## 8 Numeric Results  
Being \(\lambda^{\prime}(0)\) and \(\rho^{\prime}(1)\) given in Equations (35) and (36), respectively, we obtain:  
\[\lambda^{\prime}(0)\rho^{\prime}(1)=\lambda_{2}\sum_{j=2}^{d_{r}}(j-1)\rho_{j} \tag{40}\]  
In Tables 1-3 we report \(\lambda(x)\) and \(\rho(x)\) for the irregular LDPC codes of Table 1 and 2 in [18]. For each pair of degree distributions, we also report the values of the product \(\lambda^{\prime}(0)\rho^{\prime}(1)\) and of the parameter \(L_{2}\). Moreover, we give the \(\sigma^{*}\) values obtained in [18] by applying the density evolution analysis, jointly with the upper bound \(\sigma^{*}_{\text{jensen}}=\sqrt{2/s^{*}_{\text{jensen}}}\) defined in Equation (30). \begin{table}
\begin{tabular}{c c c c c} \hline \(\mathbf{d}_{l}\) & **4** & **6** & **7** & **8** \\ \hline \(\lambda_{2}\) & 0.38354 & 0.33241 & 0.31570 & 0.30013 \\ \(\lambda_{3}\) & 0.04237 & 0.24632 & 0.41672 & 0.28395 \\ \(\lambda_{4}\) & 0.57409 & 0.11014 & & \\ \(\lambda_{5}\) & & & & \\ \(\lambda_{6}\) & & 0.31112 & & \\ \(\lambda_{7}\) & & & 0.43810 & \\ \(\lambda_{8}\) & & & & 0.41592 \\ \(\rho_{5}\) & 0.24123 & & & \\ \(\rho_{6}\) & 0.75877 & 0.76611 & 0.43810 & 0.22919 \\ \(\rho^{\prime}\) & & 0.23389 & 0.56190 & 0.77081 \\ \(\lambda^{\prime}(0)\) & 0.38354 & 0.33241 & 0.31570 & 0.30013 \\ \(\rho^{\prime}(1)\) & 4.75877 & 5.23389 & 5.56190 & 5.77081 \\ \(\lambda^{\prime}(0)\rho^{\prime}(1)\) & 1.82518 & 1.73980 & 1.75589 & 1.73199 \\ \(\int_{0}^{1}\lambda(x)\) & 0.34942 & 0.32770 & 0.35934 & 0.29671 \\ \(L_{2}\) & 0.54883 & 0.50719 & 0.43927 & 0.50577 \\ \(\sigma^{*}\) & 0.9114 & 0.9304 & 0.9424 & 0.9497 \\ \(\sigma^{*}_{\text{jensen}}\) & 0.91160 & 0.95021 & 0.94241 & 0.95409 \\ \(\text{SNR}_{\text{gap}}\) & 0.60903 & 0.24872 & 0.32032 & 0.21333 \\ \(\text{SNR}_{\text{gap}-\text{spline}}\) & 0.61687 & 0.25656 & 0.32815 & 0.22117 \\ \hline \end{tabular}
\end{table} 
Table 1: Computed parameters and decoding threshold bound regarding rate-1/2 codes deduced from Table I of [18], with \(d_{l}=4,6,7,\) and 8. \begin{table}
\begin{tabular}{c c c c c} \hline \(\mathbf{d}_{l}\) & **9** & **10** & **11** & **12** \\ \hline \(\lambda_{2}\) & 0.27684 & 0.25105 & 0.23882 & 0.24426 \\ \(\lambda_{3}\) & 0.28342 & 0.30938 & 0.29515 & 0.25907 \\ \(\lambda_{4}\) & & 0.00104 & 0.03261 & 0.01054 \\ \(\lambda_{5}\) & & & & 0.05510 \\ \(\lambda_{6}\) & & & & \\ \(\lambda_{7}\) & & & & 0.01455 \\ \(\lambda_{8}\) & & & & \\ \(\lambda_{9}\) & 0.43974 & & & \\ \(\lambda_{10}\) & & 0.43853 & & 0.01275 \\ \(\lambda_{11}\) & & & 0.43342 & & \\ \(\lambda_{12}\) & & & & 0.40373 \\ \(\rho_{6}\) & 0.01568 & & & \\ \(\rho^{\prime}\) & 0.85244 & 0.63676 & 0.43011 & 0.25475 \\ \(\rho_{8}\) & 0.13188 & 0.36324 & 0.56989 & 0.73438 \\ \(\rho_{9}\) & & & & 0.01087 \\ \(\lambda^{\prime}(0)\) & 0.27684 & 0.25105 & 0.23882 & 0.24426 \\ \(\rho^{\prime}(1)\) & 6.11620 & 6.36324 & 6.56989 & 6.75612 \\ \(\lambda^{\prime}(0)\rho^{\prime}(1)\) & 1.69321 & 1.59749 & 1.56902 & 1.65025 \\ \(\int_{0}^{1}\lambda(x)\) & 0.28175 & 0.27276 & 0.26535 & 0.25888 \\ \(L_{2}\) & 0.49128 & 0.46020 & 0.45001 & 0.47176 \\ \(\sigma^{*}\) & 0.9540 & 0.9558 & 0.9572 & 0.9580 \\ \(\sigma^{*}_{\text{jensen}}\) & 0.97439 & 1.03314 & 1.05356 & 0.99908 \\ \(\text{SNR}_{\text{gap}}\) & 0.03046 & 0.47807 & 0.64807 & 0.18689 \\ \(\text{SNR}_{\text{gap}-\text{spline}}\) & 0.03830 & 0.47023 & 0.64023 & 0.17905 \\ \hline \end{tabular}
\end{table} 
Table 2: Computed parameters and decoding threshold bound regarding rate-1/2 codes deduced from Table I of [18], with \(d_{l}=9,10,11,\) and 12. Besides these results, we also reported the SNR\({}_{\rm gap}\) and SNR\({}_{\rm gap-spline}\) values in dB, given by:  
\[{\rm SNR}_{\rm gap}(d_{l})\,:=\,\frac{1}{C_{1}^{-1}(1/2)} \tag{41}\]  
\[{\rm SNR}_{\rm gap-spline}(d_{l})\,:=\,\frac{\frac{1}{2(\sigma_{\rm lensen}^{2}(d _{l}))^{2}}}{C_{\rm spline}^{-1}(1/2)} \tag{42}\]  
defining the gap from the Shannon limit of the bounds \(\sigma_{\rm lensen}\).