# Boosting Learning for LDPC Codes
to Improve the Error-Floor Performance

Hee-Youl Kwak
University of Ulsan
<EMAIL_ADDRESS>
&Dae-Young Yun
Seoul National University
<EMAIL_ADDRESS>
&Yongjune Kim
POSTECH
<EMAIL_ADDRESS>
&Sang-Hyo Kim11footnotemark: 1
Sungkyunkwan University
<EMAIL_ADDRESS>
&Jong-Seon No
Seoul National University
<EMAIL_ADDRESS>
Corresponding authors.

###### Abstract

Low-density parity-check (LDPC) codes have been successfully commercialized in
communication systems due to their strong error correction capabilities and
simple decoding process. However, the error-floor phenomenon of LDPC codes, in
which the error rate stops decreasing rapidly at a certain level, presents
challenges for achieving extremely low error rates and deploying LDPC codes in
scenarios demanding ultra-high reliability. In this work, we propose training
methods for neural min-sum (NMS) decoders to eliminate the error-floor effect.
First, by leveraging the boosting learning technique of ensemble networks, we
divide the decoding network into two neural decoders and train the post
decoder to be specialized for uncorrected words that the first decoder fails
to correct. Secondly, to address the vanishing gradient issue in training, we
introduce a block-wise training schedule that locally trains a block of
weights while retraining the preceding block. Lastly, we show that assigning
different weights to unsatisfied check nodes effectively lowers the error-
floor with a minimal number of weights. By applying these training methods to
standard LDPC codes, we achieve the best error-floor performance compared to
other decoding methods. The proposed NMS decoder, optimized solely through
novel training methods without additional modules, can be integrated into
existing LDPC decoders without incurring extra hardware costs. The source code
is available at https://github.com/ghy1228/LDPC_Error_Floor.

##  1 Introduction

The field of learning-based decoding for error-correcting codes began with
research on training neural networks to produce the information vector when
given a distorted codeword ABC (2018, 2018, 2018, 2018). These works assume an
arbitrary neural network with no prior knowledge of decoding algorithms, and
accordingly, face the challenge of learning a decoding algorithm. In contrast,
model-based neural decoders are designed by mapping a well-known graph-based
iterative decoding algorithm, such as belief propagation (BP) and min-sum (MS)
decoding algorithms, to a neural network and then training its weights ABC
(2018). Compared to the arbitrary network approaches or the error correction
transformer ABC (2018), model-based neural decoders offer the advantages of
guaranteeing the performance of existing iterative algorithms and using
hardware architectures ABC (2018) that are already well optimized for
iterative decoding algorithms.

LDPC codes have been incorporated into WiMAX and 5G communication systems ABC
(2018, 2018), owing to their strong error-correcting capabilities and low
decoding complexity ABC (2018, 2018). However, more advanced LDPC coding
technology needs to be developed for diverse communication environments lying
in the scope of future 6G systems. In particular, for environments that
require extremely low frame error rate (FER) such as the next generation
ultra-reliable and low-latency communications (xURLLC) ABC (2018), it is
crucial to mitigate the error-floor in the decoding of LDPC codes. The error-
floor phenomenon refers to an abnormal phenomenon where the FER does not
decrease as rapidly as in the waterfall region ABC (2018, 2018). The error-
floor phenomenon also should be addressed for systems demanding very high
reliability, such as solid-state drive (SSD) storage ABC (2018), DNA storage
ABC (2018), and cryptosystems ABC (2018). However, enhancing other features of
LDPC codes often inadvertently reinforces the error-floor phenomenon as a side
effect. For instance, the error-floor tends to be intensified when optimizing
LDPC codes for superior waterfall performance or decoding by low complexity
decoders such as quantized MS decoders ABC (2018). Therefore, research focused
on alleviating the error-floor, especially when decoding LDPC codes optimized
for performance with low decoding complexity, has become significant. Such
advancements will broaden the applications of LDPC codes.

###  1.1 Main contributions

With this need in mind, we focus on how to train a low-complexity neural MS
(NMS) decoder to prevent the error-floor in well designed LDPC codes. The main
contributions of the paper are threefold as follows.

Boosting learning using uncorrected words: We first leverage the boosting
learning technique ABC (2018, 2018) that employs a sequential training
approach for multiple classifiers, wherein subsequent classifiers concentrate
on the data samples that preceding classifiers incorrectly classify. Inspired
by this method, we divide the neural decoder into two cascaded neural decoders
and train the first decoder to be focused on the waterfall performance, while
training the second decoder to be specialized in handling the uncorrected
words that are not corrected by the first decoder due to the error-floor
phenomenon. Uncorrected words in the error-floor region mostly contain small-
error patterns related to trapping sets or absorbing sets ABC (2018), which
can be effectively corrected by weighting decoding messages. As a result, a
significant performance improvement in the error-floor region is achieved by
boosting learning.

Block-wise training schedule with retraining: To mitigate the error-floor,
iterative decoding typically requires a large number of decoding iterations,
often exceeding $50$ ABC (2018, 2018, 2018, 2018). However, NMS decoders
encompassing many iterations can undergo the vanishing gradient problem in
training ABC (2018). To address this problem, we propose a new training
schedule inspired by block-wise training methods ABC (2018, 2018). The
proposed block-wise training schedule divides the entire decoding iterations
into sub-blocks and trains these sub-blocks in a sequential manner.
Additionally, rather than fixing the weights trained from previous blocks, we
retrain them to escape from local minima. As a result, the proposed schedule
enables to train numerous weights for all $50$ iterations successfully while
outperforming both the multi-loss method ABC (2018) and the iter-by-iter
schedule ABC (2018).

Weight sharing technique with dynamic weight allocation: The weight sharing
technique is a way to reduce the number of trainable weights by grouping
specific weights to share a common value. The waterfall performance does not
severely degrade even if we bundle all weights for each iteration ABC (2018,
2018). However, our observations indicate that this does not hold true in the
error-floor region, implying that a higher degree of weight diversity is
necessary to correct error patterns causing the error-floor. To obtain
sufficient diversity with a minimal number of weights, we dynamically assign
different weights to unsatisfied check nodes (UCNs) and satisfied check nodes
(SCNs) in the decoding process. By utilizing only two weight values for SCNs
and UCNs each iteration, we achieve the performance of the NMS decoder using
different weights for every edge. This method reduces the number of weights to
be trained to only 2.6% of the original number of weights.

We employ these training methods on a range of representative LDPC codes
adopted for standards such as WiMAX ABC (2018), IEEE 802.11n ABC (2018), and
5G new radio ABC (2018). The FER point at the onset of the error-floor
diminishes by over two orders of magnitude for all codes compared to
conventional weighted MS (WMS) decoding ABC (2018). Compared to existing NMS
decoding approaches ABC (2018, 2018), our proposed scheme exhibits a notably
enhanced capability to suppress the error-floor. This scheme also achieves a
similar performance as the state-of-the-art post-processing method in ABC
(2018), with only a third of the iterations.

Table 1: Comparison between model-based neural decoders

Reference | Codes | Target region | Decoders | Training sample | Training schedule | Weight sharing
---|---|---|---|---|---|---
This work | Standard LDPC |  | Waterfall,
---
Error-floor
MS | Uncorrected words |  | Block-wise
---
with retraining
| Spatial with
---
UCN weights
ABC (2018) | BCH | Waterfall | BP, MS | Received words | One-shot | Temporal
ABC (2018) | Standard LDPC | Waterfall | BP, MS | Received words | Iter-by-Iter | Spatial
ABC (2018) | Short LDPC | Waterfall | BP | Absorbing set | One-shot | Temporal
ABC (2018) | Regular LDPC |  | Waterfall,
---
Error-floor
FAID | Received words | One-shot | Temporal
ABC (2018) | Short LDPC |  | Waterfall,
---
Error-floor
FAID | Trapping set | One-shot | Temporal
ABC (2018) | Standard LDPC | Waterfall | Layered | Received words | Iter-by-Iter | UCN weights

###  1.2 Related works

We compare the proposed training scheme with the existing schemes for NMS
decoders in Table 1. First, all works except ABC (2018, 2018) aim to improve
the waterfall performance. Although the scope of the works in ABC (2018, 2018)
includes the error-floor performance, they assumed specific conditions of
binary symmetric channels (BSC) and FAID, while we deal with the more general
situation of additive white Gaussian noise (AWGN) channels and MS decoding.
Regarding the training sample selection, the training samples can be received
words randomly taken from the AWGN channel ABC (2018, 2018, 2018, 2018), or
codewords with erroneous trapping sets (or absorbing sets) ABC (2018, 2018).
However, to use the method in ABC (2018, 2018), trapping sets should be
enumerated, which is only applicable to short LDPC codes and not feasible for
medium to large length standard LDPC codes. In contrast, the proposed boosting
method, which generates training samples through decoding with linear
complexity, can be applied even to codes of several thousand lengths.

For scheduling of training, it is common to train all weights at once (One-
shot training) ABC (2018) and some works sequentially train the weights
corresponding to a single iteration locally ABC (2018, 2018), while we train a
block of iterations with retraining. In terms of the weight sharing
techniques, we confirm that the proposed sharing technique using UCN weights
is superior to the spatial or temporal sharing technique used in ABC (2018,
2018, 2018, 2018, 2018). Meanwhile, a method of assigning different weights to
UCNs has been introduced in ABC (2018), but they applied the same UCN weight
to all CNs belonging to a proto CN when at least one CN is unsatisfied,
whereas we pinpoint specific UCNs and apply weights individually.

