## 3 Candidate Generation in Parallel Decoding of Rate-1 Nodes
### Candidate Generation Based on Partially Ordered Candidate Sets
With Theorem 4, 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 observed by (19), when \(m^{\prime}_{l,j}\) and \(L^{l}_{\text{out,m}^{\prime}}\) are replaced by \(m_{l,j}\) and \(L^{l}_{\text{out,m}}\), respectively, as

\[L_{\text{out,m}}=\sum_{l=0}^{L-1}L^{l}_{\text{out,m}}. \tag{30}\]

Fast-SSCL decoding with the use of Rate-1 nodes with PO-based candidates generation is denoted by PO-based Fast-SSCL (PO-Fast-SSCL) decoding. ### Partial Order-Based Candidate Generation with Threshold

As described in [20], fewer path forks are required to match the error-correction performance of conventional SCL decoding in practical decoder implementations. Thus, Ref. [20] introduced a threshold \(S_{\text{Rate-1}}\) to limit the number of path splits. They determined \(S_{\text{Rate-1}}\) heuristically by analyzing the error-correction performance for different \(S_{\text{Rate-1}}\) and \(\mathcal{P}(1024,512)\). For \(L\in\{2,4,8\}\), an associated threshold \(S_{\text{Rate-1}}\in\{1,1,2\}\) was set, respectively, to replace \(L-1\) in (4):

\[P=\min(N_{v},S_{\text{Rate-1}}). \tag{31}\]

POS-based candidate generation is denoted as the adoption of the threshold \(S_{\text{Rate-1}}\) in the use of Algorithm 1. POS-based candidate generation results in a vertical cutting-off of the candidates. This is shown in Figure 6a, which represents the left part of Figure 5.

Figure 6: Smallest possible candidate indices \(m_{l,j}\) for \(L=8\), calculated by **(a)** POS algorithm with \(S_{\text{Rate-1}}=2\), **(b)** ExPOS algorithm with \(S_{\text{Rate-1}}=2\), \(k_{c}=3\), **(c)** ExPOS algorithm with \(S_{\text{Rate-1}}=2\), \(k_{c}=4\), **(d)** ExPOS algorithm with \(S_{\text{Rate-1}}=1,k_{c}=8\). Fast-SSCL decoding with the use of PO-based candidate generation in Rate-1 nodes, together with (31), is denoted by POS-based Fast-SSCL (POS-Fast-SSCL) decoding. For the estimation of \(S_{\text{Rate-1}}\), we refer to the first loop of Algorithm 2 and its description in Section 3.4. ### Extended Threshold Partial Order-Based Candidate Generation

To further reduce the number of candidates for practical decoder implementations, the approach of limiting the number of paths by a threshold is extended. For that purpose, the rates of surviving the sorting step for each candidate \(j\) in the Rate-1 nodes of POS-Fast-SSCL decoders are investigated. The surviving rate \(h_{j}\) is obtained as the empirical probability of the event that candidate \(j\) from input path \(l\) survives sorting, normalized to the sum over all proceeded candidates divided by \(L\). As an example, Figure 7 shows the means of the surviving rates \(h_{j}\) of candidates \(j\) in Rate-1 nodes with \(L=32\) and \(P=S_{\text{Rate-1}}=5\) as a heat map. The values were observed by a total of \(1.5\times 10^{6}\) samples, extracted from POS-Fast-SSCL decoders for a code with \(N=1024\) and \(R\in\{1/3,1/2,5/6\}\) during decoding data received at Signal-to-Noise-Ratios (SNRs) of \(E_{b}/N_{0}\in\{0,3,6,9\}\). The normalization in the color map is divided into two slopes: The first half of the color range is mapped linearly to values from 0 to 0.05 and the second half is mapped linearly to values from 0.05 to 1.0. It can be observed that these surviving rates do not directly map to \(m_{l,j}\). Candidate \(j=0\) from input paths 0 to 9 always survives, and from input paths \(l>9\), the surviving rates decrease from 1.0 to 0.4. In general, the highest surviving rate of each \(j\) is observed for candidates originating from input path \(l=0\). The surviving rates of candidates with \(x=1\) decrease with increasing \(j\) and decreasing \(l\). An exception is \(j=2\), where a hot spot around \(l=5\) appears, but candidate \(j=2\) from input path \(l=0\) still has the highest surviving rate. For \(j\geq 4\) and \(x=1\), the surviving rates for \(l\geq 1\) are smaller than 0.002.