esa-sceva/satcom-chunk-collection huggingface dataset:
# Robust Estimators in Geodetic Networks Based on a New Metaheuristic: Independent Vortices Search
## 4 Experimental Setup  
### The GNSS Network and Simulations  
To perform the tests, we built a geodetic network from six stations in the Brazilian Network of Continuous Monitoring of Global Navigation Satellite Systems (RBMC) (Figure 3). The observation vectors were formed from the official co-ordinates of the points. We processed 6 h of data from the stations to obtain the covariance matrix (\(\mathbf{\Sigma}_{y}\)). This allowed us to compute the weight matrix (\(\mathbf{W}\)) and generate random errors for the observations. The vector \(\mathbf{l}\) was set up by adding the official values and the random errors generated from the covariance matrix.  
The network was structured without repeated observations and with independent baselines. This gives six points (one control point) and 13 vectors. As they are three-dimensional, there were a total of 15 parameters and 39 observations.  
To test the RE performances, we built nine packages with 100 error scenarios each, as presented in Table 1. Package 01 had no outlier, only random errors in all observations ranging from \([0\sigma,3\sigma)\) using a normal distribution. Trimming random errors at \(3\sigma\) should not have affected the results. Following the normal distribution, the occurrence of values greater than \(3\sigma\) was only 0.27%. All estimators were tested in this condition.  
The other packages contained at least one outlier per scenario. Outliers were also calculated from the standard deviation of each observation. The packages separated small error scenarios (with magnitude between \([3\sigma,6\sigma)\)) from large error scenarios (\([6\sigma,12\sigma]\)). Outliers were randomly distributed over \(\mathbf{l}\). Packages with four outliers represented scenarios with approximately 10% contaminated observations. In the most critical cases, the network should be resistant to two simultaneous outliers between the same vertices, as we had at least four vectors connecting each station.  
Figure 3: Network of six stations of the Brazilian Network for Continuous Monitoring of GNSS Systems (RBMC): One control point and thirteen three-dimensional observation vectors. Google Earth V 7.3.2.5776 (13 December 2015). Sao Paulo, Brazil. \(22^{\circ}57^{\prime}30.85^{\prime\prime}\) S, \(45^{\circ}55^{\prime}16.44^{\prime\prime}\) W, Viewpoint height 297.24 km. Image Landsat/Copernicus, Data Scripps Institution of Oceanography, National Oceanic and Atmospheric Administration, U.S. Navy, NGA, General Bathymetric Chart of the Oceans [48].
# Calculation of the Dangerous Failure Rate of the Safety Function
## 3 The Dangerous Failure Rate of the SF
Such a situation is shown in Figure 4.  
In general, the rate of the transition from state \(i\) to state \(j\) for time \(\Delta t\) is understood as the derivate of the transition probability, i.e.,  
\[\lambda_{ij}=\frac{p_{ij}\left(t,t+\Delta t\right)}{\Delta t}=\frac{P(x(t+ \Delta t)=j|x(t)=i)}{\Delta t}=\frac{P((x(t+\Delta t)=j)\cap(x(t)=i))}{P(x(t) =i)\cdot\Delta t}. \tag{8}\]  
Figure 3: CTMC with one absorption state. It is also true that  
\[ \begin{array}{l}\sum\limits_{i=1}^{k-1}{{\rm{P}}(x(t)=i)}+\sum\limits_{j=k}^{n}{ {\rm{P}}(x(t)=j)}=1,\\ \sum\limits_{i=1}^{k-1}{{\rm{P}}(x(t)=i)}=1-\sum\limits_{j=k}^{n}{{\rm{P}}(x(t )=j)}.\end{array}  \tag{9}\]  
Let it be true that  
\[ \begin{array}{l}\sum\limits_{i=1}^{k-1}{{\rm{P}}(x(t)=i)}=\sum\limits_{i=1 }^{k-1}{{\rm{P}}_{i}(t)},\\ \sum\limits_{j=k}^{n}{{\rm{P}}(x(t)=j)}=\sum\limits_{j=k}^{n}{{\rm{P}}_{j}(t)}. \end{array}  \tag{10}\]  
