## 3 Candidate Generation in Parallel Decoding of Rate-1 Nodes
### Candidate Generation Based on Partially Ordered Candidate Sets  
Compared to the \(L_{\text{out,p}}\) candidates according to (6), many candidates can be excluded by the method described in the previous section. However, there are still many paths that are visited, especially when \(L\) increases. Taking into account that the flipped bit locations need to be determined anyway by identifying the \(P\) least reliable LLRs in each \(\mathbf{\alpha}^{v,l}\), the knowledge about the relations between the LLRs can be used to exclude further paths. Due to this identification by an _LLR-sorter_, as shown in Figure 2, we can consider the \(P\) least reliable LLRs of input path \(l\) as sorted values, i.e.,  
\[|\alpha_{0}^{v,l}|\leq|\alpha_{1}^{v,l}|\leq\cdots\leq|\alpha_{P-1}^{v,l}|. \tag{21}\]  
Following the proof of Theorem 1 in [21], where the Partial Order (PO) of Polar code bit-channels from [22] is restated, we describe a similar PO for PMs. Therefore, we write the candidate index \(j\) in its binary expansion and, for better readability, we then write the index \(l\) of the input path as a superscript. From (21), we deduce the following lemma:  
**Lemma 1**. : _For \(P=2\) and \(\operatorname{hw}(j)=1\), we have the PM-relation \(PM_{01}^{l}\leq PM_{10}^{l}\)._  
Replacing a more significant \(0\) with the next less significant \(1\) in the binary expansion of \(j\) corresponds to flipping a bit in a position of a more reliable LLR instead of flipping a bit in the position of the next less reliable LLR. Therefore, for any \(p^{a}\in\{0,1\}^{a}\), \(q^{b}\in\{0,1\}^{b}\) and \(a,b\geq 0\), we can generalize Lemma 1 to  
\[\operatorname{PM}_{p^{a}01q^{b}}^{l}\leq\operatorname{PM}_{p^{a}10q^{b}}^{l}. \tag{22}\]  
**Theorem 3**. : _In Fast-SSCL decoding, the PMs of all possible candidates in a Rate-\(1\) node originating from input path \(l\) are related as_  
\[PM_{p^{a}0r^{a}1q^{b}}^{l}\leq PM_{p^{a}1r^{a}0q^{b}}^{l}, \tag{23}\]  
_for all \(p^{a}\in\{0,1\}^{a}\), \(r^{c}\in\{0,1\}^{c}\), \(q^{b}\in\{0,1\}^{b}\) and \(a,c,b\geq 0\)._  
Proof. : We prove Theorem 3 by induction. For the initial case, \(c=0\), Equation (23) is equal to (22). For the induction step, we assume that (23) is true for \(c=d,d>0\) and let \(c=d+1\). Then, an additional bit \(\delta\in\{0,1\}\) is appended and \(r^{d+1}=r^{d}\delta\). There are two cases to consider for the additional bit \(\delta\):  
1. If \(\delta=0\), the relations  
Figure 3: Smallest possible candidate indices \(m^{l}_{i,j}\) for \(L=8\) and \(P=4\) with candidates \(j\in[0,2^{P}-1]\) and list index \(l\in[0,L-1]\). \[\mathrm{PM}^{I}_{p^{\prime}0\sigma^{d+1}1q^{b}} =\mathrm{PM}^{I}_{p^{\prime}0\sigma^{d}01q^{b}}\overset{(A)}{\leq} \mathrm{PM}^{I}_{p^{\prime}0\sigma^{d}10\sigma^{b}}\] \[\overset{(B)}{\leq} \mathrm{PM}^{I}_{p^{\prime}11r^{\prime}00\sigma^{b}} =\mathrm{PM}^{I}_{p^{\prime}11r^{d+1}0q^{b}},\] (24) hold. Here, (A) follows from (22) and (B) is based on the induction hypothesis for \(c=d\). 2. If \(\delta=1\), the relations \[\mathrm{PM}^{I}_{p^{\prime}0\sigma^{d+1}1q^{b}} =\mathrm{PM}^{I}_{p^{\prime}0\sigma^{d}11q^{b}}\overset{(C)}{\leq }\mathrm{PM}^{I}_{p^{\prime}11r^{\prime}01q^{b}}\] \[\overset{(D)}{\leq} \mathrm{PM}^{I}_{p^{\prime}11r^{\prime}10\sigma^{b}} =\mathrm{PM}^{I}_{p^{\prime}11r^{d+1}0q^{b}},\] (25) hold. Here, (C) is based on the induction hypothesis for \(c=d\) and (D) follows from (22). Combining cases (1) and (2), we obtain \[\mathrm{PM}^{I}_{p^{\prime}0\sigma^{d+1}1q^{b}}\leq\mathrm{PM}^{I}_{p^{\prime} 11r^{\prime}10q^{b}},\] (26) for all \(r^{d+1}\in\{0,1\}^{d+1}\), \(p^{\prime}\in\{0,1\}^{a}\), \(q^{b}\in\{0,1\}^{b}\) and \(a,b\geq 0\), which completes the induction step. The PO of PMs described in Theorem 3 is directly related to the binary expansions of the candidates indices \(j\), for which reason we call it the PO of candidates.