# Distributed Rateless Codes with Unequal Error Protection Property for Space Information Networks
## 3 Design of Distributed Unequal Error Protection Rateless Codes
### Optimizations of Rateless Coding Scheme and Network Coding Rule  
To optimize the system performance, we construct a decision variables set about degree distribution, symbol selection weight and network coding rule in (2) and (3), \(Q=(\Omega_{m},\Psi_{m,1},\Psi_{m,2},...,\Psi_{m,d_{m}},w_{m,n},P)\), where \(\Psi_{m,d_{m}}\) denotes the coefficient of degree distribution, \(w_{m,n}\) denotes symbol selection weight for different importance level, and \(P\) is the network coding rule. Since the channel state information of the links are unknown, we cannot obtain the exact number of coded symbols at \(D\) in each phase. Therefore, sub-optimal \(Q\) should be acquired for minimizing the BP DER at the destination \(D\). Therefore, we give the optimizations as follows. We first consider the secondary link in our scenario. The erasure probability \(\varepsilon_{1D}\) or \(\varepsilon_{2D}\) is much too high, which may lead to the received coded symbols being less than the information symbols at the destination \(D\), i.e., \(n_{1}<k_{1}\) and \(n_{2}<k_{2}\). Thus, as a supplementation to enhance the decoding performance of the whole transmission session, the optimization problem of \(\Omega_{1}(x)\) or \(\Omega_{2}(x)\) is to guarantee that the received coded symbols can recover part of the information symbols instead of recovering the total original information symbol sets or UEP, and similar optimal formulations can be found in [25] and [27]. The optimization problem of \(\Omega_{1}(x)=\sum_{m_{1}=1}^{M_{1}}\Omega_{1,m_{1}}x^{m_{1}}\) to minimize the decoding error probability can be formulated as follows:  
\[\min_{\Omega_{1}} \exp\big{(}-\alpha_{1}\Omega_{1}^{\prime}(1-y_{l-1,1})\big{)},\] (11) s.t. \[\sum_{i~{}=~{}1}^{M_{1}}\Omega_{1,i}=1,\] \[y_{0,1}=1,l\rightarrow\infty,\] \[0\leq\Omega_{1,i}\leq 1.\]  
Without the exact knowledge of \(n_{1}\), the values of \(\alpha_{1}\) for different channel erasure probabilities are unknown. Problem (11) is thus a constrained nonlinear optimization problem and generally non-convex. Therefore, it is a simplified method to solve this problem instead, by considering the most important coded symbols at the LT BP-decoder. In each BP decoding iteration, degree-1 and degree-2 symbols are most important, as they can help other unrecovered coded symbols to reduce the number of edges connected with them. In this case, we could restrict the maximum degree \(M_{1}\) and \(M_{2}\) to 2, i.e., \(\Omega_{1}(x)=\Omega_{1,1}x^{1}+\Omega_{1,2}x^{2}\) and \(\Omega_{2}(x)=\Omega_{2,1}x^{1}+\Omega_{2,2}x^{2}\). In [27], a theoretical analysis is given to demonstrate that the partial decoding performance of the degree distribution with only degree-1 and degree-2 nodes is acceptable, when the received overhead \(\frac{n}{k}\) is lower than 1. In addition, these low-degree coded symbols are to be jointed with the coded symbols on the primary links to assist in full decoding. Therefore, to minimize the DER of the joint decoding at the destination \(D\) with a pre-selected decoding overhead \(\gamma=\frac{n_{1}+n_{2}+n_{NC}}{k_{1}+k_{2}}\), and the And-Or tree asymptotic performances of \(y_{l,1}\) and \(y_{l,2}\) in (2) and (3) can be easily computed by choosing the remaining parameters \(\Omega_{1,1}\), \(\Omega_{1,2}\), \(\Omega_{2,1}\), \(\Omega_{2,2}\), \(p_{1}\), \(p_{2}\) and \(p_{3}\), with \(n_{1}:n_{2}:n_{NC}\), and \(\Psi_{m,n}(x)\) is known beforehand. It is not difficult to show that \(y_{l,1}\) and \(y_{l,2}\) are two conflicting objective functions by investigating (2) and (3). Therefore, we have a multi-objective optimization (MOP) problem about the objective function \(y(S)=(y_{1}(S),y_{2}(S))\) to minimize them concurrently, where \(S\) is the set of decision variables, i.e., \(S=(\Omega_{1},\Omega_{2},p_{1},p_{2},p_{3})\)  
\[\min y=(y_{1},y_{2}),\] (12) s.t.