There have been studies adding hypernetworks to the Vanilla NMS decoder ABC
(2018, 2018, 2018) or using a transformer architecture ABC (2018) to improve
the waterfall performance at the expense of increased training and decoding
costs. While the proposed training techniques are broadly applicable to these
augmented neural decoders, this work primarily aims to improve the error-floor
performance of the Vanilla NMS decoder under practical conditions.

##  2 Preliminaries

###  2.1 LDPC codes

In this paper, we consider quasi-cyclic (QC) LDPC codes, which have been
adopted in various applications due to their implementation advantages ABC
(2018, 2018). The Tanner graph of a QC-LDPC code, consisting of $n=Nz$ VNs and
$m=Mz$ CNs, can be obtained by lifting the protograph, composed of $M$ proto
CNs and $N$ proto VNs, with a lifting factor $z$ ABC (2018, 2018). Let $E$ be
the total number of edges in the protograph. As a running example, we use the
WiMAX QC-LDPC code of length $n=576$ and code-rate $3/4$ with $N=24,M=6,E=88$,
and $z=24$ ABC (2018) .

###  2.2 Neural min-sum decoding

For iteration $\ell$, let $m^{(\ell)}_{c\rightarrow v}$ represent the message
from CN $c$ to VN $v$ and let $m^{(\ell)}_{v\rightarrow c}$ represent the
message from VN $v$ to CN $c$. The neighboring nodes of $x$ are represented by
$\mathcal{N}(x)$. The initial conditions are $m^{(0)}_{v\rightarrow
c}=m_{v}^{\rm ch}$, $m^{(0)}_{c\rightarrow v}=0$ for the channel LLR
$m_{v}^{\rm ch}$ of VN $v$. For $\ell={1,\ldots,\overline{\ell}}$, the NMS
decoding algorithm ABC (2018) updates the messages as follows

$\displaystyle m^{(\ell)}_{v\rightarrow c}$
$\displaystyle=\overline{w}^{(\ell)}_{v}m_{v}^{\rm
ch}+\sum_{c^{\prime}\in\mathcal{N}(v)\setminus
c}m^{(\ell-1)}_{c^{\prime}\rightarrow v}$ (1) $\displaystyle
m^{(\ell)}_{c\rightarrow v}$ $\displaystyle={w}^{(\ell)}_{c\rightarrow
v}\left(\prod_{v^{\prime}\in\mathcal{N}(c)\setminus v}{\rm
sgn}\left(m^{(\ell)}_{v^{\prime}\rightarrow
c}\right)\right)\min_{v^{\prime}\in\mathcal{N}(c)\setminus
v}|m^{(\ell)}_{v^{\prime}\rightarrow c}|,$ (2)

where $\overline{w}^{(\ell)}_{v}$ and ${w}^{(\ell)}_{c\rightarrow v}$ are
called the VN weight and CN weight, respectively. At the last iteration
$\overline{\ell}$, output LLRs $m_{v}^{\rm o}$ are computed as $m_{v}^{\rm
o}=m_{v}^{\rm
ch}+\sum_{c^{\prime}\in\mathcal{N}(v)}m^{({\ell})}_{c^{\prime}\rightarrow v}$.

By quantizing $m^{(\ell)}_{v\rightarrow c}$, $m^{(\ell)}_{c\rightarrow v}$,
and $m_{v}^{\rm ch}$, the quantized NMS decoding algorithm is obtained. The
quantized decoders are widely used in practical applications due to its low
complexity and commonly employed in existing error-floor researches ABC (2018,
2018, 2018, 2018). Therefore, we use it to ensure a fair comparison.
Specifically, we use 5-bit uniform quantization with a maximum magnitude of
$7.5$ and a step size of $0.5$ for the quantized NMS decoder as in ABC (2018,
2018, 2018).

###  2.3 Training weights for the NMS decoder

If all the weights in (1) and (2) are set to $1$, NMS decoding is equivalent
to MS decoding ABC (2018), or if VN weights $\overline{w}^{(\ell)}_{v}$ are
$1$ and CN weights ${w}^{(\ell)}_{c\rightarrow v}$ have the same value, the
decoder operates as the WMS decoder ABC (2018). The NMS decoder gains
performance improvement over the WMS or MS decoder by greatly increasing the
diversity of weights. However, the full diversity weights increase the
training complexity and require a large amount of memory to store the weights.
Therefore, previous studies used weight sharing techniques by assigning the
same value to weights with the same attributes. First, since this paper deals
with QC-LDPC codes, we use the protograph weight sharing technique ABC (2018)
by default, assigning the same weight value to the VNs (or CNs) belonging to a
proto VN (or CN). Then, the weights to be trained are represented by
$\\{\overline{w}_{v_{p}}^{(\ell)},w_{c_{p}\rightarrow v_{p}}^{(\ell)}\\}$ for
a proto VN $v_{p}$ and a proto CN ${c_{p}}$. The total number of weights is
then $(N+E)\overline{\ell}$. If we employ spatial weight sharing in ABC
(2018), only one VN weight and one CN weight remain for each iteration, and
the weights are
$\\{\overline{w}^{(\ell)},w^{(\ell)}\\}_{\ell=1}^{\overline{\ell}}$, with a
total number of $2{\overline{\ell}}$. On the other hand, by using temporal
weight sharing ABC (2018) to eliminate differences between iterations, the
weights are $\\{\overline{w}_{v_{p}},w_{c_{p}\rightarrow v_{p}}\\}$, and the
number is $(N+E)$.

The neural network in Fig. LABEL:Fig:Graph(b) corresponds to NMS decoding of
the Tanner graph in Fig. LABEL:Fig:Graph(a) with $\overline{\ell}=2$. The
input to this neural network is the channel LLR vector $(m_{1}^{\rm
ch},\ldots,m_{n}^{\rm ch})$, and the output is the output LLR vector
$(m_{1}^{\rm o},\ldots,m_{n}^{\rm o})$. For each iteration, two hidden layers
are arranged, and each hidden layer has a number of nodes equal to the number
of edges in the Tanner graph. In the odd hidden layers, the VN to CN message
operation in (1) is performed, while in the even hidden layers, the CN to VN
message operation in (2) is performed. The input layer is also connected to
the odd hidden layers, which corresponds to the addition of the channel LLR in
(1). The messages from the $2\ell$-th hidden layer to the $(2\ell+1)$-th
hidden layer are weighted by ${w}^{(\ell)}_{c\rightarrow v}$, and the messages
from the input nodes to the $(2\ell+1)$-th hidden layer are weighted by
$\overline{w}^{(\ell)}_{v}$. As our goal is to reduce the FER in the error-
floor region, we use the FER loss, $\frac{1}{2}\bigg{[}1-{\rm sgn}\Big{(}{\rm
min}_{1\leq v\leq N}m_{v}^{\rm o}\Big{)}\bigg{]}$ABC (2018).

##  3 Proposed training method

In this section, we introduce the proposed training methods through three
subsections. The organized training algorithm is shown in Fig.
LABEL:Fig:Algorithm.

###  3.1 Boosting learning using uncorrected words

For the boosting learning approach, we first divide the entire decoding
process into two stages: the base decoding stage $\\{1,\ldots,\ell_{1}\\}$ and
the post decoding stage
$\\{\ell_{1}+1,\ldots,\ell_{1}+\ell_{2}=\overline{\ell}\\}$. Training of base
decoder follows the conventional training method: the received words sampled
from ${\rm E}_{\rm b}/{\rm N}_{0}$ region ${\mathcal{R}}_{1}$ are used as
training samples. Specifically, we set ${\ell_{1}}=20$ and
${\mathcal{R}}_{1}=\\{2.0,2.5,3.0,3.5,4.0\\}$, which corresponds the waterfall
region of WMS decoding. We use the spatial sharing technique (i.e., ${\bf
w}_{B}=\\{\overline{w}^{(\ell)},w^{(\ell)}\\}_{\ell=1}^{\ell_{1}}$) since this
achieves comparable performance to the full diversity weights in the waterfall
region.

In Fig. LABEL:Fig:FER_iter30(a), the base NMS decoder is compared with the MS
decoder and the WMS decoder with a single weight of $0.75$ for
$\overline{\ell}=20$. The WMS decoding performance has a severe error-floor
even though its waterfall performance is better than the MS decoding
performance. Compared to the MS and WMS decoders, the NMS decoder for
$\overline{\ell}=20$ performs better over the training range
${\mathcal{R}}_{1}$. On the other hand, the NMS decoder performs worse than
the MS decoder in the error-floor region (e.g., $4.5$ dB), which is outside
${\mathcal{R}}_{1}$. To improve the performance in the error-floor region, a
straightforward approach is extending the training range ${\mathcal{R}}_{1}$
to include the error-floor region. However, the FER of the base decoder for
received words from the error-floor region is very low, resulting in an almost
negligible FER loss. Consequently, integrating the error-floor region into the
training range does not impact the weight update process.

