# 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  
In one transmission session, \(D\) will receive multi-path coded blocks in _Phase 1_ and _Phase 2_, and a joint decoding to restore the sources' original symbols via the belief propagation (BP) algorithm. In this paper, we assume that the original symbols in one source node are divided into more important bits (MIB) and less important bits (LIB), which means that \(n=2\) in the flowchart. Based on the And-Or tree analysis [26], let \(\delta(x)\) and \(\psi(x)\) denote the edge distributions of the input node and the output node in a rateless codes, respectively. The DER after \(l\) BP decoding iterations is expressed as \(y_{l}=\delta\big{(}1-\psi(1-y_{l-1})\big{)}\), where \(y_{0}=1\). When \(k\) approaches infinity, the edge distribution of input nodes become a Poisson distribution, and the DER becomes \(y_{l}=\exp\big{(}-\gamma\times\Psi^{\prime}(1-y_{l-1})\big{)}\), where \(\Psi^{\prime}(x)\) is the derivative of the output degree distribution, and \(\gamma\) is the decoding overhead defined as the ratio of the number of information symbols recovered by the destination decoder to the number of coded symbols. Thus, the decoding performance is only determined by \(\gamma\) and \(\Psi(x)\). With the addition of the UEP property, the decoding overhead of the \(j\)-th importance level in the GURC [24] becomes \(\gamma_{j}=\gamma\times w_{j}/\pi_{j}\), where \(j\in\{1,2,...,n\}\). The success decoding probability after \(l-1\) iterations becomes \((1-\sum_{i=\,1}^{n}p_{i}y_{l-1,n})\). Thus, the DER \(y_{l,j}\) of message symbols from the \(j\)-th importance level for an ensemble \((\Psi(x),\{w_{j}\}_{j=1,2,\ldots,n},\{\pi_{j}\}_{j=1,2,\ldots,n})\) at \(l\)-th iteration is expressed as  
\[y_{l,j}=\exp\big{(}-\gamma_{j}\times\Psi^{\prime}(1-y_{l-1,j})\big{)}=\exp[- \gamma\frac{w_{j}}{\pi_{j}}\Psi^{\prime}(1-\sum_{j=1}^{n}p_{j}y_{l-1,j})], \tag{1}\]  
where \(y_{0,j}=1\). In order to analyze the decoding performance of our DURC scheme, we should derive the relationship between degree distribution \(\Omega_{m}(x)\), \(\Psi_{m}(x)\) and network coding rule \(P\). We consider an And-Or tree as shown in Figure 3, and the received coded symbols can be divided into five groups. The first two groups are received on the secondary links \(S_{1}-D\) and \(S_{2}-D\), termed \(C_{1}^{\prime}\) and \(C_{2}^{\prime}\). The other three groups are received on the primary links, termed \(C_{1}\), \(C_{2}\) and \(C_{3}\). \(C_{1}\) and \(C_{2}\) are the coded blocks forwarded by \(R\) with the probabilities \(p_{1}\) and \(p_{2}\), respectively. \(C_{3}\) is the coded symbol transmitted from \(R\) after the XOR operation with the probability \(p_{3}\). \(C_{1}^{\prime}\) and \(C_{2}^{\prime}\) are generated with degree distributions \(\Omega_{1}(x)\) and \(\Omega_{2}(x)\), respectively. \(C_{1}\) and \(C_{2}\) are generated with degree distributions \(\Psi_{1}(x)\) and \(\Psi_{2}(x)\), respectively. \(C_{3}\) is generated with the convolution degree distribution \(\Psi_{1}(x)\times\Psi_{2}(x)\). The input nodes can be divided into four groups, termed \(X_{11}\), \(X_{12}\), \(X_{21}\) and \(X_{22}\). \(X_{11}\) and \(X_{12}\) are the MIB and LIB of \(S_{1}\), respectively. Similarly, \(X_{21}\) and \(X_{22}\) are the MIB and LIB of \(S_{2}\), respectively. Thus, \(T_{l,11}\) with depth \(2l\) as shown in Figure 3, with the root \(X_{11}\). Similarly, we can construct And-Or trees \(T_{l,12}\), \(T_{l,21}\) and \(T_{l,22}\) with the roots \(X_{12}\), \(X_{21}\) and \(X_{22}\) by using the same method. Theorem 3.1: _Let \(y_{l,1,n}\) (or \(y_{l,2,n}\)) be the probability that the root of \(S_{1}\) (or \(S_{2}\)) evaluates to 0, indicating one input node not being recovered after \(l\)-th BP decoding iterations. Then, we have_  
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