## 2 Asymmetric Turbo Code (ATC)
### Code Matched Interleaver
The polynomials of weight \(\beta=4\) input sequence \(\mathbf{q}_{1,4}\) and permuted sequence \(\mathbf{q}_{1,4}^{1}\) are written as:  
\[\mathbf{q}_{1,4}(\text{F})=(1+\text{F}^{\varepsilon_{1}\text{C}^{\prime}_{1}} )\text{F}^{\gamma^{\prime}_{1}}+(1\ +\ \text{F}^{\varepsilon_{1}\text{C}^{\prime}_{2}})\text{F}^{\gamma^{\prime}_{2}} \ \ \mathbf{q}_{1,4}^{1}(\text{F})=(1\ +\ \text{F}^{\varepsilon_{2}\text{C}^{\prime}_{3}})\text{F}^{\gamma^{\prime}_{3}} +(1\ +\ \text{F}^{\varepsilon_{2}\text{C}^{\prime}_{4}})\text{F}^{\gamma^{\prime}_{4}}, \tag{7}\]  
where \(\text{C}^{\prime}_{1}\), \(\text{C}^{\prime}_{2}\), \(\text{C}^{\prime}_{3}\), \(\text{C}^{\prime}_{4}\)\(\ =\ 1,2,\ldots\), \(\gamma^{\prime}_{1}\), \(\gamma^{\prime}_{2}\), \(\gamma^{\prime}_{3}\) and \(\gamma^{\prime}_{4}\) denote time delays such that \(\gamma^{\prime}_{2}\ >\ \gamma^{\prime}_{1}\ +\ \varepsilon_{1}\text{C}^{\prime}_{1}\) and \(\gamma^{\prime}_{4}\ >\ \gamma^{\prime}_{3}\ +\ \varepsilon_{2}\text{C}^{ \prime}_{3}\). The total weight of the ATC codeword generated by weight-4 input sequence \(\mathbf{q}_{1,4}\) is given as:  
\[\mathbf{w}^{\beta=4}=12+(\text{C}^{\prime}_{1}\ +\ \text{C}^{\prime}_{2})(1_{ \text{min},1}-\ 2)\ +\ (\text{C}^{\prime}_{3}\ +\ \text{C}^{\prime}_{4})(1_{\text{min},2}-2) \tag{8}\]  
It is assumed that \(\mathbf{g}_{1}\), \(\mathbf{g}_{2}\), \(\mathbf{g}_{3}\) and \(\mathbf{g}_{4}\) denote the positions of \"1\" in the input sequence \(\mathbf{q}_{1,4}\), where \(\mathbf{g}_{1}\ <\ \mathbf{g}_{2}\ <\ \mathbf{g}_{3}\ <\ \mathbf{g}_{4}\). Let \(\pi(\mathbf{g}_{1})\), \(\pi(\mathbf{g}_{2})\), \(\pi(\mathbf{g}_{3})\) and \(\pi(\mathbf{g}_{4})\) be the positions of \"1\" in the interleaved sequence \(\mathbf{q}_{1,4}^{1}\). If the mapping conditions meet the following form:  
\[\left\{ \begin{array}{c}\text{whenever}\left|g_{1}-\ \mathbf{g}_{2}\right| \bmod\varepsilon_{1}=\ 0\ \text{and}\ \left|g_{3}-\ \mathbf{g}_{4}\right|\bmod\varepsilon_{1}=\ 0\\ \left|\pi(\mathbf{g}_{1})\ -\ \pi(\mathbf{g}_{3})\right|\bmod\varepsilon_{2}=\ 0\ \text{and}\ \left|\pi(\mathbf{g}_{2})\ -\ \pi(\mathbf{g}_{4})\right|\bmod \varepsilon_{2}=\ 0\end{array} \right.\]  
or  
\[\left\{ \begin{array}{c}\text{whenever}\left|g_{1}-\ \mathbf{g}_{2}\right| \bmod\varepsilon_{1}=\ 0\ \text{and}\ \left|g_{3}-\ \mathbf{g}_{4}\right|\bmod\varepsilon_{1}=\ 0\\ \left|\pi(\mathbf{g}_{1})\ -\ \pi(\mathbf{g}_{4})\right|\bmod\varepsilon_{2}=\ 0\ \text{and}\ \left|\pi(\mathbf{g}_{2})\ -\ \pi(\mathbf{g}_{3})\right|\bmod \varepsilon_{2}=\ 0\end{array} \right. \tag{9}\]  
then the interleaver will map the input sequence \(\mathbf{q}_{1,4}\) into \(\mathbf{q}_{1,4}^{1}\), which produces finite weight parity check sequence, as illustrated in Figure 3. Due to the fact that both \(\mathbf{q}_{1,4}\) and \(\mathbf{q}_{1,4}^{1}\) generate finite weight parity check sequences, the overall ATC codeword weight given by Equation (8) is finite. In order to avoid the generated scenario, the following mapping conditions should be satisfied:  
\[\left\{ \begin{array}{c}\text{whenever}\left|g_{1}-\ \mathbf{g}_{2}\right| \bmod\varepsilon_{1}=\ 0\ \text{and}\ \left|g_{3}\ -\ \mathbf{g}_{4}\right|\bmod\varepsilon_{1}=\ 0\\ \left|\pi(\mathbf{g}_{1})\ -\ \pi(\mathbf{g}_{3})\right|\bmod\varepsilon_{2}\ \
eq\ 0\ \text{and}\ \left|\pi(\mathbf{g}_{2})\ -\ \pi(\mathbf{g}_{4})\right|\bmod \varepsilon_{2}\ \
eq\ 0\end{array} \right.\]  
or  
\[\left\{ \begin{array}{c}\text{whenever}\ \left|g_{1}-\ \mathbf{g}_{2}\right| \bmod\varepsilon_{1}=\ 0\ \text{and}\ \left|g_{3}-\ \mathbf{g}_{4}\right|\bmod \varepsilon_{1}=\ 0\\ \left|\pi(\mathbf{g}_{1})\ -\ \pi(\mathbf{g}_{4})\right|\bmod\varepsilon_{2}\ \
eq\ 0\ \text{and}\ \left|\pi(\mathbf{g}_{2})\ -\ \pi(\mathbf{g}_{3})\right|\bmod \varepsilon_{2}\ \
eq\ 0\end{array} \right. \tag{10}\]  
Figure 2: A weight-2 input sequence is mapped to another weight-2 input sequence after passing through the interleaver. It is worth mentioning that not all finite weight sequences are bad.