# Hamming Distance Optimized Underwater Acoustic OTFS-IM Systems
## 3 Hamming Distance Optimized Index Mapping Model  
### Index Mapping Method  
There are two methods of mapping index bits to active sub-grid positions, including the lookup table method and the combined model method. The lookup table method involves creating an indexed mapping table of size \(2^{p_{1}}\) at the sender and receiver sides and mapping the \(p_{1}\) bits at the sender side to the locations of active sub-grids by looking up the table. Additionally, at the receiver side, the \(p_{1}\) bits of the corresponding input are obtained by querying the location of the detected active sub-grid. Take \((n,k)=(4,2)\) as an example where \(p_{1}=\left\lfloor\log_{2}C_{n}^{k}\right\rfloor=2\) and the table size is \(2^{p_{1}}=4\). As shown in Table 1, when the index bit is \(\left[\begin{matrix}0&0\end{matrix}\right]\), the corresponding active sub-grid position is \(\left[\begin{matrix}1&2\end{matrix}\right]\), and the sub-grid group \(\left[\begin{matrix}s_{\beta,1}&s_{\beta,2}&0&0\end{matrix}\right]^{T}\) is obtained by mapping the constellation symbol to the active sub-grid. The table lookup method is simple and easy to implement when \(C_{n}^{k}\) is small.  
However, when \(C_{n}^{k}\) is large, the size of the table also increases, making the table query more complex. The combinatorial model method solves the problem of an excessive  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Index Bit**} & \multicolumn{2}{c}{**Active Sub-Grid**} & \multicolumn{2}{c}{**Sub-Grid Group**} \\ \hline \(\left[\begin{matrix}0&0\end{matrix}\right]\) & \(\left[\begin{matrix}1&2\end{matrix}\right]\) & \(\left[\begin{matrix}s_{\beta,1}&s_{\beta,2}&0&0\end{matrix}\right]^{T}\) \\ \(\left[\begin{matrix}0&1\end{matrix}\right]\) & \(\left[\begin{matrix}1&3\end{matrix}\right]\) & \(\left[\begin{matrix}s_{\beta,1}&0&s_{\beta,2}&0\end{matrix}\right]^{T}\) \\ \(  
complexity of the table lookup method when \(C_{n}^{k}\) is large by the method of combinatorial numbers in mathematical calculations by converting \(p_{1}\) bits into decimal numbers \(R\) where there is a one-to-one mapping relationship with \(k\) combinatorial values in the combinatorial numbers. For any \(R\), there exists a unique decreasing sequence \(K=\{c_{k},\cdots,c_{1}\}\), \(c_{k}~{}>~{}\cdots>~{}c_{1}\geq 0\) of length \(k\), such that (16) holds.  
\[R=C_{c_{k}}^{k}+C_{c_{k-1}}^{k-1}+\cdots+C_{c_{2}}^{2}+C_{c_{1}}^{1} \tag{16}\]  
In the combined model approach, the maximum value of \(c_{k}\) that satisfies \(C_{c_{k}}^{k}\leq R\) is first chosen, followed by the maximum value of \(c_{k-1}\) that satisfies \(C_{c_{k-1}}^{k-1}\leq R-C_{c_{k}}^{k}\), and so on until the \(c_{1}\) is finally selected and the index sequence \(I=K+1\) corresponding to the position of the activated sub-grid is obtained from sequence \(K\). Taking \((n,k)=(8,3)\) as an example, each sub-grid group size is set to 8, where the number of active sub-grids is 3. There are \(C_{n}^{k}=56\) different ways of combining the sub-grids:  
\[\begin{array}{ccccc}55=C_{7}^{3}+C_{6}^{2}+C_{5}^{1}&\rightarrow&K=\{7,6,5 \}&I_{\beta}=\{8,7,6\}\\ 54=C_{7}^{3}+C_{6}^{2}+C_{4}^{1}&\rightarrow&K=\{7,6,4\}&I_{\beta}=\{8,7,5\}\\ \vdots&&\vdots&\vdots\\ 1=C_{3}^{3}+C_{1}^{2}+C_{0}^{1}&\rightarrow&K=\{3,1,0\}&I_{\beta}=\{4,2,1\}\\ 0=C_{2}^{3}+C_{1}^{2}+C_{0}^{1}&\rightarrow&K=\{2,1,0\}&I_{\beta}=\{3,2,1\} \end{array} \tag{17}\]