## 3 Candidate Generation in Parallel Decoding of Rate-1 Nodes
### Path Exclusion Based on PM Relations  
We start our considerations by investigating the relations between PMs. The PM calculation in (7) represents a cost-function, thus PMs are monotonically increasing. Therefore, in a Rate-1 node, candidates originating from input path \(l\) and generated by flipping bits compared to \(\beta^{l,0}\) can never obtain a lower PM than \(\beta^{l,0}\) itself. Since the input paths are sorted, i.e.,  
\[\text{PM}_{0}\leq\text{PM}_{1}\leq\cdots\leq\text{PM}_{L-1}, \tag{8}\]  
the codeword estimated by HDD of input path \(l\) always remains more probable than the one of input path \(l+1\), which also holds for equivalence since we use a stable sorter. **Theorem 1**: _In Fast-SSCL decoding with list size \(L\), there is no need for path splitting in the least reliable input path \(L-1\) of a Rate-1 node._  
We prove Theorem 1 by contradiction. Let us assume that the two most probable paths originating from input path \(L-1\) survive sorting. These paths are obtained by the HDD estimate of \(\boldsymbol{a}^{v,L-1}\) and its fork with one flipped bit at the least reliable bit position,respectively. The corresponding LLR is \(\alpha_{min}^{v,L-1}\). After sorting, these paths have the indices \(p\) and \(q\), respectively. The corresponding PMs are  
\[\text{PM}_{p} =\text{PM}_{L-1,0}, \tag{9}\] \[\text{PM}_{q} =\text{PM}_{L-1,0}+|\alpha_{min}^{v,L-1}|=\text{PM}_{L-1,1}, \tag{10}\]  
where \(0\leq p<q<L\) must apply due to the assumption that these candidates survive. Now, we show that there are at least \(L\) paths with PMs which are less than or equal to PM\({}_{q}\). Let us consider a path that originates from input path \(l\) and corresponds to the HDD estimate of the LLR vector \(\mathbf{a}^{v,l}\). After sorting, this path has the index \(s\) and its PM is  
\[\text{PM}_{s}=\text{PM}_{l,0}. \tag{11}\]  
With (8), we can conclude that  
\[\text{PM}_{s}\leq\text{PM}_{q}, \tag{12}\]  
which results in \(s<q\). Since we have \(L\) input paths, \(s\) can assume \(L\) values (including \(p=L-1\)), for which (12) is true. Thus, \(q\geq L\), which contradicts the assumption that path \(q\) survives. Theorem 1 is consequently proven. From the considerations of Theorem 1, it can be deduced that the minimum index of the candidate partial sum \(\mathbf{\beta}^{l,0}\) is \(l\). To a priori exclude further paths without the knowledge of the LLR values themselves, a metric is required, which rates all \(L_{\text{out,p}}\) candidate paths. For that purpose, we introduce \(m_{l,j}^{\prime}\) as the smallest possible index that a candidate \(j\) originating from input path \(l\) can have after estimating and sorting all the \(L_{\text{out,p}}\) possible candidates. The proof of Theorem 2 will show that \(m_{l,j}^{\prime}\) can be calculated by  
\[m_{l,j}^{\prime}=l+2^{\text{hw}(j)}-1. \tag{13}\]  
**Theorem 2**. : _In Fast-SSCL decoding with list size \(L\), the number of required candidates originating from input path \(l\) to a Rate-\(1\) node of length \(N_{v}\) is_  
\[L_{\text{out,m}^{\prime}}^{l}=\sum_{ \begin{subarray}{c}j=0\\ m_{l,j}^{\prime}<\L\end{subarray} }^{2^{p}-1}1. \tag{14}\]  
Proof. : We start this proof by discussing why \(m_{l,j}^{\prime}\) is the smallest possible index of candidate \(j\) originating from path \(l\). It follows from (4) that the number of candidates originating from path \(l\) is \(2^{p}\). These candidates, indexed by \(j\), are generated by the HDD estimate of \(l\)-th LLR vector (\(j=0\)) and flipping the bits in the \(P\) least reliable positions in all possible combinations. We now consider the candidates where exactly one, two and \(x\) bits are flipped:  
1. _One flipped bit:_ According to (7), all PMs of the candidates with exactly one flipped bit are greater than or equal to PM\({}_{l,0}\) of the HDD estimation candidate of path \(l\) because the one corresponding \(|\alpha_{i_{flip},i_{flip}}^{v,l}|\) is added. Thus, the minimal index these candidates can have after sorting is \(l+1\).