Before training the post decoder, we first collect uncorrected words that the
trained base decoder fails to correct among the received words sampled from
region ${\mathcal{R}}_{2}$. Then, the uncorrected words serve as training
samples for the post decoder, which is distinct from the conventional training
methods. The post decoder trains the weights
$\\{\overline{w}^{(\ell)}_{v_{p}},w^{(\ell)}_{c_{p}\rightarrow
v_{p}}\\}_{\ell=\ell_{1}+1}^{\overline{\ell}}$ with the aim of correcting the
uncorrected words. After completing the training, the trained weights are used
for the NMS decoding algorithm in (1)–(2). From the perspective of the NMS
decoder, it performs continuous decoding up to iteration $\overline{\ell}$
using the trained weights, but for the sake of discussion, we assume as if
there are two cascaded decoders (base and post) in the perspective of
training. Note that we employ the full diversity weights for the post decoder
to confirm the best performance but we will introduce the shared weights ${\bf
w}_{P}$ (used in Fig. LABEL:Fig:Algorithm) in the next subsection. We also set
$l_{2}=10$, $\overline{\ell}=30$ for this experiment, and subsequently extend
the maximum number of iterations in the following subsection.

To analyze the effectiveness of the proposed boosting learning, we compare the
following three cases.

  * Case 1

Uncorrected words sampled at $4.5$ dB in the error-floor region (i.e.,
${\mathcal{R}}_{2}=4.5$).

  * Case 2

Uncorrected words sampled at $3.5$ dB in the waterfall region (i.e.,
${\mathcal{R}}_{2}=3.5$).

  * Case 3

Received words sampled at $4.5$ dB without filtering.

Regarding Case 1 and Case 2, we collect a total of $60{,}000$ uncorrected
words, allocating $50{,}000$ for training, $5{,}000$ for validation, and
remaining $5{,}000$ for test. Training is conducted for $100$ epochs. Fig.
LABEL:Fig:FER_iter30(b) shows the distribution of the number of errors after
base decoding and post decoding for the test samples used in Case 1 and Case
2. For Case 1, the uncorrected words collected in the error-floor region
mainly have a small number of errors since most of decoding errors are trapped
in small trapping sets, so the distribution is concentrated on small numbers
(see Case 1 after base decoding). For ease of use, we refer to codewords with
fewer than $11$ remaining errors as small-error words. The post decoder, which
is mainly trained on these small-error words, corrects a significant number of
small-error words (see Case 1 after post decoding). Out of the total $5{,}000$
test samples, $68.5\%$ of samples are corrected by the post decoder, resulting
that the test FER for the test samples is $0.315$. This means that, when
decoding for received words of the AWGN channel, the resulting FER at ${\rm
E}_{\rm b}/{\rm N}_{0}=4.5$ dB after post decoding is $0.315$ times of the FER
after base decoding as shown in Fig LABEL:Fig:FER_iter30(a). In other words,
the post decoder is successfully trained to correct small-error words inducing
the error-floor.

On the other hand, Case 2, where the samples are collected from the waterfall
region, has a distribution that is widespread across all areas (see Case 2
after base decoding). In addition, the distribution remains almost the same
after post decoding (see Case 2 after post decoding), which means that the
post decoder fails to reduce the FER sufficiently. For the test samples, the
test FER at ${\rm E}_{\rm b}/{\rm N}_{0}=3.5$ dB after post decoding is $0.77$
times of the FER after base decoding whose difference is not noticeable as
shown in Fig LABEL:Fig:FER_iter30(a). Comparing the results of the two cases,
we conclude that composing mainly with small-error words facilitates the post
decoder to learn to correct small-error words more effectively. As a result,
Fig. LABEL:Fig:FER_iter30(a) shows that Case 1 mitigates the error-floor more
than Case 2. Meanwhile, for Case 3, where all received words are used as
training samples without filtering, almost all of them are corrected during
base decoding. Since the post stage training is mainly performed on codewords
without errors, the loss function becomes almost $0$. Then, the weights of the
post decoder are unchanged from the initial value $1$, and accordingly, the
performance approaches the MS decoding performance, as shown in Fig.
LABEL:Fig:FER_iter30(a).

###  3.2 Block-wise training schedule

In the previous subsection, the number of iterations for the post decoder is
set to $\ell_{2}=10$. To lower the error-floor further, a large number of
iterations is required, so we set $\ell_{2}=30,\overline{\ell}=50$. However,
deep neural decoders with a large iteration number are prone to suffer from
the vanishing gradient problem. In order to tackle this issue, we propose a
block-wise training schedule which is shown in Fig. LABEL:Fig:Schedule(a). The
proposed training schedule locally trains the weights corresponding to a block
of $\Delta_{1}$ iterations at each training stage. In the first stage, the
weights belonging to the first block are trained, and in the next stage, the
weights of the subsequent $\Delta_{1}$ iterations are trained. At this point,
the weight values of previous $\Delta_{2}$ iterations, which are already
trained in the first stage, are further trained by taking the result of the
first stage training as its initial state. This retraining, which is not used
in the iter-by-iter training schedule ABC (2018), assists in preventing the
learning process from falling into a local minimum. Note that the method of
one-shot training ABC (2018) corresponds to the case of
$\Delta_{1}=\ell_{2},\Delta_{2}=0$, and the iter-by-iter training schedule ABC
(2018) is equivalent to the case of $\Delta_{1}=1,\Delta_{2}=0$.

We employ a greedy approach to determine the optimal values for $\Delta_{1}$
and $\Delta_{2}$, resulting that $\Delta_{1}=5,\Delta_{2}=10$ offers superior
performance in terms of the test FER. Fig. LABEL:Fig:Schedule(b) shows the
evolution of test FER as a function of the iteration number. In the case of
one-shot training ABC (2018), the vanishing gradient problem hinders training
the weights of earlier iterations, so the test FER stays roughly the same
until iteration $40$ and starts to fall thereafter. The same behavior is
observed for $\overline{\ell}=40$. Thus, the test FERs of $\overline{\ell}=40$
and $\overline{\ell}=50$ are almost the same. Next, since the iter-by-iter
training schedule ABC (2018) finds the optimal weight for each individual
iteration, the test FER falls even at the first iteration of the post decoder
(i.e., $\ell=21$) without encountering the vanishing gradient problem.
However, this local optimization leads to a degraded local minimum, and
consequently, the test FER gradually and slowly decreases with each iteration.
Likewise, the multi-loss method ABC (2018) shows a similar result. In
contrast, the block-wise training schedule with $\Delta_{1}=5,\Delta_{2}=0$
shows a superior test FER at iteration $25$ compared to the other training
schedules because it results in a better solution by training the weights of
multiple iterations simultaneously. Moreover, the schedule with retraining
(i.e., $\Delta_{1}=5,\Delta_{2}=10$) outperforms the schedule without
retraining (i.e., $\Delta_{1}=5,\Delta_{2}=0$) at iteration $30$ though it
shows a worse result at iteration $25$. This implies that through retraining,
the weights of intermediate iterations have been adjusted to preprocess error
patterns thereby leading to stronger correction capabilities in the final
iteration. As a result, at the maximum of $50$ iterations, the proposed
training schedule with $\Delta_{1}=5,\Delta_{2}=10$ provides the better test
FER value compared to the other training schedules as shown in Fig.
LABEL:Fig:Schedule(b): $0.11$ for the block-wise, $0.16$ for the multi-loss,
$0.18$ for the one-shot, $0.37$ for the iter-by-iter, .

###  3.3 Weight sharing technique using UCN weights

| Test FER | Number of weights
---|---|---
Full diversity | $0.112$ | $(N+E)\ell_{2}=3360$
Spatial sharing | $0.168$ | $2\ell_{2}=60$
Temporal sharing | $0.186$ | $(N+E)=112$
Proposed sharing | $0.111$ | $3\ell_{2}=90$

Figure 2: Illustration of the proposed weight sharing technique and comparison
with other sharing techniques.

Assuming the techniques proposed thus far (in detail, uncorrected words at
${\rm E}_{\rm b}/{\rm N}_{0}=4.5$ dB and training schedule with
$\Delta_{1}=5,\Delta_{2}=10$) are used, we compare the weight sharing
techniques in Fig. 2. Compared to the full diversity weights, the spatial and
temporal sharing techniques significantly reduce the number of distinct
weights, but cause performance degradation. In contrast, the proposed sharing
technique that introduces a new weight type called UCN weight shows almost
identical performance while using only about 2.6% of the weights compared to
the full diversity weights. The proposed sharing technique assigns different
weights to UCNs and SCNs as shown in Fig. 2. This is feasible because the
decoder knows whether a CN satisfies the check equation or not. Using the
spatial sharing technique and distinguishing between SCN weight $w^{(\ell)}$
and UCN weight $\hat{w}^{(\ell)}$, the proposed sharing technique can be
represented as
$\\{\overline{w}^{(\ell)},w^{(\ell)},{\hat{w}}^{(\ell)}\\}_{\ell}$ for
iteration $\ell$ and the total number of distinct weights becomes $3\ell_{2}$.

Techniques using different weights for UCNs and SCNs have also been proposed
in ABC (2018, 2018). However, the work ABC (2018) uses only one suppression
factor $\rho$ to represent the UCN weight (i.e.,
${\hat{w}}^{(\ell)}=(1+\rho)w^{(\ell)}$), making the UCN weight dependent on
the CN weight. As a result, due to the limited degree of freedom for the UCN
weight, it is difficult to obtain the decoding diversity for effectively
removing various types of error patterns. Moreover, in ABC (2018), if at least
one of the CNs belonging to a single proto CN is unsatisfied, all $z$ CNs from
the proto CN are weighted by the UCN weight. This approach, which applies the
same weight to a large number of CNs tied together at the proto level, is not
suitable for correcting words with a small number of UCNs, because it does not
separately handle individual CNs like the proposed method.

