## 3 Candidate Generation in Parallel Decoding of Rate-1 Nodes
### Path Exclusion Based on PM Relations
The candidate index \(j\) in this case is a power of \(2\) because \(\text{hw}(2^{n})=1\) for all \(n\in\mathbb{N}_{0}\). 2. _Two flipped bits:_ The PMs of the candidates generated by flipping exactly two bits are greater than or equal to PM\({}_{l,0}\) and greater than or equal to the two PMs of the candidates where the bits at the same two positions were flipped separately. To clarify, let us suppose three different candidates with indices \(\sigma,\tau,\omega\) exhibit \(\text{hw}(\sigma)=\text{hw}(\tau)=1\) and \(\text{hw}(\omega)=2\), respectively. These two bits in which candidates \(\sigma\) and \(\tau\) differ from the HDD candidate \(0\) are flipped simultaneously in candidate \(\omega\). Then,\[\text{PM}_{l,\sigma} =\text{PM}_{l,0}+|\alpha_{i_{e}}^{v,l}|\] (15) \[\text{PM}_{l,\omega} =\text{PM}_{l,0}+|\alpha_{i_{e}}^{v,l}|+|\alpha_{i_{r}}^{v,l}|\] (16) \[\text{PM}_{l,\tau} =\text{PM}_{l,0}+|\alpha_{i_{r}}^{v,l}|\] (17) and \[\text{PM}_{l,\sigma} \leq\text{PM}_{l,\omega}\geq\text{PM}_{l,\tau}.\] (18) Considering a new candidate \(\rho\) with \(\text{hw}(\rho)=1\), the relation between candidates \(\omega\) and \(\rho\) is an open question. A priori, the relation between \(\text{PM}_{l,\omega}\) and \(\text{PM}_{l,\rho}\) is unknown, since \(|\alpha_{i_{e}}^{v,l}|\) can be smaller than, greater than, or equal to \(|\alpha_{i_{e}}^{v,l}|+|\alpha_{i_{r}}^{v,l}|\). Therefore, we can only state with certainty that the three candidates \(j=0\), \(\sigma\) and \(\tau\) are always more likely than candidate \(\omega\). Consequently, the minimal index candidates with exactly two flipped bits can have after sorting is \(l+3\). 3. \(x\)_flipped bits:_ In general, the number of PMs known to be less than or equal to \(\text{PM}_{l,j}\) with \(\text{hw}(j)=x\) equals \(2^{x}-1\). This is the case because the number of candidates generated by the combinations of flipping the same \(x\) bits is \(2^{x}\) and candidate \(j\) is the one where all \(x\) bits are flipped simultaneously. It follows that the minimal index candidate \(j\) can have after sorting all \(2^{x}\) forks is \(l+2^{x}-1\). It can be seen that (13) describes the smallest possible index of a candidate \(j\), depending on the number of flipped bits and the index \(l\) of its input path. With the knowledge of \(m_{l,j}^{\prime}\) being the smallest possible index of candidate \(j\), we can conclude that each path with \(m_{l,j}^{\prime}>L-1\) will be discarded, and therefore it is unnecessary to consider these paths. Theorem 2 is consequently proven. **Remark 1**. : _Theorem 1 is a corollary to Theorem 2, but it was described first to build on._  
As an example, Figure 3 shows all the \(m_{l,j}^{\prime}\) for \(L=8\) and \(P=4\). All the candidates corresponding to a red colored \(m_{l,j}^{\prime}\) can be excluded a priori. With Theorem 2, the total number of required candidates in a Rate-1 node of length \(N_{v}\) in Fast-SSCL decoding with list size \(L\) consequently is  
\[L_{\text{out},m^{\prime}}=\sum_{l=0}^{L-1}L_{\text{out},m^{\prime}}^{l}. \tag{19}\]  
Furthermore, the number of required candidates from input path \(l\) as given in (14) is restated as a function of \(l\), \(L\) and \(P\) by  
\[L_{\text{out},m^{\prime}}^{l}=\sum_{a=0}^{|\text{Id}(L-l)|}\binom{P}{a}. \tag{20}\]  
There are two different approaches for the candidate generation, a serial one in [11] and a parallel one in this work. In a parallel implementation of a Rate-1 node, it is not known a priori which candidate paths would survive the intermediate sorting steps in [11] and, therefore, would be further expanded. However, with Theorem 1, for every bit estimation step in [11], the number of paths can be reduced from \(2L\) to \(2L-1\). Theorem 2 shows that the number of paths can be further reduced, but the theorem is not directly applicable to the serial approach, since it is based on a global view.