj), Tj , where vi (vj, respectively) differs

from v by only the ith (jth, respectively)

element. b. Set vi→v and exclude the ith position from T as

T=T\{i}. Go to step 2.a) until T=Φ.

3. Substitute each q-tuple trial solution obtained in the

search into b̂ over the q least reliable bit positions

so as to form a subset 1ε of 1+q(q+1)/2 K-tuple

candidate vectors

Assuming that q=4 and 3210

ˆ,ˆ,ˆ,ˆˆjjjj bbbbv , an example

of 1+q(q+1)/2=11 candidate vectors generated by

performing the 1-step Lin-Kernighan algorithm over the q

least reliable positions is given by,

)ˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

)ˆˆˆˆˆˆ(

10

10

10

10

10

10

10

10

10

10

10

1

3210

3210

3210

3210

3210

3210

3210

3210

3210

3210

3210

Kjjjj

Kjjjj

Kjjjj

Kjjjj

Kjjjj

Kjjjj

Kjjjj

Kjjjj

Kjjjj

Kjjjj

Kjjjj

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

bbbbbb

ε

(7)

where each row in 1ε denotes a K-tuple candidate vector.

After forming 1ε , the next step is to select a vector

kbb , k=0,,K-1, in 1ε that corresponds to the largest

metric as a hard decision of the transmitted bit vector, i.e,

bbεb

1

maxarg (8)

and form another subset 2ε that consists of K-q

neighboring vectors each of which differs from b over

exactly one reliable bit position. With q=4, an example of

2ε is given by,

)(

)(

)(

)(

1210

1210

1210

1210

2

3210

3210

3210

3210

KKjjjj

KKjjjj

KKjjjj

KKjjjj

bbbbbbbb

bbbbbbbb

bbbbbbbb

bbbbbbbb

ε (9)

The LLR calculation given in (5), which is based on

the union 21 Uεεε , thus has a computational

complexity of O(q2/2-q/2+K+1) per iteration. Note that

for large values of q or K, the proposed detector requires

much fewer candidate vectors as compared with the

Chase-II detector. For example, for the value of q=8 and

K=20, the proposed detector forms 49 candidate vectors

per iteration, which is only 18.3% of the number of 268

vectors required by the Chase-II detector [5].

The proposed iterative receiver for LDPC coded

CDMA systems operates in an iterative manner. At the

first iteration, no a priori information is available. Hence,

the proposed detector is replaced by a linear minimum-

mean-square-error (MMSE) detector with time-invariant

coefficients. The MMSE output, which is assumed

Gaussian, is forwarded to channel decoders to produce

the initial solution b̂ and the a priori LLR λ2, which will

be used in Chase multiuser detection starting from the

second iteration. Extensive simulation results have shown

that the proposed scheme may be viewed as a general

approach to reduce the computational complexity of

SISO detection/decoding algorithms and can be extended

to more general systems such as asynchronous channels,

multiple-input multiple-output (MIMO) fading channels,

coded intersymbol interference (ISI) channels, and

decoding turbo product codes (TPC) [12].

IV.

For the simulation results of the proposed receiver for

highly-correlated LDPC coded systems over AWGN

channels. all users are transmitting with the same power

and the same rate-1/2 (504, 252). LDPC code based on

random construction is used as the channel code [13]. For

each multiuser iteration between the multiuser detector

and single-user LDPC decoders, 5 sum-product decoding

iterations [14] are used inside LDPC decoders.

First, we consider a system with user number K=10

and identical cross-correlation coefficients between users

ρ=0.6. The simulation is tested for the first ten multiuser

iterations. For clarity, we only present BER results

obtained at the 10th iteration in Fig. 1. For comparison

purposes, the performance of the iterative receiver based

on the full-complexity (FC) APP multiuser detection [3]

and the single-user (SU) performance of LDPC decoding

over AWGN channels at the 5th sum-product iterations

are also included as well as the performance of the



PERFORMANCE RESULTS

iterative receiver that employs the soft interference

cancellation and time-varying MMSE (SICMMSE)