##  4 Performance evaluation

In this section, we compare the proposed and other conventional decoding
schemes in terms of the decoding performance. All simulations are performed by
NVIDIA GeForce RTX 3090 GPU and AMD Ryzen 9 5950X 16-Core Processor CPU. For
training, the weights are trained by the Adam optimizer ABC (2018) with a
learning rate of $0.001$. We evaluate the decoding performance using Monte
Carlo methods with at least $500$ uncorrected words for each FER point. Fig.
LABEL:Fig:FER_iter50(a) shows the FER performances of the proposed scheme for
the WiMAX LDPC code. The proposed scheme incorporates the i) boosting learning
with uncorrected words, ii) block-wise training schedule with retraining, and
iii) spatial weight sharing with UCN weights. The performance is compared with
MS decoding, WMS decoding, and existing NMS decoding schemes in ABC (2018,
2018). Among the neural decoder studies listed in Table 1, we exclude
comparison with the studies that use FAID and layered decoding, or that
require enumerating trapping sets and absorbing sets. In addition, we choose
not to compare with augmented neural networks ABC (2018, 2018) since our
approach does not increase model complexity to deal with the low-error-rate
region of long codes. A comparative analysis for short codes in the waterfall
region can be found in the appendix.

For the NMS decoding schemes in ABC (2018, 2018), the base decoder is used for
iterations from $1$ to $20$ like the proposed scheme, and the training methods
introduced in ABC (2018, 2018) are employed for the post stage. The full
diversity weights are used for the schemes in ABC (2018, 2018) to observe
their best performance. For the scheme in ABC (2018), received words in the
waterfall region (${\rm E}_{\rm b}/{\rm N}_{0}$ $2$-$4$ dB) are used as
training samples, and the weights for the post stage are trained all at once
without a training schedule. For the scheme in ABC (2018), received words from
the ${\rm E}_{\rm b}/{\rm N}_{0}$ points where the MS decoder achieves bit
error rate of $10^{-3}$ are used as training samples, and the iter-by-iter
training schedule is employed. The remaining hyper-parameters are set in the
same way as in the proposed scheme. As shown in Fig. LABEL:Fig:FER_iter50(a),
the conventional NMS decoders in ABC (2018, 2018) show good performance in the
waterfall region (${\rm E}_{\rm b}/{\rm N}_{0}$ $2$-$4$ dB), but the error-
floor occurs from $4$ dB. This is because the training samples are composed of
received words without filtering. In contrast, the proposed scheme shows
excellent performance in both the waterfall and error-floor regions, and the
error-floor phenomenon is barely noticeable down to FER of $10^{-7}$. In
particular, comparing the results of $\overline{\ell}=20$ and
$\overline{\ell}=50$, it is confirmed that the post decoder successfully
removes the error-floor.

Table 2: Complexity comparison for the WiMAX LDPC of
$(N,M,E,z,\alpha)=(24,6,88,24,15.6)$, where $\alpha$ is the comparison count
for each CN (i.e., $\alpha=d_{c}+\lceil{\rm log}d_{c}\rceil-2$ for the CN
degree $d_{c}$ ABC (2018)).

| Complexity per iteration | Total complexity | Memory
---|---|---|---
Addition $A$ | Comparison $C$ | Multiplication $M$ | $(A+2C+M)\overline{\ell}$ | for weights
MS, $\overline{\ell}=50$ |  $2Ez$ $=4224$ |  $\alpha Mz$ $=2256$ | $Ez=2112$ | $542400$ | 0
NMS ABC (2018, 2018), $\overline{\ell}=50$ |  $(2E+N)z$ $=4800$ | $676800$ | $3760$
Proposed NMS, $\overline{\ell}=50$ | $130$
| Post processing ABC (2018)
---
$\overline{\ell}=150$
$Ez=2112$ | $1627200$ | $0$

In addition, we compare the proposed scheme with the state-of-the-art post
processing scheme in ABC (2018). We directly reference the simulation results
from ABC (2018). As shown in Fig. LABEL:Fig:FER_iter50(a), the scheme in ABC
(2018) shows a similar or worse performance to the proposed scheme, but it has
disadvantages of having very high decoding complexity and latency since it
consumes a large number of iterations $\overline{\ell}=150$. Table 2 compares
the schemes in terms of the decoding complexity. The NMS decoder has more
multiplications than the MS decoder by $(E+N)z$ due to the weighting
operation. The number of other operations is the same as in the MS decoder.
Total complexity is evaluated with assumption that the comparison $C$ is as
twice as complex than the addition $A$ and multiplication $M$ ABC (2018). The
additional memory for storing the weights of the proposed scheme is
$3\overline{\ell}$ which is much lower than those of ABC (2018, 2018) which
exploit full weight diversity. Since the scheme in ABC (2018) does not use
weighting, the complexity per iteration is lower than the proposed NMS scheme,
but the total complexity is more than twice as high as the proposed NMS scheme
due to the higher number of iterations. Moreover, additional complexity is
required for the error path detector ABC (2018). In Fig.
LABEL:Fig:FER_iter50(b), (c), similar results are observed for the IEEE802.11n
LDPC and 5G LDPC codes, where the proposed scheme outperforms the other
schemes and achieves an FER of $10^{-7}$ without a severe error-floor.

##  5 Conclusions

This paper proposed training methods for the NMS decoder of LDPC codes to
enhance the error-floor performance. Using uncorrected words from the base
decoder, we trained the post decoder to be specialized for error patterns
causing the error-floor, promoting decoding diversity in the cascaded base and
post decoders. We also proposed a training schedule to circumvent the
vanishing gradient and local minimum problems, and a weight sharing technique
that significantly reduces the number of distinct weights without sacrificing
performance. The proposed NMS decoder using the trained weights showed the
excellent waterfall and error-floor performances for several standard LDPC
codes. Along with the performance improvement, the proposed training scheme
has the advantage of being flexibly applicable regardless of the types of
channel, code, and decoding algorithm. This scheme can also be implemented
directly on hardware architectures without additional costs, and can be
directly utilized with no prior analysis of the target code and decoding
algorithm.

##  6 Acknowledgments

This work was supported by Samsung Electronics Co., Ltd (IO230411-05859-01),
by Electronics and Telecommunications Research Institute (ETRI) grant funded
by the Korean government [2021-0-00746, Development of Tbps wireless
communication technology], by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (No. RS-2023-00212103), and by the
National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIT) (No. RS-2023-00247197).

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##  7 Appendix

###  7.1 A better trade-off between the waterfall and error-floor performances

Typically, LDPC code design assumes the availability of a floating-point
decoder and optimizes the code to maximize waterfall performance. In Fig.
LABEL:Fig:FER_Quantization(a), it’s evident that the optimized WiMAX LDPC code
outperforms the non-optimized regular LDPC code in terms of the waterfall
performance. Additionally, the WMS decoder enhances the performance by
assigning weights. As shown in Fig. LABEL:Fig:FER_Quantization(a), WMS
decoding of the WiMAX LDPC code exhibits the best waterfall performance when
floating-point operations are possible. However, for practical hardware
implementation, quantization of decoding messages is essential, but it leads
to performance degradation. Notably, WMS decoding suffers substantial
performance degradation and an early onset of the error-floor. A side-by-side
comparison of the quantized WMS decoding performances between the non-
optimized and WiMAX LDPC codes reveals that the performance loss due to
quantization is more pronounced in optimized LDPC codes. Consequently, there’s
a need for a method that ensures high performance in both the waterfall and
error-floor regions with low complexity. This work accomplishes this goal by
leveraging machine learning techniques. In Fig. LABEL:Fig:FER_Quantization(a),
the proposed quantized NMS decoder achieves the floating WMS decoding
performance in both waterfall and error-floor regions.

###  7.2 Boosting learning of two networks v.s. conventional learning of a
single large network

In Fig. LABEL:Fig:FER_Quantization(b), we compare the proposed method with the
method that uses a single decoder, which is trained by a large single network
at once. For the proposed method, the boosting learning is employed with two
networks. Both have the same model size. Fig. LABEL:Fig:FER_Quantization(b)
shows that simply increasing the model size doesn’t significantly mitigate the
error-floor. In contrast, the two-stage network using boosting learning
achieves greater decoding diversity and demonstrates superior efficiency in
alleviating the error-floor.

###  7.3 Optimization of parameters of the block-wise training schedule

Table 3: Test FER values of the block-wise training with various $\Delta_{1}$ and $\Delta_{2}$. $\Delta_{1}$ $\backslash$ $\Delta_{2}$ | $0$ | $5$ | $10$ | $15$ | $30$
---|---|---|---|---|---
$1$ | $0.376$ | $0.144$ | $0.123$ | $0.122$ | $0.150$
$5$ | $0.193$ | $0.134$ | ${\bf 0.112}$ | $0.127$ | $0.137$
$10$ | $0.142$ | $0.142$ | $0.13$ | $0.128$ | $0.13$
$30$ | $0.179$ |  |  |  |

To fine-tune the parameters $\Delta_{1}$ and $\Delta_{2}$ used in the block-
wise training schedule, we compare the test FER values across a range of
$\Delta_{1}$ and $\Delta_{2}$. The comparison is done for the post decoder
training with $\ell_{2}=30$ and is summarized in Table 3. For the case where
$\Delta_{2}=0$ (i.e., without retraining), the performance improves as
$\Delta_{1}$ increases from $1$ to $10$. This can be attributed to the
training of more weights simultaneously, aiding in escaping from local minima.
However, if $\Delta_{1}$ becomes too large, the earlier weights are not
adequately trained due to the vanishing gradient problem, causing performance
degradation. In addition, as the retraining iteration number $\Delta_{2}$
grows for a given $\Delta_{1}$ value, a similar trend is noted: performance
enhances up to a certain point and then plateaus. According to Table 3, the
best performance is achieved with $\Delta_{1}=5$ and $\Delta_{2}=10$.