The rate of transition to state \(j\) (state from the set of absorption states \(j\
i A\)) from the set of transition states  
\[\lambda_{j}=\sum\limits_{i=1}^{k-1}{\lambda_{ij}}=\frac{\sum_{i=1}^{k-1}{{\rm{ P}}((x(t+\Delta t)=j)\cap(x(t)=i))}}{\sum_{i=1}^{k-1}{{\rm{p}}(x(t)=i)}\cdot \Delta t}=\frac{p_{j}(t)}{\sum_{i=1}^{k-1}{{\rm{p}}(x(t)=i)}\cdot\Delta t}= \frac{\frac{dp_{j}(t)}{dt}}{1-\sum_{j=k}^{n}{{\rm{p}}_{j}(t)}}. \tag{11}\]  
If the system contains one absorption state (state D), then it follows from (11) that  
\[\lambda_{D}(t)=\frac{\frac{dp_{D}(t)}{dt}}{1-p_{D}(t)} \tag{12}\]  
Equation (12) corresponds to Equation (7). If the system contains two absorption states (state D and state S), then it follows from (11) that  
\[\lambda_{D}(t)=\frac{\frac{dp_{D}(t)}{dt}}{1-(p_{D}(t)+p_{S}(t))}, \tag{13}\]  
\[\lambda_{S}(t)=\frac{\frac{dp_{S}(t)}{dt}}{1-(p_{D}(t)+p_{S}(t))}. \tag{14}\]  
Another approach can be chosen for the CTMC in Figure 4. This CTMC can be described by a transition rate matrix (15) and a set of differential Equation (16)  
\[\mathbb{A}=\left( \begin{array}{ccc}-(\lambda_{D}(t)+\lambda_{S}(t))&\lambda _{D}(t)&\lambda_{S}(t)\\ 0&0&0\\ 0&0&0\\ \end{array} \right). \tag{15}\]  
\[\begin{array}{l}\frac{dp_{X}(t)}{dt}=-(\lambda_{D}(t)+\lambda_{S}(t)).\ p_
# Design and Analysis of a Hybrid MPPT Method for PV Systems Under Partial Shading Conditions
## 2 Materials and Methods  
### Design of the Proposed PV System  
A solar energy system using the proposed Perturb & Observe and Particle Swarm Optimization hybrid-based MPPT algorithm has been developed in the MATLAB/Simulink environment. The system consists of a solar panel array, a DC-DC boost converter, and DC-DC boost converter control circuit sections.  
In a PV system, the voltage and current values produced from the PV array are measured. The measured voltage and current values are used by the MPPT algorithms.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**MPPT Algorithm** & **Merits** & **Demerits** \\ \hline \multirow{4}{*}{SSA-P\&O Hybrid MPPT [54]} & Fast MPP tracking, low MPP tracking & \multirow{4}{*}{Computational complexity, special parameter selection required.} \\  & oscillation, appropriate to changes in weather conditions, easy to implement, & \\  & needs only one sensor. & \\ \hline \multirow{4}{*}{Musical Chair MPPT [55]} & High performance, low computational & \multirow{4}{*}{Special parameter selection required.} \\  & time, low oscillation in steady state, high & \\ \cline{1-1}  & MPPT accuracy. & \\ \hline \multirow{2}{*}{Modified Krill Swarm Algorithm-FLC} & Precise MPP tracking, fast response to & \multirow{2}{*}{Computational complexity, special parameter selection required.} \\ \cline{1-1}  & changes in weather conditions, & \\ \cline{1-1}  & dependable power source. & \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._The determined MPPT algorithms are prepared as MATLAB code in the MATLAB Function block. At this stage, MPP tracking is performed, and the function produces a reference voltage. This voltage is transmitted to the DC-DC boost converter control circuit. The error between the reference voltage and the PV array voltage is detected. This error signal is processed by the Proportional-Integral (PI) controller and then transmitted to the comparator. Here, a comparison process is made with the triangle wave signal, and a switching signal is produced at the output. Lastly, the switching element on the DC-DC boost converter is operated with switching signal, which is named Pulse Width Modulation (PWM) signal. DC voltage generated in the PV array is increased using this switching signal. The block diagram of the implemented PV system is shown in Figure 1.