multiuser detector [7] and the performance of the full κ-

opt local-search (LS) detector [9]. Fig. 1 shows that the

SICMMSE receiver has a performance gap of 0.6 dB

from that of the FC receiver at the BER of 10-3. In this

case, we can resort to multiuser detectors based on search

methods to minimize performance degradation at low-to-

moderate SNR. The full κ-opt LS multiuser detector [9],

which performs the 1-step Lin-Kernighan algorithm over

all K bit positions and hence requires a total of

1+K(K+1)/2=56 vectors for LLR calculation per iteration,

is shown to achieve performance very close to that of the

APP throughout the simulated SNR range. The Chase-II

detector [5], which forms 2q+K-q=68 vectors over the

q=6 least reliable bit positions, performs very close to the

APP detector within 0.1dB at the BER of 10-3. The

computational complexity of Chase-II detector, however,

can be significantly reduced without suffering from

performance degradation. The proposed Chase detector

with q=6, which forms only q2/2-q/2+K+1=26 candidate

vectors per iteration, is shown to approach closely the

performance of the Chase-II detector with q=6

throughout the simulated SNR range. In Fig. 2, we

present the BER performance of the proposed detector at

Eb/No=4.6dB versus the number of multiuser iterations in

comparison with the others, the proposed detector

produces almost identical BER results from the 1st

iteration to the 10th iteration. It is also observed that the

SICMMSE detector converges most slowly to the

performance of the FC detector among various detection

schemes. At the 5th and the 10th iterations, the proposed

detector outperforms the SICMMSE detector by more

than two orders of magnitude.

0 1 2 3 4 5 610

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

BE

R

SU

FC

SICMMSE

Full k-opt

Chase-II,q=6

Proposed,q=6

Fig. 1. Performance of the proposed iterative receiver for a rate-1/2

LDPC coded system with K=10 and =0.6.

Next, we consider a larger system with K=20 and

=0.6. Only the performance of various low-complexity

schemes is presented in Fig. 3, where the full κ-opt LS

detector achieves the best BER performance by

generating a total of 1+K(K+1)/2=211 out of 220

candidate vectors as required by the APP algorithm. The

Chase-II detector with q=6, which forms 2q+K-q=78

vectors, has a performance loss of 0.3 dB at the BER of

10-2. Hence, the proposed detector can significantly

reduce the complexity of the Chase-II algorithm without

compromising BER performance. The proposed detector

with q=6 performs very close to the Chase-II detector

throughout the simulated SNR range by generating only

q2/2-q/2+K+1=36 candidate vectors. Fig. 3 also shows

that the performance of the Chase detector can be

effectively improved by using a larger value of q. The

Chase-II detector with q=8, which forms 2q+K-q=268

vectors per iteration, provides a performance gain of 0.2

dB over the same detector with q=6 at the BER of 10-2. A

more efficient approach, however, is to use the proposed

detector that can afford searching over a larger number of

q=11 least reliable bit positions and yet has

approximately the same computational complexity for the

considered system as that of the Chase-II detector with

q=6. Fig. 3 shows that the proposed detector with q=11,

which forms 76 candidate vectors for LLR calculation per

iteration, performs better than the Chase-II detector with

q=6 (corresponding to 78 vectors) by 0.3dB. Interestingly,

compared with the Chase-II detector with q=8 (268

vectors), the proposed detector with q=11 also provides a

non-negligible performance gain at the BER of 10-2

though it has a much lower computational complexity

(28.4% of the former). Moreover, even compared with

the full κ-opt LS detector that searches 211 vectors per

iteration, the proposed detector with q=11 is shown to

produce almost identical BER results by requiring only

76 candidate vectors for LLR calculation.

1 2 3 4 5 6 7 8 9 1010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Iteration

BE

R

FC

Full k-opt

Chase-II,q=6

Proposed,q=6

SICMMSE

SU

Eb/No=4.6 dB

Fig. 2. Convergence of the proposed iterative receiver for a rate-1/2


0 1 2 3 4 5 610

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

BE

R

SU

Full k-opt

SICMMSE

Chase-II,q=6

Proposed,q=6

Chase-II,q=8

Proposed,q=11


LDPC coded system with K=20 and =0.6



Finally, we consider a larger system with K=30 and

=0.6. Fig. 4 shows that the proposed detector with q=11

performs very close to the full κ-opt LS detector by

forming only 86 as compared with 466 candidate vectors

required by the latter and outperforms the Chase-II

detector with q=8 that requires 278 vectors per multiuser

iteration. For clarity purposes, complexity comparison of

systems with 10, 20, 30 users is quantified in Table I.