###  7.4 Discussion on the trained weights

Table 4: Trained weights | $\ell$ | Weights
---|---|---
VW $\overline{w}^{(\ell)}$ | $1\sim 10$ | $0.74$ | $0.98$ | $0.96$ | $1.10$ | $1.20$ | $1.15$ | $1.22$ | $1.19$ | $1.17$ | $1.09$
$11\sim 20$ | $1.15$ | $1.09$ | $1.07$ | $1.04$ | $1.05$ | $1.07$ | $1.00$ | $0.98$ | $0.90$ | $0.81$
$21\sim 30$ | $1.14$ | $0.97$ | $0.93$ | $0.69$ | $0.66$ | $0.58$ | $0.57$ | $0.51$ | $0.49$ | $0.45$
$31\sim 40$ | $0.48$ | $0.41$ | $0.37$ | $0.38$ | $0.38$ | $0.39$ | $0.32$ | $0.33$ | $0.32$ | $0.34$
$41\sim 50$ | $0.39$ | $0.37$ | $0.31$ | $0.31$ | $0.34$ | $0.45$ | $0.39$ | $0.37$ | $0.42$ | $0.42$
CW $w^{(\ell)}$ | $1\sim 10$ | $0.74$ | $0.71$ | $0.66$ | $0.69$ | $0.67$ | $0.70$ | $0.75$ | $0.71$ | $0.73$ | $0.76$
$11\sim 20$ | $0.73$ | $0.70$ | $0.77$ | $0.79$ | $0.84$ | $0.84$ | $0.86$ | $0.82$ | $0.81$ | $0.97$
$21\sim 30$ | $0.19$ | $0.30$ | $0.54$ | $0.58$ | $0.62$ | $0.59$ | $0.73$ | $0.73$ | $0.74$ | $0.78$
$31\sim 40$ | $0.72$ | $0.78$ | $0.79$ | $0.81$ | $0.88$ | $0.82$ | $0.88$ | $0.85$ | $0.97$ | $0.79$
$41\sim 50$ | $0.81$ | $0.82$ | $0.85$ | $0.91$ | $1.19$ | $0.86$ | $0.96$ | $1.10$ | $1.18$ | $1.62$
UCW $\hat{w}^{(\ell)}$ | $1\sim 10$ | $0.74$ | $0.71$ | $0.66$ | $0.69$ | $0.67$ | $0.70$ | $0.75$ | $0.71$ | $0.73$ | $0.76$
$11\sim 20$ | $0.73$ | $0.70$ | $0.77$ | $0.79$ | $0.84$ | $0.84$ | $0.86$ | $0.82$ | $0.81$ | $0.97$
$21\sim 30$ | $0.58$ | $0.80$ | $0.83$ | $0.73$ | $0.81$ | $0.86$ | $0.72$ | $0.78$ | $0.73$ | $0.73$
$31\sim 40$ | $0.59$ | $0.71$ | $0.64$ | $0.67$ | $0.67$ | $0.72$ | $0.65$ | $0.65$ | $0.66$ | $0.70$
$41\sim 50$ | $0.77$ | $0.70$ | $0.72$ | $0.81$ | $0.92$ | $0.58$ | $0.71$ | $0.76$ | $0.87$ | $1.57$

Table 4 shows the weights trained using the proposed training methods. Owing
to the spatial sharing technique, only three types of weights, VW
$\overline{w}^{(\ell)}$, CW $w^{(\ell)}$, and UCW $\hat{w}^{(\ell)}$, exist
for each iteration. Thus, weight variations across iterations can be depicted
in two dimensions, as shown in Fig. LABEL:Fig:Weights(a), enabling analysis of
the trained results. Without the weight sharing technique, there would be
$(N+E)$ weights per iteration, making a 2D graphical representation
unfeasible. Fig. LABEL:Fig:Weights(a) shows that the weights for the base
decoding stage, where the CW and VW are roughly $0.75$ and $1$, are similar to
the weights of conventional WMS decoding. As a result, the waterfall
performance of WMS decoding and NMS decoding are nearly indistinguishable. The
NMS decoding technique offers minimal advantage in the waterfall region.

On the other hand, for the post decoding stage, the weights undergo
significant changes. At iteration $21$, the VW increases while the CW
decreases substantially. This means that the channel LLR values are given more
weight when performing the sum operation at each VN, while the messages coming
from CNs are attenuated. As a result, the decoding process reverts to the
initial decoding state (with a high error rate). This leads to an increase in
the number of samples having more than $6$ errors (denoted by $e>6$), as shown
in Fig. LABEL:Fig:Weights(b), and a decrease in the number of samples with
$e\leq 6$. This intentional regression aims to break free from trapped error
patterns. From iteration $22$ onward, the CW gradually increases, the VW
decreases, and the count of $e\leq 6$ samples starts to increase again. After
iteration $27$, the CW and VW show less variations. We term the period from
iteration $21$ to $26$ as Phase 1 and the period following iterations as
Phase2. At iteration $20$, the numbers of samples with $e\leq 6$ and $e>6$ are
roughly equal, but through Phase 1, samples with $e>6$ transform into samples
with $e\leq 6$. Consequently, by the end of Phase 1, there are more samples
with $e\leq 6$ than samples with $e>6$. The small-sized error samples
diminished steadily in Phase 2.

The result without Phase 1 is also shown in Fig. LABEL:Fig:Weights(b). Without
Phase 1, the process of transforming into small-sized error samples is
omitted. Consequently, the decoding proceeds in a situation where the number
of samples with $e\leq 6$ and $e>6$ is similar. Although Phase 2 effectively
correct most of the $e\leq 6$ samples, the $e>6$ samples remain in significant
numbers as they are initially abundant. This indicates the need for the pre-
processing of Phase 1.

###  7.5 Multiple stage decoders

It is possible to extend to the case of more than two decoders in the same
manner. For instance, in the WiMax LDPC code result (Fig.
LABEL:Fig:FER_iter50(a)), uncorrected words from the error-floor region of the
base + post decoder (i.e., ${\rm E}_{\rm b}/{\rm N}_{0}$ 5dB) can be collected
to train a third decoder. In Fig. LABEL:Fig:Multiple(a), we present numerical
results that demonstrate the error-floor’s further reduction with the aid of
this third decoder (labeled as “Base+Post+Third”). In addition we add a graph
showing the finite length performance limit [44]. It is worth mentioning that
introducing this third decoder might lead to increased decoding latency.
Moreover, the process of collecting uncorrected words would be time-consuming,
especially in the very low FER region. Addressing these challenges could be a
promising direction for subsequent research.

In addition, we want to note that the concept of multi-stage decoding has been
employed in the previous works [45, 46, 47], particularly in the field of
coding theory. Our study has similarities with theses previous approaches, as
we also carries out a two-stage decoding. However, unlike physically divided
decoders in [45, 46, 47], our method utilizes a single LDPC decoder. While the
proposed decoder is conceptually divided into two stages (base/post) based on
iterations, in practice, we employ only a single decoder with distinct weight
parameter sets. Moreover, our approach has a novelty in that the post decoder
is ‘trained’ dependent on the results of the base decoder.

###  7.6 Application to a different channel

The concept of training using uncorrected words can be consistently applied
regardless of the channel type. In Fig. LABEL:Fig:Multiple(b), we include the
results for the Rayleigh channel with the scale parameter $\alpha=1$, showing
the effectiveness of our proposed method holds across different channel types.

###  7.7 Application to long length codes

Figure 9: FER performance for the long 5G LDPC code of length 1248.

We show the results for the (1248, 1056) 5G LDPC code in Fig. 9. This result
demonstrates the proposed method is also effective for long codes. There is
little improvement in waterfall performance with the NMS technique at such
long lengths. However, the performance improvement in the error-floor region
using our proposed method is clearly evident.

###  7.8 Comparison with the augmented neural decoders

Table 5: Comparison with [5], [34], [6] in terms of the waterfall performance of short codes. The results are measured by the negative natural logarithm of BER. Higher is better. Architecture |  | Vanilla NBP | NBP+HyperNet. | Transformer
---|---|---|---|---
Method | BP |  Orig. NBP [5] | Proposed |  Hyper [34] |  ECCT [6]
${\rm E}_{\rm b}/{\rm N}_{0}$ | 4 | 5 | 6 | 4 | 5 | 6 | 4 | 5 | 6 | 4 | 5 | 6 | 4 | 5 | 6
Polar (64,48) | 4.74 | 5.94 | 7.42 | 4.70 | 5.93 | 7.55 | 4.93 | 6.64 | 8.77 | 4.92 | 6.44 | 8.39 | 6.36 | 8.46 | 11.09
BCH (63,51) | 4.58 | 5.82 | 7.42 | 4.64 | 6.21 | 8.21 | 4.72 | 6.42 | 8.96 | 4.80 | 6.44 | 8.58 | 5.66 | 7.89 | 11.01
MacKay (96,48) | 8.15 | 11.29 | 14.29 | 8.66 | 11.52 | 14.32 | 8.26 | 11.83 | 15.85 | 8.90 | 11.97 | 14.94 | 8.39 | 12.24 | 16.41

In Table 5, we compare the waterfall performance of various neural decoders
for short length codes. To ensure a fair comparison with other works, we
employ the NBP decoder and utilize the soft-BER loss function for BER
performance optimization while making use of full diversity weights. Out of a
total of $50$ iterations, $20$ iterations are allocated for base decoding and
$30$ iterations for post decoding.