## 3 Materials and Methods
### Deep Gated Neural Network Model
Second GRU layer:  
\[{h_{t}^{(GRU2)}}={GRU}({h_{t}^{(GRU1)}},{h_{t-1}^{(GRU2)}}) \tag{9}\]  
Here, \({h_{t}^{(GRU1)}}\) is the output from the first GRU layer at time step \(t\), and \({h_{t-1}^{(GRU2)}}\) is the hidden state from the previous time step in the second GRU layer. This layer further refines the temporal representations, capturing more complex patterns. 5. Output layer:  
\[y={Softmax}\;(W.\;{h_{t}^{(GRU2)}}+b) \tag{10}\]  
In the output layer, the final hidden state \({h_{t}^{(GRU2)}}\) from the second GRU layer is passed through a dense layer with a Softmax activation function. Here:  
\(W:\) weight matrix;  
\(b:\) bias vector;  
\(y:\) output probabilities for each class (low, average, high workload). The Softmax activation function ensures that the model's output represents a probability distribution, enabling clear classification of cognitive workload levels. The training process of the model is crafted to ensure not just efficiency but also robustness in its learning. The model is trained using the categorical cross-entropy loss function and optimized using the Adam optimizer. During the training process, the model undergoes 50 epochs, with a batch size of 32, effectively learning and reaching convergence. The Adam optimizer utilizes a default learning rate of 0.001, ensuring stable and efficient training dynamics. Additionally, dropout layers are incorporated to prevent overfitting, and 5-fold cross-validation is utilized to ensure robust model evaluation. Upon completion of training, the performance evaluation of the DGNN model is conducted using a set of metrics. The confusion matrix offers a clear overview by comparing the actual workload levels (low, average, high) with the model's predictions. In addition, we use fundamental evaluation metrics like accuracy [65], precision [66], recall [67], and F1 score [68] to thoroughly evaluate the model's ability to accurately classify workload levels.
## 2 Results
### The Semi-Covalent Approach
#### 2.2.2 Binding Isotherms  
Uptake of **1** was observed to be rapid, with optimal binding noted after 10 min for both MIPs. Binding isotherms (Supplementary Figure S19E) were generated across a 0.3-10 mM concentration range (of **1**) in CHCI3, using 30 mg of polymer and a 60 min rebinding time in order to be consistent with the self-assembly systems. The binding constant, \(K_{\text{d}}\), and \(\text{B}_{\text{max}}\) were calculated using non-linear regression of the Langmuir binding isotherms generated from GraphPad Prism, with the best-fit values obtained from the one-site-total-binding equation presented in Table 1. Since the NIPs showed negligible binding of **1**, only the MIPs were analysed. The \(K_{\text{d}}\) value for T1**6**MIFCHCI3 (0.02 uM) is 5 times lower than E1**6**MIFCHCI3 (0.09 uM), as is consistent with earlier observations (Figure 6) that the former has a greater affinity for **1**. These \(K_{\text{d}}\) values are also 3 and 33 times lower than the equivalent self  
Figure 6: Rebinding results for E1**6**MIFCHCI3 and T1**6**MIFCHCI3 in CHCI3, using 1 mL 0.8 mM solution of **1** with a binding time of 30 min. All experiments were performed in triplicate. Numerical values above the bars indicate imprinting factor (IF) values.  
assembly MIPs (E7\({}_{\text{1-MIPCHCI3}}\) and T7\({}_{\text{1-MIPCHCI3}}\)), respectively. The binding capacities (Bmax) of these semi-covalent MIPs are close in values and comparable to T7\({}_{\text{1-MIPCHCI3}}\) but 3 times lower than E7\({}_{\text{1-MIPCHCI3}}\).
# Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures
## 3 Elastic Metasurfaces
Elastic MS can be applied to manipulate not only flexural waves but also Rayleigh and Love waves. Rayleigh waves are formed when longitudinal and transverse motions are combined and an elliptical retrograde motion emerges from a vertical plane along the direction of propagation. Love waves are formed when the horizontal particle motion is perpendicular to the direction of propagation. In seismology, both Rayleigh and Love waves are surface seismic waves produced on the Earth. MSs can applied as protective devices against seismic waves. An elastic MS which can create elastic rainbow trapping and support mode conversion of Rayleigh waves to bulk shear waves has been designed and experimentally demonstrated [40] based on the meta-wedge design [39] and the theoretical dispersion properties of an MS [36]. One elastic MS that manipulates Love waves has been demonstrated using resonators [151]. By adjusting the interaction coefficient which controls the strength of interaction between resonators and the Love waves, phase speed and \(n\) are controlled. By using this, a Luneburg and a Maxwell lens were also demonstrated in this study. An adaptive and tunable MS can manipulate  
Figure 11: **(a)** Schematic of MS design; **(b)** Left side: the curl of the displacement field showing anomalous refraction of a normal incident SV-wave (upper) and focusing a cylindrical SV-wave source at 60 kHz (lower). Right side: the trace of the strain tensor showing anomalous refraction of a normal incident SV-wave (upper) and cylindrical P-wave incidence (lower) at 60 kHz; **(c)** Schematic of the source shifter (lower) and displacement field produced by different heights \(h\) of three zigzag unit cells (upper); **(d)** Left side: Source shifter; the A0 wave field at 12 kHz with the source placed at (0, 0) mm (upper) and (25, 0) mm (lower). Center side: source transformer; the A0 wave field at 8 kHz with the source placed at (0, 0) mm (upper) and (\(-\)15, 0) mm (lower). Right side: source splitter; the target profile (upper) and the experimentally-measured A0 wave field at 8 kHz (lower). **(a,b)** adapted from [145]. **(c,d)** adapted from [148]. elastic waves by using piezoelectric circuit integration without change in the host structure [152]; the proposed elastic MS consists of arrayed piezoelectric devices with individually-connected elements that have negative capacitance. By adjusting the negative capacitances online, discontinuous phase profiles can be obtained. A full transmission MS for in-plane longitudinal waves has also been introduced [153]; its unit cell is composed of separated substructures, so that each property is decoupled by its own geometrical parameter (Figure 12d); therefore, each substructure embedded in elastic solids behaves as a mass or as a spring element. Full transmission was achieved by breaking an intrinsic proportionally-coupled relationship between density and stiffness. The corresponding values of the discrete phase by both theoretical and experimental approach agreed well (Figure 12e).
## 2 Materials and Methods
### Metabolites Quantification  
High-performance liquid chromatography (HPLC) was used to determine the concentration of itaconic anhydride (ITAN), maleic anhydride (MA), and 5-hydroxymethylfurfural (5-HMF). Synthetic versions (Sigma-Aldrich, Saint Louis, MO, USA) of these three compounds were used as standards to make the calibration curve. The chromatographic system HPLC 1100 consisted of a quaternary system of pumps (Agilent Technologies G1310A, Waldbronn, Germany) connected to an automated sample injector (Agilent Technologies G1313A, Waldbronn, Germany). The compounds were detected using a wavelength of 210 nm (Agilent Technologies G1314A, Hachioji, Tokyo, Japan), and a reverse phase C18 column (Polaris 5 \(\upmu\)m, 4.6 mm inner diameter \(\times\) 250 Santa Clara, CA, USA). For the detection of ITA, MA, and 5-HMF, the mobile phase was H\({}_{2}\)O/acetic acid at 0.05%. The flow rate for all of the samples was 0.625 mL/min and the column temperature was kept constant at 25 \({}^{\circ}\)C. The water used was previously degassed and filtered (ultrapure water; Milli-Q\({}^{\circledR}\) ZMQS6V001).  
Molecular Docking of the Mycobacterium tuberculosis Isocitrate Lyase (MitCL) with 5-HMF, ITAN, and MA  
The crystal structure of the MtCL (PDB code = 6 XPP) enzyme was used for docking analysis. The three-dimensional (3D) structure was downloaded from the Protein Data Bank database ([[https://www.rcsb.org/](https://www.rcsb.org/)]([https://www.rcsb.org/](https://www.rcsb.org/)), accessed on 9 February 2022). The 3D structure was characterized and represented using X-ray crystallography [16].  
A molecular docking experiment was conducted to evaluate the docking properties of four compounds (5-HMF, ITAN, and MA) with the MtCL enzyme. Docking was carried out with the HDOCK server ([[http://hdock.phys.hust.edu.cn/](http://hdock.phys.hust.edu.cn/)]([http://hdock.phys.hust.edu.cn/](http://hdock.phys.hust.edu.cn/)), accessed on 11 February 2022), a multi-component integrated package [17; 18]. The molecular ligand formulas were downloaded from PubChem ([[https://pubchem.ncbi.nlm.nih.gov/](https://pubchem.ncbi.nlm.nih.gov/)]([https://pubchem.ncbi.nlm.nih.gov/](https://pubchem.ncbi.nlm.nih.gov/)), accessed on 11 February 2022), and a blind molecular docking grid box was used to define the docking region. The parameters were the default values in HDOCK, and the structures had a root mean square deviation (RMSD) of up to 3 A. All of the results were analyzed and visualized using the UCSF Chimera 1.14 Molecular Graphics Systems [19].
