## 3 Code Matched Interleaver (CMI)
The scenario is depicted in \[d=\omega+\omega(y_{p1})+\omega(y_{p2}) \tag{11}\]4 The worst case for weight-4 information sequence \(x_{i1}^{4}\) is given as follow:  
\[\left\{ \begin{array}{l}\text{If }|P_{1}-P_{2}|\text{mod }\alpha=0\text{ and }|P_{3}-P_{4}|\text{mod }\alpha=0\\ \text{Then }|\pi(P_{1})-\pi(P_{3})|\text{mod }\alpha=0\\ \text{and }|\pi(P_{2})-\pi(P_{4})|\text{mod }\alpha=0\end{array} \right\}\]  
OR  
\[\left\{ \begin{array}{l}\text{If }|P_{1}-P_{2}|\text{mod }\alpha=0\text{ and }|P_{3}-P_{4}|\text{mod }\alpha=0\\ \text{Then }|\pi(P_{1})-\pi(P_{3})|\text{mod }\alpha=0\\ \text{and }|\pi(P_{2})-\pi(P_{3})|\text{mod }\alpha=0\end{array} \right\} \tag{22}\]  
However such a bad scenario can be avoided if the interleaver incorporates the following mapping criterion  
\[\left\{ \begin{array}{l}\text{If }|P_{1}-P_{2}|\text{mod }\alpha=0\text{ and }|P_{3}-P_{4}|\text{mod }\alpha=0\\ \text{Then }|\pi(P_{1})-\pi(P_{3})|\text{mod }\alpha\
eq 0\\ \text{and }|\pi(P_{2})-\pi(P_{4})|\text{mod }\alpha\
eq 0\end{array} \right\}\]  
OR  
\[\left\{ \begin{array}{l}\text{If }|P_{1}-P_{2}|\text{mod }\alpha=0\text{ and }|P_{3}-P_{4}|\text{ mod }\alpha=0\\ \text{Then }|\pi(P_{1})-\pi(P_{4})|\text{ mod }\alpha\
eq 0\\ \text{and }|\pi(P_{2})-\pi(P_{3})|\text{ mod }\alpha\
eq 0\end{array} \right\} \tag{23}\]  
As mentioned earlier that not all the weight-2 sequences are bad, the same is the case with weight-4 sequences. There are such a weight-4 sequences which does not satisfy (23) but still they are not considered as bad sequences. Similarly, there is a threshold parameter \(d_{max}^{\gamma=4}\) in given [16]. Applying this parameter \(d_{max}^{\gamma=4}\) to the weight-4 input sequences results in the following inequality  
\[12+(\text{k}_{1}^{\prime}+\text{k}_{2}^{\prime}+\text{k}_{3}^{ \prime}+\text{k}_{4}^{\prime})+(Q_{min}-2)\leq d_{max}^{\gamma=4} \tag{24}\] \[(\text{k}_{1}^{\prime}+\text{k}_{2}^{\prime}+\text{k}_{3}^{ \prime}+\text{k}_{4}^{\prime})\leq\frac{d_{max}^{\gamma=4}-12}{(Q_{min}-2)} \tag{25}\]  
The weight-4 sequence will only be classified as bad weight-4 sequence if and only if it satisfies both the inequality (23) and (25) simultaneously. Otherwise weight-4 sequence will not degrade the performance of turbo code. The effect of input sequence with input weight \(\text{w}>4\) on over all turbo code performance is very negligible. Therefore, we will not consider them in the interleaver design. Figure 3: Weight-4 information sequence is mapped to weight-4 information sequence with similar pattern after permuted through interleaver. A four-state turbo code having two identical recursive systematic convolutional encoders each with generator polynomial \(G(1,5/7)_{8}\) is considered in this manuscript. The values of threshold parameters such as \(\hat{d}_{max}^{\gamma=2}\), \(\hat{d}_{max}^{\gamma=3}\) and \(\hat{d}_{max}^{\gamma=4}\) are found in [21]. The parameters values are \(\hat{d}_{max}^{\gamma=2}=\hat{d}_{max}^{\gamma=4}=20\) and \(\hat{d}_{max}^{\gamma=3}=13\). The weight-2 information sequence that causes minimum parity sequence is given as (... 010010... ). The parity sequence that is produced by considered weight-2 information sequences by RSC-1 is given as (... 011110... ). The value of \(\alpha=3\) and \(Q_{min}=4\) is considered for this specific code. To design CMI all the design mapping criteria that were aforementioned in this section must be taken into an account.