## 4 Simulation and Results  
In this section, the simulation approach of this work is presented to evaluate the efficiency of the three main schemes: syndrome-based error estimation, rate-adaptive information reconciliation, and polynomial-based hash subblock confirmation. All the initial sifted keys used in this experiment were generated by a pseudo-random number generator (PRNG) based on the observed error rates in a QKD system. The parameters employed for the proposed schemes in this approach are presented in Table 1.  
First, the performance of the syndrome-based QBER estimation is presented to demonstrate its efficiency in channel error estimation. In this step, the syndrome information from both legitimate parties is derived using the maximum code rate \(R_{\text{C}}^{\text{(}max{\text{)}}}\). This rate signifies the highest value within the set of mother code rates \(\mathcal{R}\). In the proposed scheme, the irregular LDPC codes with two specific block lengths were implemented for \begin{table}
\begin{tabular}{l l l} \hline \hline
**Procedure Step** & **Parameter** & **Value** \\ \hline \multirow{4}{*}{Syndrome-based error estimation} & Sifted key size & \multirow{2}{*}{64,800/16,200} \\  & (primary/additional round) [bit] & \\ \cline{2-3}  & Maximum code rate (\(R_{\text{C}}^{\text{(}max{\text{)}}}\)) & 9/10 for block length \(N\) \\ \cline{2-3}  & Maximum QBER threshold (\(q_{threshold}\)) & 0.25 \\ \hline \multirow{4}{*}{Rate-adaptive information reconciliation} & Block length of LDPC codes: & \multirow{2}{*}{\(N\) = 64,800/\(N_{sb}\) = 16,200} \\  & \(N\) for ordinary round/ & \\ \cline{2-3}  & \(N_{sb}\) for additional round [bit] & \\ \cline{2-3}  & & For block length \(N\): \\ \cline{2-3}  & & [9/10,8/9, 154/180, 5/6, 4/5, 7/9, 3/4, 22/30, 128/180, 25/36, 2/3, 116/180, 28/45, 3/5, 26/45, 11/20, \\ \cline{2-3}  & Set of mother code rates (\(\mathcal{R}\)) & 96/180, 1/2] [36; 37] \\ \cline{2-3}  & & For block length \(N_{sb}\): \\ \cline{2-3}  & & [8/9, 5/6, 4/5, 3/4, 32/45, 2/3, 3/5, 26/45, 8/15, 1/2] [36; 37] \\ \hline \multirow{2}{*}{Baseline metric (\(\eta_{\text{IR}}^{\text{(}base{\text{)}}}\))} & \multirow{2}{*}{1.08} \\ \cline{2-3}  & Total number of punctured and & 3200 for block length \(N\) \\ \cline{2-3}  & shortened bits (\(n_{d}\)) [bit] & 1200 for block length \(N_{sb}\) \\ \hline \multirow{2}{*}{Maximum number of decoding iterations [iteration]} & \multirow{2}{*}{100} \\ \cline{2-3}  & Increment factor for updating & \multirow{2}{*}{0.2} \\ \cline{2-3}  & the efficiency metric (\(\delta\)) & \\ \hline \multirow{4}{*}{Subblock confirmation} & Subblock size (\(N_{sb}\)) [bit] & 16,200 \\ \cline{2-3}  & Number of subblocks (\(m\)) & 4 \\ \cline{1-1} \cline{2-3}  & Hash value length (\(l_{hash}\)) [bit /byte] & 64/8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter setup for the experimental approach used in the proposed schemes: syndrome-based error estimation, rate-adaptive information reconciliation, and subblock confirmation. syndrome-based QBER estimation. Specifically, the primary round used the block length of \(N\) = 64,800 bits with \(R_{C}^{(max)}\) = 9/10, while the additional rounds employed the block length of \(N_{sb}\) = 16,200 bits with \(R_{C}^{(max)}\)= 8/9. Notably, the syndrome information was generated from the sifted keys of both _Alice_ and _Bob_ using the original parity-check **H** of irregular LDPC codes. This syndrome encoding was operated without any adjustments to the code rate, effectively bypassing the puncturing and shortening bits. Consequently, the value of the quantum bit error rate (QBER) was estimated by the maximum-likelihood estimator, as defined in Equation (11). Based on this parameter setup, Figure 4 presents the comparison results of the proposed syndrome-based QBER estimation and the traditional key sampling method with 5% and 10% sampling rates of the sifted keys for QBER estimation.