###### Abstract
Classification is one of the functionalities of machine-learning algorithms in which the algorithm learns a classification model on a given set of data [9]. However, each machine-learning algorithm performs differently with the data set, leading to the need to test different algorithms to find the best classifier [10]. The use of machine learning applied to spectral data has been widely explored, especially in soybean seeds, as it is a non-destructive method used in the identification of varieties [11] or in the classification of vigor [12]. It is also possible to use both technologies to distinguish soybean genotypes using spectral bands using public Landsat-8 satellite images, providing advances in soybean mapping [13] using such technology in the classification of five genetic genotypes. Thus, the objectives of this work are (i) to classify soybean cultivars under different irrigation managements using hyperspectral data, looking for the best machine-learning algorithm for classification and the input that improves the performance of the models, and (ii) to find the most accurate machine-learning algorithm for classifying soybean cultivars and to determine the input data for the models that improve the performance of the algorithms.
## 3 MPC Controller Design and Longitudinal Speed Compensation
### Control Model  
The state space model of path tracking system can be derived from Equations (2), (5) and (6) as:  
\[\left\{\begin{array}{l}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}(t)+\mathbf{B}u( t)+\mathbf{E}k(s)+\mathbf{d}\\ \mathbf{y}=\mathbf{C}\mathbf{x}+\mathbf{f}\end{array}\right. \tag{7}\]  
Figure 5: The relationship between lateral deviation and heading deviation during path tracking.  
\[\mathbf{A}=\left[\begin{array}{cccc}0&a_{12}&a_{13}&0\\ 0&0&0&a_{24}\\ 0&0&a_{33}&a_{34}\\ 0&0&a_{43}&a_{44}\end{array}\right]\mathbf{B}=\left[\begin{array}{c}0\\ 0\\ b_{31}\\ b_{41}\end{array}\right],\mathbf{E}=\left[\begin{array}{c}0\\ 0\\ -v_{\mathrm{x}}\\ 0\end{array}\right],\mathbf{d}=\left[\begin{array}{c}0\\ 0\\ \frac{2(\overline{F}_{\mathrm{r}}+\overline{C}_{\mathrm{r}}\overline{a}_{ \mathrm{r}})}{m}\\ \frac{2(\overline{F}_{\mathrm{r}}+\overline{C}_{\mathrm{r}}\overline{a}_{ \mathrm{r}})}{m}\\ \frac{2(\overline{F}_{\mathrm{r}}+\overline{C}_{\mathrm{r}}\overline{a}_{ \mathrm{r}})}{l_{\mathrm{Z}}}\\ \end{array}\right],\mathbf{C}=\left[\begin{array}{c}0\\ 1\\ 1\\ 0\\ 0\end{array}\right]^{T},\mathbf{f}=\left[\begin{array}{c}0\\ \beta\end{array}\right],\]  
\(a_{12}=v_{\mathrm{x}},\)\(a_{13}=1,\)\(a_{14}=1,\)\(a_{33}=-\frac{2\overline{C}_{\mathrm{r}}}{mv_{\mathrm{x}}},\)\(a_{34}=\frac{2\overline{b}\overline{C}_{\mathrm{r}}}{mv_{\mathrm{x}}}-v_{\mathrm{x}},\)\(a_{43}=\frac{2\overline{b}\overline{C}_{\mathrm{r}}}{l_{\mathrm{Z}}v_{\mathrm{x}}},\)\(a_{44}=-\frac{2b^{2}\overline{C}_{\mathrm{r}}}{l_{\mathrm{Z}}v_{\mathrm{x}}},\)\(b_{31}=\frac{2}{m},\)\(b_{41}=\frac{2a}{l_{\mathrm{Z}}}.\)  
where \(\mathbf{x}=[e,\)\(\theta_{\mathrm{e}},\)\(v_{\mathrm{y}},\)\(r]^{T}\) is the state vector, \(\mathbf{y}=[e,\)\(\Delta\varphi]^{T}\) is the path tracking error chosen as the output vector, \(u=[F_{\mathrm{y}\mathrm{f}}]\) is the lateral tire force of the front axle chosen as the control input vector.  