The result reveals a notable performance improvement over the original NBP
[5], which employs the same vanilla NBP architecture. The improvement is
particularly pronounced at high SNR. We achieve even better performance at
high SNR than the work [34], which adds hyper-networks to the vanilla NBP
architecture. In comparison to the ECCT [6] using the transformer
architecture, our method shows inferior performance for high density codes
(Polar, BCH), yet it achieves fairly comparable performance for LDPC codes.
Owing to its intricate transformer structure, the ECCT requires higher
training and decoding complexity than NMS decoders [6], implying its current
limited practicality. Nevertheless, it is worth emphasizing that our proposed
training method can be adaptable across any architecture. In other words,
there’s promising potential in future research that integrates the proposed
training methods with the transformer architecture.

###  7.9 Ablation study

Table 6: Ablation analysis. O: Boosting | O: Block-wise | O: Proposed sharing |
---|---|---|---
X: Conv. | X: One-shot | X: Full diversity |  FER (at ${\rm E}_{\rm b}/{\rm N}_{0}$ 5.0dB)
X | X | X | $3.14\times 10^{-6}$
O | X | X | $2.77\times 10^{-7}$
O | O | X | $1.84\times 10^{-7}$
O | O | O | $1.85\times 10^{-7}$

In Table 6, we provide the ablation study for the WiMAX LDPC code. The result
shows that the boosting learning is the core technique to reduce the FER
performance and the block-wise training also contributes the FER reduction,
while the proposed sharing technique does not involve performance degradation.


methReferences

# The Hidden Variables: Harnessing Half-Shell Potentials for Enhanced
Precision in Nuclear Reaction Calculations

Hao Liu  School of Physics Science and Engineering, Tongji University,
Shanghai 200092, China.  Jin Lei<EMAIL_ADDRESS>School of Physics Science
and Engineering, Tongji University, Shanghai 200092, China.  Zhongzhou Ren
School of Physics Science and Engineering, Tongji University, Shanghai 200092,
China.

###### Abstract

We explore the impact of half-shell components on nuclear reaction
calculations, focusing on nonelastic breakup cross sections within the
Ichimura-Austern-Vincent (IAV) model. By advocating for the use of a
consistent Single Folding Model (SFM) for all optical potentials in IAV
calculations, we aim to reduce the uncertainties associated with half-shell
components and enhance agreement with experimental data. We present results
from deuteron-induced reactions on 60Ni and 208Pb, which serve as surrogate
targets for neutron-induced reactions on short-lived nuclei. The application
of consistent optical potentials derived from the SFM shows improved alignment
with experimental data compared to traditional global phenomenological
potentials. Furthermore, we investigate the 59Co(6Li,$\alpha X$) reaction,
which reveals that the half-shell $T$-matrix plays a pivotal role in
accurately modeling nuclear reactions. Our findings suggest that a unified
approach to optical potentials, accounting for half-shell effects, is critical
for a precise understanding of complex nuclear reactions. This work highlights
the significance of the internal dynamics of the wave function, particularly
in lighter targets, and underscores the importance of the half-shell
$T$-matrix as a previously underappreciated variable in reaction calculations.

###### pacs:

24.10.Eq, 25.70.Mn, 25.45.-z

##  I Introduction

The optical potential is a fundamental concept in the study of nuclear
reactions, offering a method to describe the complex interactions between a
projectile and its target. In practice, the optical potential is often
represented by a Woods-Saxon shape with parameters that are adjusted to align
with experimental data. This fitting process effectively calibrates the on-
shell component of the optical potential, corresponding to situations where
the projectile’s momentum $k$ remains unchanged in magnitude during the
collision. Half-shell components complicate the nucleus-nucleus interaction
models. They are crucial in few-body nuclear systems, exemplified by the
Phillips line, which connects triton binding energy with neutron-deuteron
scattering length, revealing the sensitivity of three-nucleon systems to the
details of nucleon-nucleon (NN) interactions Faddeev (1961); Alt  _et al._
(1967); Hadizadeh  _et al._ (2020); Phillips (1968); Witała  _et al._ (2003).
Polyzou and Glöckle demonstrated that half-shell NN interactions, including
three-nucleon forces (3NFs), can replicate the effects of distinct NN
interactions Polyzou and Glöckle (1990).

Understanding and accounting for these half-shell components are therefore
essential for accurate theoretical descriptions of nuclear systems,
particularly when extending beyond the simplest two-body interactions. In this
paper, we investigate the impact of half-shell components on nuclear reaction
calculations, focusing particularly on nonelastic breakup (NEB) cross
sections. We employ the IAV (Ichimura, Austern, and Vincent) model Ichimura
_et al._ (1985), a theoretical framework crafted to unravel the intricacies of
NEB, a phenomenon occurring when a composite two-body projectile ($a=b+x$)
engages with a target $A$, leading to the detection of one projectile fragment
while the other engages nonelastically with the target. This interaction may
entail excitations, particle exchanges, or fragment absorption by the target.
The IAV model, leveraging the Distorted Wave Born Approximation (DWBA),
adeptly addresses the complexities of such NEB events and has shown efficacy
in reactions induced by weakly bound projectiles Potel  _et al._ (2017);
Carlson  _et al._ (2016); Lei and Moro (2015a, b, 2017, 2018, 2019).

In the practical application of the IAV model, different optical potentials
are required for each pair system involved. The inconsistent use of these
optical potentials within a single IAV model calculation can introduce
additional systematic errors due to varying behaviors in the interior part of
the scattering wave function or differing half-shell properties of the
$T$-matrix.

To address these issues, we propose the use of a consistent single folding
model as the starting point for all optical potentials in the IAV
calculations. This model, which uses the nucleon-A interaction of KD02 Koning
and Delaroche (2003); Lu  _et al._ (2023), is applied to deuteron and 6Li
induced reaction systems. We argue that this approach can reduce the
uncertainties associated with half-shell components. Establishing a consistent
baseline for the optical potentials minimizes the systematic discrepancies
that arise from different interior wave function behaviors and half-shell
$T$-matrix properties. Our results demonstrate that this consistent single
folding potential approach leads to a better agreement with experimental data,
reinforcing the importance of a unified framework in complex nuclear reaction
analysis.

The structure of the paper is as follows: Section II introduces the importance
of the half-shell component and provides a thorough derivation of the IAV
model. In Section III, we apply the formalism to inclusive reactions induced
by deuterons and 6Li. Lastly, Section IV summarizes the main findings of this
study and provides an overview of potential future developments.

##  II Theoretical framework

Figure 1:  Nuclear scattering dynamics and two-body interactions: (a) Elastic
scattering of a deuteron with initial momentum $k$ off a target nucleus,
resulting in an outgoing momentum $k^{\prime}$. (b) A contour plot
illustrating the real part of the two-body scattering T-matrix, with the half-
shell region delineated by the intersection of two solid black lines, and the
on-shell point is located at their intersection. This on-shell point dictates
the asymptotic behavior of the wave function. (c) The two-body scattering wave
function is influenced by the on-shell point at larger inter-nuclear distances
and undergoes changes due to half-shell interactions as the nuclei approach
one another. (d) A focused representation of the half-shell region from (b),
highlighting its impact on the scattering process.

###  II.1 Half-shell $T$-matrix and scattering wave function

In this subsection, we explore the properties of the half-shell $T$-matrix and
elucidate its relation to the scattering wave function. The $T$-matrix, a
fundamental component in quantum mechanics, satisfies the Lippmann-Schwinger
(LS) equation QM2 (1995):

$\displaystyle T(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},k_{0})$
$\displaystyle=V(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},k_{0})$
$\displaystyle+\int
d^{3}k^{\prime\prime}V(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime\prime},k_{0})\frac{1}{E^{+}-\frac{{{k}^{\prime\prime}}^{2}}{2\mu}}T(\mbox{\boldmath$k$}^{\prime\prime},\mbox{\boldmath$k$}^{\prime},k_{0}),$
(1)

where $E^{+}=\lim_{\epsilon\to 0}\frac{k_{0}^{2}}{2\mu}+i\epsilon$ and $k_{0}$
denotes the on-shell point. The term
$V(\mbox{\boldmath$k$},\mbox{\boldmath$k$}^{\prime},k_{0})$ stands for
$\langle\mbox{\boldmath$k$}|V(k_{0})|\mbox{\boldmath$k$}^{\prime}\rangle$,
indicating an energy-dependent potential. The half-shell $T$-matrix is defined
by

$T(\mbox{\boldmath$k$},\mbox{\boldmath$k$}_{0},k_{0})=\langle\mbox{\boldmath$k$}|T(k_{0})|\mbox{\boldmath$k$}_{0}\rangle=\langle\mbox{\boldmath$k$}|V(k_{0})|\chi^{(+)}\rangle.$
(2)

In a pure two-body scenario, only the on-shell $T$-matrix
$T(\mbox{\boldmath$k$}_{0},\mbox{\boldmath$k$}_{0},k_{0})$ is utilized to
calculate observables, as the half-shell part accounts for the internal
dynamics of the colliding entities. Consider a scenario where the projectile
is a deuteron; the optical potential must encapsulate the potential for the
deuteron to be excited into its continuum post-collision, which may result in
a change in the magnitude of the outgoing momentum compared to the incoming
one, as depicted in Fig. 1 (a). The shadow of the deuteron indicates that
after the collision, the deuteron may get excited into its continuum, and
therefore the magnitude of outgoing momentum differs from the incoming one.
These half-shell components act as hidden variables that are not directly
observable in two-body elastic scattering but become pivotal in reactions
involving greater degrees of freedom.

