## 5 Upper Bounds on LDPC Codes Decoding Thresholds
### Further Upper Bounds on LDPC Codes Decoding Thresholds Holding for \(i_{1}=2\)  
With \(a_{j}:=\text{log}A_{j}=-2\text{log}((j-1)\lambda_{i_{1}})\), Equation (18) can be rewritten as:  
\[f(s,t)=2\sum_{j=2}^{d_{r}}\rho_{j}\text{W}\Big{(}z(s,t)\text{e}^{z(s,t)+a_{j }}\Big{)}+O(t^{-1}). \tag{22}\]  
Assuming that the following simplified approximation holds:  
\[\text{W}\Big{(}z(s,t)\text{e}^{z(s,t)+a_{j}}\Big{)}\simeq\text{W}\Big{(}(z(s, t)+a_{j})\text{e}^{z(s,t)+a_{j}}\Big{)}, \tag{23}\]  
calling \(x:=z_{s}(t)+a_{j}\), and being \(W(x\text{e}^{x})\equiv x\) for \(x>0\), we find a much simpler asymptotic expression for \(f(s,t)\) (simpler than the one of Equation (18)):  
\[f(s,t)\simeq 2\sum_{j=2}^{d_{r}}\rho_{j}\text{W}\Big{(}(z(s,t)+a_{j})\text{e}^{ z(s,t)+a_{j}}\Big{)}=2\sum_{j=2}^{d_{r}}\rho_{j}(z(s,t)+a_{j})=\sum_{j=2}^{d_{r}} \rho_{j}(s+(i_{1}-1)t-4\text{log}((j-1)\lambda_{i_{1}})). \tag{24}\]Thus, the asymptotic performance of \(f(s,t)\) is given by:  
\[f(s,t)=s+(i_{1}-1)t-4\text{log}\lambda_{i_{1}}-4\sum_{j=2}^{d_{r}}\rho_{j}\text{ log}(j-1)+O(t^{-1}). \tag{25}\]  
This is the same asymptotic expression obtained, following a different derivation, in [11], where the approximation in Equation (23) was implicitly used in the proof of Lemma 3. Thus, ignoring the \(O(t^{-1})\), for large \(t\) Equation (17) can be rewritten as:  
\[\left\{ \begin{array}{l}s+(i_{1}-2)t-4\text{log}\lambda_{i_{1}}-4\sum_{j=2}^{ d_{r}}\rho_{j}\text{log}(j-1)=0\\ i_{1}-2=0\end{array} \right.. \tag{26}\]  
The solution of Equation (26) is:  
\[s^{*}_{\text{approx}}=4\text{log}\lambda_{i_{1}}+4\sum_{j=2}^{d_{r}}\rho_{j} \text{log}(j-1) \tag{27}\]  
and \(\sigma^{*}_{\text{approx}}=\sqrt{\frac{2}{s^{*}_{\text{approx}}}}\) gives the second upper bound of [10]. Applying the [PERSON]'s inequality:  
\[\prod_{j=2}^{d_{r}}(j-1)^{\rho_{j}}\leq\sum_{j=2}^{d_{r}}\rho_{j}(j-1) \tag{28}\]  
and ignoring the \(O(t^{-1})\), for large \(t\) Equation (17) can be rewritten as:  
\[\left\{ \begin{array}{l}s+(i_{1}-2)t-4\text{log}\lambda_{i_{1}}-4\text{log} \Big{(}\sum_{j=2}^{d_{r}}(j-1)\rho_{j}\Big{)}=0\\ i_{1}-2=0\end{array} \right. \tag{29}\]  
The solution of Equation (29) is:  
\[s^{*}_{\text{ [PERSON]}}=4\text{log}\lambda_{i_{1}}+4\text{log}\Big{(}\sum_{j=2}^{d_ {r}}(j-1)\rho_{j}\Big{)} \tag{30}\]  
Taking \(\sigma^{*}_{\text{ [PERSON]}}=\sqrt{\frac{2}{s^{*}_{\text{ [PERSON]}}}}\) we obtain the third upper bound of [10]. Since, for the [PERSON]'s inequality for Equation (28):  
\[\sum_{j=2}^{d_{r}}\rho_{j}\text{log}(j-1)\leq\text{log}\Big{(}\sum_{j=2}^{d_{ r}}(j-1)\rho_{j}\Big{)} \tag{31}\]  
it results \(s^{*}_{\text{approx}}\leq s^{*}_{\text{ [PERSON]}}\) and thus \(\sigma^{*}_{\text{ [PERSON]}}\leq\sigma^{*}_{\text{approx}}\), i.e., \(\sigma^{*}_{\text{ [PERSON]}}\) gives a tighter upper bound. Role of the Product \(\lambda^{\prime}(0)\rho^{\prime}(1)\) in Determining the Weight Distribution of LDPC Codes  
In Reference [9] it was demonstrated that \(\lambda^{\prime}(0)=\lambda_{2}\) occupies a key position in determining LDPC codes performance, both theoretically and in practice. In particular, the minimum distance growth rate, namely, whether it is linear or not, was shown to depend only on the product \(\lambda^{\prime}(0)\rho^{\prime}(1)\). Namely, if \(\lambda^{\prime}(0)\rho^{\prime}(1)>1\), the minimum distance growth rate is sublinear with the block length, otherwise, i.e., if \(\lambda^{\prime}(0)\rho^{\prime}(1)<1\), it is linear with the block length.