## 3 Parallel Distribution Matching Based on CCDM
### Paired Optimization Principle  
When the length n of CCDM output amplitude sequence is small, the probability distribution of each amplitude must be quantized--that is, \(P_{A}\to P_{A^{\prime}}\), to ensure that the number of occurrences of each amplitude is an integer and the quantization should try to satisfy that \(P_{A}\) and \(P_{A^{\prime}}\) are as close as possible.  
When the output amplitude sequence of the CCDM is \(A^{n}=a_{1}a_{2}\ldots a_{n}\), the length is n, the amplitude setting is \(\chi=\{\alpha_{1},\alpha_{2},\ldots,\alpha_{i}\}\), each amplitude value in the sequence \(A^{n}\) is taken from \(\chi\), the number of occurrences of the amplitude \(\alpha_{i}\) is \(n_{\alpha_{i}}\), and \(n_{\alpha_{i}}=n\cdot P_{A^{\prime}}(\alpha_{i})\); the set \(C=\{n_{\alpha_{1}},n_{\alpha_{2}},\ldots,n_{\alpha_{i}}\}\) is called \(P_{A^{\prime}}\) composition with a probability distribution a, and the size of the set C can be determined using the permutation and combination formula as shown in (4), which is equivalent to \(\Gamma_{P_{A}}^{n}\).  
\[M(C)=\frac{n!}{n_{\alpha_{1}}!n_{\alpha_{2}}!...n_{\alpha_{i}}!} \tag{4}\]  
\(C_{typ}=\{n_{\alpha_{1}},n_{\alpha_{2}},...,n_{\alpha_{i}}\}\) represents a typical composition of CCDM output, and obeys the probability distribution \(P_{A^{\prime}(\alpha_{i})}\). The pairwise optimization means that the single output amplitude sequence of the CCDM does not follow the probability distribution \(P_{A^{\prime}(\alpha_{i})}\)--that is, \(C\
eq C_{typ}\). However, the average composition of many output amplitude sequences is equal to \(C_{typ}\), with all output amplitude sequences obeying the target probability \(P_{A^{\prime}(\alpha_{i})}\) as a whole. The purpose of pairwise optimization is to find components that satisfy (5) [20], where 1 represents the \(N_{comp}\) possible composition of the output amplitude sequence and \(c_{l}\) represents the number of occurrences of the sequence whose composition is \(C_{l}\) in the output sequence; its value does not exceed \(M(C_{l})\) at most. Here, we only consider the case of pairwise complementarity, not the case of three or more compositions of complementarity.  
\[\frac{\sum\limits_{\begin{subarray}{c}l\\ l\end{subarray}}^{N_{comp}}c_{l}\cdot C_{l}}{\sum\limits_{\begin{subarray}{c}l \\ l\end{subarray}}^{N_{comp}}c_{l}}=C_{typ} \tag{5}\]  
For example, the length of the output amplitude sequence of CCDM is n = 10; the amplitude is taken from \((a_{1},a_{2},a_{3},a_{4})\), assuming the probability distribution after quantization is \(P_{A^{\prime}}=(0.4,0.3,0.2,0.1)\); \(C_{typ}=(4,3,2,1)\); and the information entropy H(A) = 1.85 bit/symbol. If only \(C_{typ}\) is considered in the output composition of CCDM, \(M(C_{typ})=\) 12,600, it can map \(k=\left\lfloor\log_{2}(C_{typ})\right\rfloor=13\) input bits, the rate is 1.3 bit/symbol, and the rate loss is 0.55 bit. When considering the composition of \(C_{1}=(4,2,3,1)\) and \(C_{2}=(4,4,1,1)\), although a single output amplitude sequence is not equal to \(P_{A^{\prime}}\), but \(M(C_{1})=M(C_{2})\) = 6300, the output amplitude sequences of \(C_{1}\) and \(C_{2}\) can be complementary, and the overall obeys the probability distribution \(P_{A^{\prime}}\); thus, the total output amplitude sequence is equal to 12,600 + 6300 + 6300 = 25,200, which can map 14-bit input binary sequence. Compared with only considering \(C_{typ}\), the rate is increased by 0.1 bit/symbol [20].