However, the half-shell $T$-matrix
$T(\mbox{\boldmath$k$},\mbox{\boldmath$k$}_{0},k_{0})$, depicted in Fig. 1 (d)
and representing the half-shell part of the whole $T$-matrix as shown in Fig.
1 (b), is for the $s$-wave reaction system of $d+^{60}$Ni at an incident
energy of 17 MeV in the Lab frame. The potential used in the calculations is
taken from Ref. Han  _et al._ (2006). This half-shell $T$-matrix can be used
to compute the scattering wave function, given by

$\langle\mbox{\boldmath$r$}|\chi^{(+)}\rangle=\langle\mbox{\boldmath$r$}|\mbox{\boldmath$k$}_{0}\rangle+\int
d\mbox{\boldmath$k$}\langle\mbox{\boldmath$r$}|\mbox{\boldmath$k$}\rangle
G_{0}^{(+)}(k)T(\mbox{\boldmath$k$},\mbox{\boldmath$k$}_{0},k_{0}),$ (3)

where $\langle\mbox{\boldmath$r$}|\mbox{\boldmath$k$}\rangle$ is the plane
wave. This relationship indicates that the half-shell $T$-matrix is integral
to deriving the scattering wave function, with the on-shell point determining
the asymptotic behavior and the off-shell points affecting the interior of the
wave function.

As depicted in Fig. 1 (c), the characteristics of the scattering wave function
can be explored by computing the half-shell $T$-matrix. On-shell points,
defined by $k=k^{\prime}=k_{0}$, dictate the asymptotic behavior of the
scattering wave function, as represented by the $S$-matrix. The half-shell
components, denoted by $k=k_{0}\neq k^{\prime}$, play a crucial role in the
detailed structure of the wave function within the interaction region. This
highlights the significance of half-shell interactions in shaping the wave
function. Consequently, they potentially impact nuclear reaction outcomes,
especially in scenarios that require consideration of more complex internal
structures and degrees of freedom.

###  II.2 The IAV model

We briefly review the IAV model Ichimura  _et al._ (1986, 1985) here. The
inclusive breakup reaction under study is described by the equation

$a(=b+x)+A\rightarrow b+B^{*},$ (4)

where the projectile $a$ has a two-body structure $(b+x)$, $b$ is the detected
particle, and $B^{*}$ denotes any possible final state of the $x+A$ system. In
the IAV model, the fragment $b$ is referred to as the spectator, while the
fragment $x$ is considered the participant.

The IAV model provides the NEB cross section as

$\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega_{b}\mathrm{d}E_{b}}\Big{|}_{\textbf{post}}^{\textbf{NEB}}=-\frac{2}{\hbar
v_{a}}\rho_{b}(E_{b})\langle\psi_{x}(\boldsymbol{k}_{b})|W_{x}|\psi_{x}(\boldsymbol{k}_{b})\rangle,$
(5)

where $v_{a}$ is the projectile-target relative velocity,
$\rho_{b}(E_{b})=\frac{\mu_{b}k_{b}}{(2\pi)^{3}\hbar^{2}}$ is the density of
states for particle $b$, with $\mu_{b}$ and $k_{b}$ being the reduced mass and
wave number, respectively. The term $W_{x}$ represents the imaginary part of
$U_{x}$, which characterizes the $x+A$ elastic scattering, and $\psi_{x}$ is
the $x$-channel wave function obtained by solving the inhomogeneous
differential equation

$(E_{x}-K_{x}-U_{x})\psi_{x}(\boldsymbol{k}_{b},\boldsymbol{r}_{x})=\langle\boldsymbol{r}_{x}\chi_{b}^{(-)}(\boldsymbol{k}_{b})|V_{\mathrm{post}}|\chi_{a}^{(+)}\phi_{a}\rangle,$
(6)

where $E_{x}=E-E_{b}$. $K_{x}$ is the kinetic energy operator for the relative
motion between fragment $x$ and target $A$, $\chi_{b}^{(-)}$ is the scattering
wave function with incoming boundary conditions, describing the scattering of
$b$ in the final channel with respect to the $x+A$ subsystem. The post-form
transition operator $V_{\mathrm{post}}=V_{bx}+U_{bA}-U_{bB}$ includes
$V_{bx}$, the potential binding the clusters $b$ and $x$ in the initial
composite nucleus $a$, $U_{bA}$, the fragment-target optical potential, and
$U_{bB}$, the optical potential in the final channel. Furthermore,
$\chi_{a}^{(+)}$ is the distorted wave describing the $a+A$ elastic scattering
with an outgoing boundary condition, and $\phi_{a}$ is the initial ground
state of the projectile $a$.

##  III Result

###  III.1 Application to $(d,pX)$

In the present research, we focus on deuteron-induced reactions $(d,pX)$ under
standard kinematic conditions, employing isotopes $\mathrm{{}^{60}Ni}$ and
$\mathrm{{}^{208}Pb}$ as target nuclei. To facilitate reliable comparisons
between different systems and to reduce systematic errors, we have chosen
incident energies that align with the peaks of the Coulomb barriers for each
nucleus. Specifically, we analyzed the $\mathrm{{}^{60}Ni}(d,pX)$ reaction at
an incident energy of 23 MeV and the $\mathrm{{}^{208}Pb}(d,pX)$ reaction at
55 MeV, with both energies referenced in the laboratory frame. The
experimental results pertaining to the $\mathrm{{}^{60}Ni}(d,pX)$ reaction
have been previously reported and are available for review in the referenced
literature Burtebaev  _et al._ (1984).

To underscore the impact of the effective $d+A$ interaction in IAV model, we
meticulously evaluate and compare the results derived from employing the
Single Folding Model (SFM) with the KD02 global nucleon-target interaction
against those obtained using two sets of established global phenomenological
optical model potentials for deuteron-induced reactions: the Han-Shi-Shen
(HSS) Han  _et al._ (2006) and the An-Cai (AC) An and Cai (2006) potentials.
Due to a lack of experimental data, the renormalization factors of SFM
potential were determined by fitting to the results of elastic scattering data
obtained through the Continuum Discretized Coupled Channels (CDCC) method and
experimental data Takei  _et al._ (1987); Matsuoka  _et al._ (1986). For the
$d-\mathrm{Ni}$ system, we determined the real part to be $Nr=0.669$ and the
imaginary part as $Ni=1.125$. Similarly, for the $d-\mathrm{Pb}$ system, the
real and imaginary parts were found to be $Nr=0.899$ and $Ni=1.169$,
respectively.

To elucidate the influence of these diverse optical potentials within the IAV
model, we showcase the absolute value of radial components of the scattering
wave function for $l=8$, corresponding to the relative angular momentum
between $d+\mathrm{{}^{60}Ni}$, in Fig. 2 (a), and for $l=19$, correlating
with the relative angular momentum between $d+\mathrm{{}^{208}Pb}$, in Fig. 2
(b). These particular values of angular momentum were selected due to their
predominant contribution to the NEB cross section. The findings and a more
detailed discussion on this matter are presented subsequently. The scattering
wave functions are depicted using solid lines for the SFM results, dashed
lines for the HSS potential, and dotted lines for the AC potential, allowing
for a clear comparison of the implications these potentials have.

Figure 2:  The absolute value of the radial wave functions and the half-shell
$T$-matrix ($T_{\text{half-shell}}$) for $d+\mathrm{{}^{60}Ni}$ at an incident
energy of 23 MeV in the Lab frame for $l=8$ are depicted in Panel (a) and
Panel (c), respectively. Similarly, the absolute value of the radial wave
functions and the half-shell $T$-matrix for $d+\mathrm{{}^{208}Pb}$ at an
incident energy of 55 MeV in the Lab frame for $l=19$ are plotted in Panel (b)
and Panel (d), respectively. Calculated results with SFM, HSS, and AC
potentials are depicted as solid black lines, dashed blue lines, and dotted
red lines, respectively, with on-shell points indicated by yellow circles.

Observations from Fig. 2(a) and (b) confirm that all the optical potentials
under consideration are phase-equivalent, indicating their equal aptitude in
describing elastic scattering data. However, notable differences are
discernible in the inner regions of the wave functions. To delve deeper into
these variations, we plot the absolute value of the half-shell $T$-matrix for
the same reaction system and partial waves in Fig. 2 (c) for
$d+\mathrm{{}^{60}Ni}$ and Fig. 2 (d) for $d+\mathrm{{}^{208}Pb}$. The half-
shell $T$-matrix is a critical component in the IAV calculation, accounting
for off-energy-shell effects that are pivotal for the comprehension of breakup
reactions where the deuteron disintegrates, leading to the relative energy of
the $np$ pair with respect to the target being off the energy shell.
Furthermore, the half-shell $T$-matrix is instrumental in calculating the
scattering wave function through the Lippmann-Schwinger equation. It is
evident that the on-shell $T$-matrix calculated with these different
potentials concurs, while the left part exhibits a marked divergence. The on-
shell point is related to the scattering matrix, which is used to generate the
scattering wave function asymptotically, whereas the left part of the half-
shell $T$-matrix contributes to the internal part of the wave function, which
is the region where the deuteron interacts nonelastically with the target.

