# Improving Data Communication of Enhanced Loran Systems Using 128-_ary_ Polar Codes
## 2 Materials and Methods
### The Proposed 128-ary Polar Coding Scheme for eLoran
#### 2.2.1 Polar Codes for 128-_ary_ PPM
Each element \(\widetilde{\mathbf{y}}_{m}(m\in\{1,\,2,\ldots,M\})\) is a \(q\)-dimensional vector, and \(\widetilde{\mathbf{y}}_{m}=\left(\widetilde{y}_{m}(0),\widetilde{y}_{m}(1),\ldots, \widetilde{y}_{m}(q-1)\right)\), where \(\widetilde{y}_{m}(j)\) for \(j\in\{0,1,\ldots,q-1\}\) represents the demodulation output with the possible symbol value \(j\) for the \(m\)-th received symbol. Before decoding, this sequence \(\widetilde{\mathbf{y}}_{1}^{M}\) is de-punctured to reconstruct a sequence \(\mathbf{y}_{1}^{N}=\left(\mathbf{y}_{1},\mathbf{y}_{2},\ldots\mathbf{y}_{N}\right)\) of length \(N\), corresponding to the mother code. Each element \(\mathbf{y}_{i}(i\in\{1,\,2,\ldots,N\})\) is a \(q\)-dimensional vector, \(\mathbf{y}_{i}=\left(y_{i}(0),y_{i}(1),\ldots,y_{i}(q-1)\right)\). This involves inserting erasure information at the punctured positions (1 st and 17 th). For these positions, the corresponding vectors \(\mathbf{y}_{i}=\varnothing(i\in\{1,17\})\) for all \(q\) possible symbol values, indicating that these symbols were not transmitted. For the non-punctured positions, \(\mathbf{y}_{i}=\widetilde{\mathbf{y}}_{k_{i}}(i\in\{1,2,\ldots,N\}\backslash\{1,17\})\), where \(k_{i}\in\{1,2,\ldots,M\}\) maps to the \(i\)-th non-punctured index. For the decoder, let \(W_{N}^{(i)}\left(\mathbf{y}_{1}^{N},\hat{u}_{1}^{i-1}\Big{|}u_{i}\right)(i\in\{1, \,2,\ldots,N\})\) denote the transition probability of \(u_{i}\) under \(\mathbf{y}_{1}^{N}\) and the previously estimated symbols \(\hat{u}_{1}^{i-1}=(\hat{u}_{1},\hat{u}_{2},\ldots,\hat{u}_{i-1})\) of \(u_{1}^{i-1}\). Let \(L_{N,j}^{(i)}\left(\mathbf{y}_{1}^{N},\hat{u}_{1}^{i-1}\right)\) denote the Log-Likelihood Ratio (LLR) of \(u_{i}\) with symbol value \(j\in\{0,1,\ldots,q-1\}\) under the demodulation outputs \(\mathbf{y}_{1}^{N}\) and the previously estimated symbols \(\hat{u}_{1}^{i-1}\). The \(L_{N,j}^{(i)}\left(\mathbf{y}_{1}^{N},\hat{u}_{1}^{i-1}\right)\) is computed as follows:  
\[L_{N,j}^{(i)}\left(\mathbf{y}_{1}^{N},\hat{u}_{1}^{i-1}\right)=ln\frac{W_{N}^{(i) }\left(\mathbf{y}_{1}^{N},\hat{u}_{1}^{i-1}\Big{|}u_{i}=0\right)}{W_{N}^{(i)}\left( \mathbf{y}_{1}^{N},\hat{u}_{1}^{i-1}\Big{|}u_{i}=j\right)}, \tag{6}\]  
The initial value of the decoder is the LLRs of \(\mathbf{y}_{i}\) with symbol value \(j\), denote as \(L_{1,j}^{(1)}(\mathbf{y}_{i})=ln\frac{W(\mathbf{y}_{i}(0))}{W(\mathbf{y}_{i}(0))}\), where \(L_{1,j}^{(1)}(\mathbf{y}_{i})=0\) if \(\mathbf{y}_{i}=\varnothing\). The estimated symbol \(\hat{u}_{i}\) is generated as follows:  
\[\hat{u}_{i}= \begin{cases}argmin\Big{(}L_{N,j}^{(i)}\left(\mathbf{y}_{1}^{N},\hat{u }_{1}^{i-1}\right)\Big{)},&if\;\;min\Big{(}L_{N,j}^{(i)}\left(\mathbf{y}_{1}^{N}, \hat{u}_{1}^{i-1}\right)\Big{)}<0\\ 0,&otherwise\end{cases} , \tag{7}\]  
For the decoding method, the SC decoding method for \(q\)-_ary_ polar codes described in [23] is referred and extended to work with a \(2\times 2\) non-binary kernel:  
\[ \begin{array}{rl}L_{N,j}^{(2i-1)}\left(\mathbf{y}_{1}^{N},\hat{u}_{1}^{2i-2} \right)&=\ln\frac{\Gamma e^{-(l_{N/2,0}^{(i)}/\mathbf{y}_{1}^{N/2},X)+l_{N/2,0}^{( i)}/\mathbf{y}_{N/2+1}^{(i)}})}{\Gamma e^{-(l_{N/2,0}^{(i)}/\mathbf{y}_{1}^{N/2},X)+l_{N/2,0}^{(i)}/ \mathbf{y}_{N/2+1}^{(i)}}}\\ &\approx\max_{r\in[0,1,\ldots,d-1]}\Bigl{(}-\left(L_{N/2,0}^{(i)}\left(\mathbf{y}_ {1}^{N/2},X\right)+L_{N/2,r}^{(i)}\left(\mathbf{y}_{N/2+1}^{N},X^{\prime}\right) \right)\Bigr{)}\\ &-\max_{r\in[0,1,\ldots,d-1]}\Bigl{(}-\left(L_{N/2,j}^{(i)}\left(\mathbf{y}_{1}^ {N/2},X\right)+L_{N/2,r}^{(i)}\left(\mathbf{y}_{N/2+1}^{(i)},X^{\prime}\right) \right)\Bigr{)},\end{array}  \tag{8}\]  
and  
\[ \begin{array}{rl}L_{N,j}^{(2i)}\left(\mathbf{y}_{1}^{N},\hat{u}_{1}^{2i-1} \right)&=\Bigl{(}-\left(L_{N/2,\hat{u}_{2-1}}^{(i)}\left(\mathbf{y}_{1}^{N/2},X \right)+L_{N/2,0}^{(i)}\left(\mathbf{y}_{N/2+1}^{(i)},X^{\prime}\right)\right) \Bigr{)}\\ &-\Bigl{(}-\left(L_{N/2,\hat{u}_{2-1}}^{(i)}\left(\mathbf{y}_{1}^{N/2},X\right)+L_ {N/2,j}^{(i)}\left(\mathbf{y}_{N/2+1}^{N},X^{\prime}\right)\right)\Bigr{)},\end{array}