# Minimization of _n_th Order Rate Matching in Satellite Networks with One to Many Pairings
## 2 Dynamic Capacity Allocation
### Capacity Allocation Problem and Proposed Mechanism  
We focus on finding GWs-UEs pairing matrices \(\mathbf{X}\in\{0,1\}^{\mathrm{M}\times\mathrm{N}}\) for approaching the minimization of the system's \(n\)th order rate matching and formulate separately the cases of even and odd \(n\). Setting \(b_{ij}=\bigg{(}RC_{j}-OC_{ij}\bigg{/}\sum\limits_{\omega=1}^{N}x_{i\omega} \bigg{)}^{n}\), where \(\mathrm{k}_{\mathrm{i}}=\sum\limits_{\omega=1}^{\mathrm{N}}x_{i\omega}\),\(\forall\mathrm{i}\), describes the O2M case, and applying the BE, we conclude in (1) where \(n\) has to be a positive integer to guarantee the convergence of right-hand side (RHS) ([26], Equation (5.12) and \(\bigg{(} \begin{array}{c}n\\ m\end{array} \bigg{)}\) is the binomial coefficient. Moreover, considering the identity \(|c^{\alpha}|=|c|^{\alpha}\) for every real \(c\) and \(a\) (\(c\) must be nonzero for negative \(a\)), the initial problem becomes as in (2). \[\mathrm{b}_{\mathrm{ij}}=\sum\limits_{m=0}^{n}\Bigg{[}(-1)^{m}\bigg{(} \begin{array} []{c}n\\ m\end{array} \bigg{)}RC_{j}^{n-m}OC_{ij}^{m}\bigg{(}\sum\limits_{\omega=1}^{N}x _{i\omega}\bigg{)}^{-m}\Bigg{]}, \tag{1}\]  
\[ \begin{array}{c}\min\limits_{\mathbf{X}\in\mathrm{S}}\sum\limits_{\mathrm{i }=1}^{\mathrm{M}}\sum\limits_{\mathrm{j}=1}^{\mathrm{N}}x_{i\omega}\Bigg{|} \mathrm{RC}_{\mathrm{j}}-\frac{\mathrm{OC}_{\mathrm{ij}}}{\sum\limits_{\omega= 1}^{\mathrm{M}}\sum\limits_{\mathrm{j}=1}^{\mathrm{N}}x_{i\omega}\Big{|} \mathrm{b}_{\mathrm{ij}}\Big{|}=\\ \min\limits_{(\mathbf{X},\mathrm{T})\in\mathrm{S}_{\mathrm{i}}^{\mathrm{M}} }\sum\limits_{\mathrm{i}=1}^{\mathrm{M}}\sum\limits_{\mathrm{j}=1}^{\mathrm{N} }x_{i\mathrm{j}}\mathrm{i}_{\mathrm{j}}.\end{array}  \tag{2}\]  
The minimization of the initial problem on the left-hand side (LHS) of (2) has the feasible set S, where the inequality constraint guarantees that each GW serves at least one UE and the equality constraint guarantees that each UE is served by one GW, \(\mathrm{S}=\bigg{\{}1\leq\sum\limits_{\omega=1}^{\mathrm{N}}x_{i\omega} \forall\mathrm{i},\sum\limits_{\mathrm{i}=1}^{\mathrm{M}}x_{i\overline{\mathrm{ j}}}=1,\forall\mathrm{j},\mathbf{X}\in\{0,1\}^{\mathrm{M}\times\mathrm{N}}\bigg{\}}\). The initial problem on the LHS of (2) is transformed appropriately in last relation of (2) with feasible set S\({}_{1}^{\mu}\), where \(\mu\) is even or odd, considering the problem with even and odd \(n\), respectively, and auxiliary variables \(\mathbf{T}\in\mathrm{R}^{\mathrm{M}\times\mathrm{N}}\) are added. Specifically, by setting \(\mathrm{u}=\bigg{(} \begin{array}{c}n\\ m\end{array} \bigg{)}RC_{j}^{n-m}OC_{ij}^{m}\bigg{(}\sum\limits_{\omega=1}^{N}x _{i\omega}\bigg{)}^{1-m}\) and \(\mathbf{T}\in\mathrm{R}^{\mathrm{M}\times\mathrm{N}}\), where \(\mathrm{R}^{\mathrm{M}\times\mathrm{N}}\) is the maximum achievable  
the proposed CCP scheme, the CVXPY library in PYTHON is used, and statistical averages for 1000 independent simulation scenarios are studied. Due to the dependence of the **Proposed** method's outcome on the initial feasible points, the comparison of a **Constant** feasible point and a **Random** procedure is made. In the Constant case, the (1/M)\(\mathbf{1}_{M\times N}\) point is used, where \(\mathbf{1}_{M\times N}\) is the all-ones M \(\times\) N matrix, while in the Random process, the point, among 200 random feasible points, resulting in the minimum objective in LHS of (2) is selected. Moreover, the tolerance of the Proposed iterative algorithm is \(\varepsilon=10^{-3}\) and \(r\), \(r_{j}\) are the ceiling numbers of the corresponding absolute minimum eigenvalues, as described in Section 2. Due to the fact that the proposed scheme consists of the first effort to approximate the solution of the \(n\)th order rate-matching minimization and no other known techniques exist in the literature, its performance is examined by the comparison with the corresponding optimal **Exhaustive** scheme. The latter explores all the feasible pairs to arrive at the best result.