0 1 2 3 4 5 6 710

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

BE

R

SU

Full k-opt

SICMMSE

Proposed,q=6

Proposed,q=11

Chase-II,q=6

Chase-II,q=8



TABLE I: COMPLEXITY COMPARISON

Detectors Full κ-opt LS Chase-II Proposed

Complexity

(a) 1+K(K+1)/2 2q+K -q 1+K+q2/2-q/2

(b) 211 268 76

(c) 466 278 86

where (a) Complexity comparison among various detection schemes. (b)

Complexity comparison for the 20-user system; (c) Complexity

comparison for the 30-user system. q=8 for the Chase-II detector and

q=11 for the proposed detector in (b) and (c), respectively.

V. CONCLUSION

In this paper, we have proposed a low-complexity

SISO Chase multiuser detector that performs the 1-step

Lin-Kernighan algorithm over the q least reliable bit

positions in the tentative hard estimate fed back from

LDPC decoders in the previous multiuser iteration. The

proposed detector has a polynomial computational

complexity of O(q2/2-q/2+K+1) per iteration, as

compared with the original Chase-II multiuser detector

that has an exponential complexity of O(2q+K-q).

Simulation results have shown that the proposed detector

can afford searching over a larger number of the least

reliable bit positions and provide better BER performance

than the Chase-II multiuser detector.

CONFLICT OF INTEREST

The authors declare no conflict of interest.

AUTHOR CONTRIBUTIONS

Xingming Li and Xiaowei Liu conducted the research;

Yu Qin and Yuanhao Sun implemented the simulation;

Xingming Li, Zhiliang Qin and Qidong Lu wrote the

paper; all the authors had approved the final version.

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Copyright © 2021 by the authors. This is an open access article

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or adaptations are made.

Xingming Li was born in Shandong

Province, China, in 1983. He received

the B.Eng. degree in the College of

Information Engineering and the M.Eng.

degree in the Mechanical and Electrical

Engineering from Shandong University,

Weihai, in 2009 and 2015 respectively.

He is currently the CEO in Cloud

Computing Center, Weihai, China. His research interests

include machine learning, digital signal processing, cloud

computer, computer system.

Zhiliang Qin was born on August 22,

1974. He obtained the B. Eng. degree

from the Beijing Institute of Technology

(BIT) in 1995, the M. Eng. degree from

the Graduate School of China Academy

of Engineering Physics (CAEP) in 1998,

and the Ph. D. Degree from the Nanyang

Technological University (NTU),

Singapore in 2003. From 2002 to 2019, he worked at the

Agency for Science, Technology, and Research (ASTAR) in

Singapore, a renowned government agency both on academic

researches and engineering applications, as the Scientist in the

area of algorithm developments for artificial intelligence (AI),

deep learning, machine learning, signal processing, data

analytics, optimization theories, and data storage systems. From

2019 to present, he is the Deputy Chief Engineer at the Weihai

Beiyang Electric Group. Co. Ltd. He published around 70 SCI

and EI technical papers and (co-)authored three US. Patents. He

frequently takes the role of being the reviewer of international

research journals and being the Technical Committee Member

(TPC) of international conferences on AI and signal processing,

including the ICSPS 2020, MLMI2020, ICCCR2021, etc.

Yu Qin was born in Weihai, China, in

1992. He received the B.S. and the M.S.

degree in the School of Mathematics

from Shandong University, Jinan, China,

in 2014 and 2017, respectively. He is

currently with Weihai Beiyang Electrical

Group Co., Ltd, Weihai, 264200, China.

His current research interests include

artificial intelligence, speech recognition, natural language

processing.

Yuanhao Sun was born in Weihai,

China, on August 21, 1981. He received

the B.Eng. in the College of Computer

Science and Technology from Shandong

University, JiNan, China, in 2005. He is

currently with Weihai Beiyang Electrical

Group Co., Ltd, Weihai, 264200, China.

His current research interests include

cloud computing, big data, artificial intelligence.

Qidong Lu was born in Yantai, China,

on January 17, 1992. He received the

B.Eng. in the College of Mechanical and

Electronic Engineering and the M.Eng.

degree in the College of Electrical

Engineering and Automation from

Shandong University of Science and

Technology, Qingdao, China, in 2016

and 2019, respectively. He is currently with Weihai Beiyang

Electrical Group Co., Ltd, Weihai, 264200, China. His current

research interests include fault diagnosis, artificial intelligence,

speech recognition.

Xiaowei Liu was born in 1971. He is the

master of underwater acoustic

engineering and had been as the

academic visitor in the Computer

Department of Stanford University. He

has 20 years of experience in technology

research, product development and new

business incubation, with more than 50

technology patents. From 2019 to present, he is the Chief

Technology Officer at the Weihai Beiyang Electric Group Co.,

Ltd.



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