Figure 3: Panels (a) and (b) display the differential cross-section of NEB
energy distribution with respect to the relative energies between
$n-{}^{60}\text{Ni}$ and $n-{}^{208}\text{Pb}$ targets for the reaction
systems 60Ni($d$,$pX$) at 23 MeV and 208Pb($d$,$pX$) at 55 MeV, respectively.
Panels (c) and (d) explore the contributions of the partial wave $l_{a}$,
which is the relative partial wave between the projectile and target, to the
NEB cross sections for the reaction systems 60Ni($d$,$pX$) at 23 MeV and
208Pb($d$,$pX$) at 55 MeV, respectively. The black solid lines represent the
results obtained with the SF potential, the blue dashed lines represent the
results obtained with the HSS potential, and the red dotted lines represent
the results obtained with the AC potential.

To explore these half-shell effects within the IAV model, we computed the NEB
cross section of the $(d,pX)$ reaction for targets $\mathrm{{}^{60}Ni}$ and
$\mathrm{{}^{208}Pb}$. The differential cross-section energy distributions
with respect to the relative energies between $n-\mathrm{{}^{60}Ni}$ and
$n-\mathrm{{}^{208}Pb}$ are depicted in Fig. 3 (a) and (b), respectively. The
solid, dashed, and dotted lines correspond to the results obtained with the
SFM, HSS, and AC optical potentials for $d+$target, respectively. The $n-p$
relative potential was modeled using a simple Gaussian form Ichimura  _et al._
(1986), and the left interactions in the IAV model were sourced from KD02
Koning and Delaroche (2003). Notably, in both reactions, the HSS and AC
potentials closely align with each other but deviate from the SFM potential
results for the $\mathrm{{}^{60}Ni}(d,pX)$ reaction. Conversely, all three
potentials show good agreement for the $\mathrm{{}^{208}Pb}(d,pX)$ reaction.
These observations are consistent with the hypothesis that for heavier
targets, the surface approximation is sufficient and only the asymptotic part
affects the NEB cross sections Liu  _et al._ (2023). However, for lighter
targets, the internal part of the scattering wave function plays a more
critical role.

In an effort to further understand this distinction, we display the integrated
NEB cross section as a function of angular momentum in Fig. 3 (c) for
$\mathrm{{}^{60}Ni}(d,pX)$ and Fig. 3 (d) for $\mathrm{{}^{208}Pb}(d,pX)$. It
is evident that for the heavier target, all the results concur across the
entire range of partial waves, whereas for the lighter target, discrepancies
emerge in the lower partial waves. This indicates that the surface
approximation holds validity for higher partial waves, but for lower partial
waves, particularly in lighter targets, the internal dynamics of the wave
function are more influential.

Figure 4: The double-differential cross section (DDX),
$d^{2}\sigma/(dE_{p}d\Omega_{p})$, as a function of outgoing proton energy in
the $\mathrm{{}^{60}Ni}(d,pX)$ reaction at 23 MeV and at an outgoing proton
angle of $45^{\circ}$ in the Lab frame, is presented. The yellow points
represent the experimental data Burtebaev  _et al._ (1984). The solid black
line is the sum of the NEB and EBU cross sections calculated using the SFM;
the blue dashed line shows the sum calculated using the HSS potential Han  _et
al._ (2006); and the red dotted line represents the sum calculated using the
AC potential An and Cai (2006).

Lastly, we compare the calculated inclusive breakup cross section with
experimental data. Fig. 4 displays the double differential cross-section
$d^{2}\sigma/(dE_{p}d\Omega_{p})$ as a function of the outgoing proton energy
in the lab frame for the reaction $\mathrm{{}^{60}Ni}(d,pX)$ at 23 MeV, with
the proton detected at $\theta=45^{\circ}$. The elastic breakup (EBU) is
computed by the Continuum Discretized Coupled Channels (CDCC) method, treating
the deuteron breakup as inelastic excitations to the $p-n$ continuum, which is
discretized in energy bins. For this case, $p-n$ states up to $\ell=0-2$
partial waves and a maximum excitation energy of 18 MeV were included. The
combined EBU and NEB results with different $d+\mathrm{{}^{60}Ni}$ optical
potentials show a bell-shaped distribution, peaking around half the deuteron
energy. However, the sum of these contributions does not account for the
experimental yield at low proton energies, which primarily result from
compound nucleus processes followed by evaporation and pre-equilibrium, not
covered by our formalism. Results obtained with the SFM potential demonstrate
improved agreement with the experimental data Burtebaev  _et al._ (1984)
compared to those obtained with the HHS and AC potentials. It is noteworthy
that using the IAV model with the CDCC wave function can inherently include
the correct half-shell properties in the $d-\text{target}$ channel, as
discussed in Ref. Lei and Moro (2023).

###  III.2 Application to $(\mathrm{{}^{6}Li},\alpha X)$.

Figure 5:  The differential cross section angular distribution for the
$\mathrm{{}^{59}Co(^{6}Li},\alpha X)$ reaction at 21.5 MeV is depicted. The
yellow points correspond to the experimental data Souza  _et al._ (2009). The
solid black line illustrates the results obtained using the global
phenomenological optical model potential, and the red dashed line indicates
the results using the full SFM potential.

As a second example, we turn our attention to the production of alpha
particles following the breakup of the weakly bound nucleus 6Li. We examine
the reaction 59Co(6Li,$\alpha X$) at an incident energy of 21.5 MeV in the
laboratory frame. The experimental data is taken from Ref. Souza  _et al._
(2009). To streamline our calculations, we have chosen to disregard the
internal spin of the involved particles. For the interaction between the
$\alpha$ particles and the $d$, we employ a Woods-Saxon potential
characterized by parameters: depth $V_{0}=78.46$ MeV, radius $r_{0}=1.15$ fm,
and diffuseness $a=0.7$ fm, as outlined in Ref. Nishioka  _et al._ (1984).

In our analysis, we consider two distinct sets of optical potentials. The
first set utilizes global optical potential parameters for the interactions
between 6Li and 59Co, between $d$ and 59Co, and between $\alpha$ and 59Co, as
reported in Refs. Cook (1982); Han  _et al._ (2006); Su and Han (2015),
respectively. For the second set, we apply the SFM optical potentials to all
interactions involved in both the CDCC and the IAV calculations. To ensure
that the SFM optical potentials reproduce the same phase shifts as the global
optical potentials, normalization factors for the SFM potentials are employed.
The normalization factor of $\mathrm{{}^{6}Li}$ is taken from Ref. Lu  _et
al._ (2023). For the deuteron, we use $Nr=0.899$ and $Ni=0.905$. For the
$\alpha$ potential, we use $Nr=1.34$ and $Ni=2.0$.

The differential cross section as a function of the outgoing $\alpha$ particle
angles in the laboratory frame is depicted in Fig. 5. The solid line
represents the combined results of the EBU and NEB using the global optical
potential parameter sets, while the dashed line corresponds to the combined
results of EBU and NEB with fully SFM optical potentials. It is evident that
the results utilizing SFM optical potentials significantly improve the
concordance with the experimental data. These findings suggest that the half-
shell $T$-matrix, which cannot be directly constrained by two-body scattering
data, serves as a hidden variable that can enhance the precision of reaction
calculations. This underlines the importance of the half-shell $T$-matrix in
capturing the complexities of the reaction dynamics, thereby providing a more
accurate depiction of the processes involved.

##  IV Discussion

Our investigations into the role of half-shell components in nuclear reaction
calculations have provided significant insights into the NEB cross sections
within the IAV model. In particular, the NEB mechanism is crucial for
investigating reaction processes such as knockout and surrogate reactions. Due
to numerical limitations, most of these processes are based on the DWBA. By
employing a consistent SFM for all optical potentials in the IAV calculations,
we have demonstrated a marked improvement in alignment with experimental data.
This suggests that the half-shell components of the interaction and the
interior behavior of the resultant wave functions are critical to accurately
characterizing NEB processes.

The traditional approach of fitting optical potentials to experimental data
does not fully capture the complexity of nuclear reactions. Our results
highlight the limitations of this approach, especially when discrepancies in
half-shell properties can lead to significant systematic errors in NEB cross
sections. The consistent use of SFM optical potentials, derived from the KD02
nucleon-target interaction, mitigates these errors, providing a more robust
framework for analyzing complex reactions.

Our study also highlighted the importance of lower partial waves in the
integrated NEB cross section for lighter targets. The discrepancies observed
in these waves between different optical potentials reinforce the need for a
more detailed understanding of the internal wave function dynamics to
accurately predict reaction outcomes.

In conclusion, our work presents a compelling case for the adoption of
consistent SFM optical potentials in NEB cross section calculations within the
IAV model. The enhanced agreement with experimental data achieved through this
approach underlines the necessity of a unified and comprehensive modeling of
optical potentials. Future research should focus on further validating this
approach across a broader range of targets and projectiles, as well as
integrating these findings into the development of more sophisticated and
predictive nuclear reaction models.

###### Acknowledgements.

We are grateful to Pierre Descouvemont for a critical reading of the
manuscript. This work has been supported by National Natural Science
Foundation of China (Grants No.12105204 and No.12035011), by the National Key
R&D Program of China (Contracts No. 2023YFA1606503), and by the Fundamental
Research Funds for the Central Universities.

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