## 2 Asymmetric Turbo Code (ATC)
### Code Matched Interleaver  
The quality of a code depends on its distance spectrum properties. Choosing a good interleaver contributes to the excellent distance characteristics of an ATC. In [28], it was proven that the CMI fully removes low weight sequences as compared to other types of interleavers to boost the properties of a distance spectrum. Therefore, the CMI is regarded as an optimal interleaver. Furthermore, the CMI is, first, an S-random interleaver. The S-random interleaver can either eliminate a short weight-3 input sequence with lengths up to S + 1 or expand it to another sequence with lengths greater than \(2(S+1)\)[31]. Therefore, we primarily focused on weight-2 and weight-4 sequences in the design of the CMI. We discuss the construction of the CMI for the ATC in the following section. The input sequence \(\mathbf{q}_{1}\) is encoded by the ATC encoder into codeword \((\mathbf{q}_{1},\ \mathbf{p}_{1},\ \mathbf{p}_{2})\). The weights of information and parity bit sequences are denoted by \(\beta\), \(\beta(\mathbf{p}_{1})\) and \(\beta(\mathbf{p}_{2})\), respectively. The overall weight of ATC codeword is expressed as:  
\[\mathrm{w}=\beta+\beta(\mathbf{p}_{1})+\beta(\mathbf{p}_{2}), \tag{1}\]  
Assume that weight \(\beta=2\) input sequence \(\mathbf{q}_{1,2}\) produces a finite weight ATC codeword; then, the input sequence \(\mathbf{q}_{1,2}\) and permuted sequence \(\mathbf{q}_{1,2}^{1}\) can be represented by the following polynomials:  
\[\mathbf{q}_{1,2}(\mathrm{F})=(1+\mathrm{F}^{\varepsilon_{1}\varsigma_{1}}) \mathrm{F}^{\gamma_{1}},\ \mathbf{q}_{1,2}^{1}(\mathrm{F})=(1+\mathrm{F}^{ \varepsilon_{2}\varsigma_{2}})\mathrm{F}^{\gamma_{2}}, \tag{2}\]  
where \(\varsigma_{1},\ \varsigma_{2}=1,\ 2,\ldots,\ \varepsilon_{1}\) and \(\varepsilon_{2}\) denote the minimum distances between two consecutive \"1\"s of the \(\mathbf{q}_{1,2}\) and \(\mathbf{q}_{1,2}^{1}\), respectively, and \(\gamma_{1}\) and \(\gamma_{2}\) represent time delays, \(\gamma_{1},\gamma_{2}=1,\ 2,\ldots\) The weights of parity bits given in Equation (2) can be written as:  
\[\varsigma_{1}(l_{\min,1}-2\ )+2,\ \varsigma_{2}(l_{\min,2}-2)+2, \tag{3}\]  
where \(l_{\min,1}\) and \(l_{\min,2}\) represent the lowest weights of parity check sequences \(\mathbf{p}_{1}\) and \(\mathbf{p}_{2}\) generated by the sequences \(\mathbf{q}_{1,2}\) and \(\mathbf{q}_{1,2}^{1}\), respectively. By combining Equations (1) and (3), the overall weight of the ATC codeword produced by the weight-2 input sequence \(\mathbf{q}_{1,2}\) can be given as:  
\[\mathrm{w}^{\beta=2}=6+\varsigma_{1}(l_{\min,1}-\ 2)+\varsigma_{2}(l_{\min,2}-2) \tag{4}\]  
Let \(g_{1}\) and \(g_{2}\) represent the positions of \"1\"s in the input sequence \(\mathbf{q}_{1,2}\). The positions of \"1\"s in the interleaved sequence \(\mathbf{q}_{1,2}^{1}\) are denoted by \(\pi(g_{1})\) and \(\pi(g_{2})\). If the mapping condition satisfies the following form:  
\[\left|g_{1}-g_{2}\right|\mathrm{mod}\ \varepsilon_{1}=\ 0\ \mathrm{and}\ \left|\pi(g_{1})-\pi(g_{2})\right|\mathrm{mod}\ \varepsilon_{2}=\ 0, \tag{5}\]  
The interleaver maps the input sequence \(\mathbf{q}_{1,2}\) to the interleaved sequence \(\mathbf{q}_{1,2}^{1}\) that produces a finite weight parity check sequence, as shown in Figure 2. Note that this case is detrimental because both  
Figure 1: The concatenation of the asymmetric turbo code (ATC). and \(\mathbf{q}_{1,2}^{1}\) generate finite weight parity check sequences to cause a degradation in system performance. To avoid this phenomenon, the mapping condition is required to meet the following expression:  
\[\left|g_{1}-g_{2}\right|\bmod\varepsilon_{1}=\ 0\ \text{and}\ \left|\pi(g_{1})\ -\ \pi(g_{2})\right|\bmod\varepsilon_{2}\ \
eq\ 0, \tag{6}\]  
Now, we consider the weight-4 input sequence. The weight-4 input sequence is comprised of two weight-2 sequences. Thus, a finite weight ATC codeword is generated.