# Distributed Rateless Codes with Unequal Error Protection Property for Space Information Networks
## 3 Design of Distributed Unequal Error Protection Rateless Codes
### Analysis of RC and NC Degree Distributions by the And-Or Tree Technique
\begin{split} y_{l,1,n}&=\delta_{1}(1-\omega_{1}(1- \sum_{n=1}^{2}w_{1,n}y_{l-1,1,n}))\times\tau_{1,n}\big{\{}1-P_{1}\psi_{1}(1- \sum_{n=1}^{2}w_{1,n}y_{l-1,1,n})\\ &\quad-P_{3}\sum_{i=1}^{D_{1}+D_{2}-1}\sum_{j=0}^{i-1}[\psi_{1,j} (1-\sum_{n=1}^{2}w_{1,n}y_{l-1,1,n})^{j}\psi_{2,i-j}(1-\sum_{n=1}^{2}w_{2,n}y_ {l-1,2,n})^{i-j}]\big{\}},\end{split}  \tag{2}\]  
\[ \begin{split} y_{l,2,n}&=\delta_{2}(1-\omega_{2}(1- \sum_{n=1}^{2}w_{2,n}y_{l-1,2,n}))\times\tau_{2,n}\big{\{}1-P_{2}\psi_{2}(1- \sum_{n=1}^{2}w_{2,n}y_{l-1,2,n})\\ &\quad-P_{4}\sum_{i=1}^{D_{1}+D_{2}-1}\sum_{j=0}^{i-1}[\psi_{2,j }(1-\sum_{n=1}^{2}w_{2,n}y_{l-1,2,n})^{j}\psi_{1,i-j}(1-\sum_{n=1}^{2}w_{1,n}y_ {l-1,1,n})^{i-j}]\big{\}},\end{split}  \tag{3}\]  
Figure 3: And-Or tree illustration of the edges connection of \(T_{l,11}\). _where \(y_{0,1,n}=y_{0,2,n}=1\), and_  
\[\delta_{1}(x)=\sum_{i\;=\;0}\delta_{1,i}x^{i}=\exp(\alpha_{1}(x-1)),\;\alpha_{1}= \frac{\lambda_{1}n_{1}}{k_{1}}, \tag{4}\]  
\[\delta_{2}(x)=\sum_{i\;=\;0}\delta_{2,i}x^{i}=\exp(\alpha_{2}(x-1)),\;\alpha_{2} =\frac{\lambda_{2}n_{2}}{k_{2}}, \tag{5}\]  
\[\tau_{1,n}(x)=\sum_{i\;=\;0}\tau_{1,n,i}x^{i}=\exp(\beta_{1,n}(x-1)),\;\beta_{1,n}=\frac{\mu_{1}n_{NC}(p_{1}+p_{3})w_{n}}{k_{1}\pi_{n}}, \tag{6}\]  
\[\tau_{2,n}(x)=\sum_{i\;=\;0}\tau_{2,n,i}x^{i}=\exp(\beta_{2,n}(x-1)),\;\beta_{2,n}=\frac{\mu_{2}n_{NC}(p_{2}+p_{3})w_{n}}{k_{2}\pi_{n}}, \tag{7}\]  
\[\left\{ \begin{aligned} P_{1}&=\frac{p_{1}}{1-p_{2}}, P_{3}=1-P_{1}\\ P_{2}&=\frac{p_{2}}{1-p_{1}},P_{4}=1-P_{2}\end{aligned}  \right.. \tag{8}\]  
Proof of Theorem 1.: In (2) and (3), the first product term means \(y_{l}\) of _Phase 1_, and the second one means \(y_{l}\) of _Phase 2_. For every importance level of the information symbols in a single source node, their input edge distribution and average degree are different. Thus, the parameters used in (2) and (3) can be defined as follows. The input edge distribution of primary link is expressed as \(\tau_{m,n}(x)=\exp(\beta_{m,n}(x-1))\), where \(\beta_{m,n}\!=\!\mu_{m}n_{NC}(p_{i}+p_{3})w_{m,n}/(k_{i}\pi_{m,n})\) is the input average degree, and where \(\mu_{m}=\sum_{d_{m}=\;1}^{D_{m}}d_{m}\Psi_{md_{m}}\) is its output average degree. The input edge distribution of secondary link is expressed as \(\delta_{m}(x)=\exp(\alpha_{m}(x-1))\), where \(\alpha_{m}=\lambda_{m}n_{m}/k_{m}\), and the output edge distribution of secondary link is expressed as \(\lambda_{m}(x)=\Omega_{m}{}^{\prime}(x)/\Omega_{1}\). \(\psi_{1,i}\) is defined as the probability that \(C_{1}\) has \(i\) children of input nodes, and \(\psi_{2,i}\) is defined as the probability that \(C_{2}\) has \(i\) children of input nodes. Moreover, \(\psi_{1,i}\) also refers to the probability that \(C_{3}\) has \(i\) children of \(X_{1}\) (including \(X_{11}\) and \(X_{12}\)), and \(\psi_{2,i}\) refers to the probability that \(C_{3}\) has \(i\) children of \(X_{2}\) (including \(X_{21}\) and \(X_{22}\)). Every \(C_{1}\) only has the children of \(X_{1}\), and the probability of any \(X_{1}\) evaluating to \(1\) is \(1-\sum_{n\;=\;1}^{2}w_{1,n}y_{l-1,1,n}\). Thus, if a \(C_{1}\) has \(i\) children of \(X_{1}\), the probability of \(C_{1}\) evaluating to \(1\) is \(\sum_{i\;=\;0}^{D_{1}-1}\psi_{1,i}(1-\sum_{n\;=\;1}^{2}w_{1,n}y_{l-1,1,n})^{i}\). In the same way, if a \(C_{3}\) node has \(i\) children of \(X_{1}\) and \(j\) children of \(X_{2}\), the probability of \(C_{3}\) evaluating to \(1\) is \(\sum_{i\;=\;1}^{D_{1}+D_{2}-1}\!\sum_{j\;=\;0}^{i-1}\left[\psi_{1,j}(1\!-\! \sum_{n\;=\;1}^{2}w_{1,n}y_{l-1,1,n})^{j}\psi_{2,i-j}(1\!-\!\sum_{n\;=\;1}^{2} w_{2,n}y_{l-1,2,n})^{i-j}\right]\). In the And-Or tree \(T_{l,11}\), \(P_{1}\) and \(P_{3}\) are the proportions of \(X_{1}\) connecting with \(C_{1}^{\prime}+C_{1}\) and \(C_{3}\), respectively. \(P_{2}\) and \(P_{4}\) are the proportions of \(X_{2}\) connect with \(C_{2}^{\prime}+C_{2}\) and \(C_{3}\), respectively. The relationships are expressed as (8).