To facilitate controller design and implementation, the continuous time state space model is converted into a discrete time system using the zero-order hold method, which is expressed as:  
\[\left\{\begin{array}{l}\mathbf{x}(k+1)=\mathbf{A}_{\mathrm{c}}\mathbf{x}(k) +\mathbf{B}_{\mathrm{c}}u(k)+\mathbf{E}_{\mathrm{c}}k(s)+\mathbf{d}_{\mathrm{ c}}\\ \mathbf{y}(k)=\mathbf{C}\mathbf{x}(k)+\mathbf{f}\end{array}\right. \tag{8}\]
## 4 Methodology
### The PIN Process in the Case of a Block-by-Block (BbB) Basis  
Four steps are considered for a PIN process on a block-by-block basis (Figure 5). The first step is to identify the building blocks of the information. For example, in the case of a video, we can consider that each image is independent of the others and can be treated separately without affecting the others.  
The second step is to add a control packet at the beginning of the transmission of a block, which indicates the block identifier and the total number of packets that are part of the block. This scenario is similar to a session establishment mechanism. Each time that a node receives a packet PIN control, it opens the corresponding block and becomes ready to receive packets for that block.  
The third step is to add a field of \"block ID\" to each packet to identify packages that are part of the same block.  
The fourth step is to initiate the process of the block as soon as it is complete. A node can have opened several blocks but only begins PIN processing when all of the packets are received for that block.
# An Efficient Advantage Distillation Scheme for Bidirectional Secret-Key Agreement
## 3 Bidirectional Secret-Key Agreement
### Proposed Advantage Distillation scheme
#### 3.1.2 SKA from [PERSON] to [PERSON]  
**(A1. 1)** [PERSON] uses the received sequence \(T_{1}\) to calculate  
\[U_{1}=T_{1}\oplus M^{1}_{ab}, \tag{12}\]  
where \(U_{1}=(u^{1}_{0},u^{1}_{1},\cdots,u^{1}_{n-1}),u^{1}_{k}=t^{1}_{k}\oplus m^{1 }_{ab_{k}}\). [PERSON] sends \(U_{1}\) over the main channel and [PERSON] wiretaps \(U_{1}\) through the eavesdropper's channel.  
**(A1. 2)** [PERSON] and [PERSON] receive the noisy version of \(U_{1}\) as \(V_{1}\) and \(W_{1}\), respectively, as  
\[V_{1}=U_{1}\oplus E^{1}_{ab}=M^{1}_{ab}\oplus Q\oplus E^{0}_{ba }\oplus E^{1}_{ab}, \tag{13}\] \[W_{1}=U_{1}\oplus E^{1}_{ae}=M^{1}_{ab}\oplus Q\oplus E^{0}_{ba }\oplus E^{1}_{ae}. \tag{14}\]  
**(A1. 3)** [PERSON] knows \(Q\), so he can XOR \(Q\) to \(V_{1}\) as  
\[\bar{M}^{1}_{ab}=V_{1}\oplus Q=M^{1}_{ab}\oplus E^{0}_{ba}\oplus E ^{1}_{ab}, \tag{15}\]  
where \(\bar{M}^{1}_{ab}=(\bar{m}^{1}_{ab_{1}},\bar{m}^{1}_{ab_{1}},\cdots,\bar{m}^{1 }_{ab_{n-1}})\) is the received raw key. [PERSON] only knows the noisy version of \(Q\), which is \(TA_{1}\). Therefore, she can only XOR \(TA_{1}\) to \(W_{1}\) as  
\[Z^{1}_{ae}=W_{1}\oplus TA_{1}=M^{1}_{ab}\oplus E^{0}_{ba}\oplus E ^{0}_{be}\oplus E^{1}_{ae}, \tag{16}\]  
where \(Z^{1}_{ae}=(z^{1}_{a_{0}},z^{1}_{a_{0}},\cdots,z^{1}_{a_{n-1}})\) in the eavesdropped raw key.