## 3 Candidate Generation in Parallel Decoding of Rate-1 Nodes
### Candidate Generation Based on Partially Ordered Candidate Sets
The aforementioned hot spot also appears for \(j=3\), but here the candidate originating from input path \(l=0\) does not exhibit the highest surviving rate. Similar observations were also made for other list sizes, as well as for the separate samples of the parameters \(R\) and \(E_{B}/N_{0}\) that were averaged in Figure 7. In summary, the investigation of the surviving rates revealed that:

* Candidates with \(j=3\) are the only needed candidates with \(x=2\). All other candidates with \(x\geq 2\) can be discarded. Figure 7: Empirical probability of surviving the sorting step \(h_{j}\) for candidates \(j\) in Rate-1 nodes with \(L=32\) and \(P=5\). * The number of candidates with equal \(j\) can be decreased linearly until only the candidate originating from path \(l=0\) needs to be considered. We approximate an exclusion of the candidates with the smallest surviving rates by a new threshold \(\theta_{j}\) to replace \(L\) in (29) and in Algorithm 1, line 4. The observed linear decrease is implemented by a gradient \(k_{c}\). The threshold for the exclusion of candidate \(j\) with \(x=\mathrm{hw}(j)\) flipped bits and originating from input path \(l\) is determined by

\[\theta_{j}=\min(L,\max(m_{0,j},L-k_{c}\cdot j+x)+1). \tag{32}\]

The gradient \(k_{c}\) needs to be determined for each combination of \(N\), \(R\), \(L\) and \(S_{\text{Rate-1}}\). To consider the described findings about the surviving rates, Algorithm 1 is adapted at the given lines as follows:

**line 4**: _Set \(x=1\) instead of looping over the number of flipped bits._
**line 8**: _Replace the stopping condition by \(m_{l,j}<\theta_{j}\)._
**line 19**: _Append candidate \(j=3\) when \(P>=1\) and \(m_{l,3}<\theta_{3}\) according to (32)._

We call the adapted algorithm ExPOS-based candidate generation. Fast-SSCL decoding with the use of ExPOS-based candidate generation in Rate-1 nodes is denoted by ExPOS-based Fast-SSCL (ExPOS-Fast-SSCL) decoding. To determine the thresholds in ExPOS-Fast-SSCL decoding, an iterative approximation is provided in Algorithm 2. The iterative approximation is based on Monte Carlo error-correction performance simulations. The first loop of Algorithm 2 determines \(S_{\text{Rate-1}}\) by comparing the error-correction performance of POS-Fast-SSCL decoding and SCL decoding. \(S_{\text{Rate-1}}\) is initialized with 0 and increased until the performance loss at a specified target Frame Error Rate (FER) or Bit Error Rate (BER) is smaller than a specified \(\Delta\). Once a sufficient \(S_{\text{Rate-1}}\) is found, it is used in the second loop to determine \(k_{c}\) as the relevant parameter for the calculation of the internal threshold \(\theta_{j}\) for the candidate selection. The approximation proceeds in the same way as for \(S_{\text{Rate-1}}\), with \(k_{c}\) being initialized with \(L\) and decreased until the error-correction performance of ExPOS-Fast-SSCL decoding is close enough to SCL decoding. ```
1:procedureThresholdApproximation(\(\mathcal{P}\), \(L\), target FER/BER, \(\Delta\))
2: Run error-correction performance simulation with SCL decoding. 3:for\(S_{\text{Rate-1}}\gets 0\), \(L-1\)do\(\triangleright\) Start with minimal number of path splits
4: Run error-correction performance simulation with POS-Fast-SSCL decoding. 5:if Performance loss at target FER/BER \(<\Delta\)then
6:break
7:endif
8:endfor
9:for\(k_{c}\gets L\), 0 do\(\triangleright\) Start with maximal gradient for candidate exclusion
10: Run error-correction performance simulation with ExPOS-Fast-SSCL decoding. 11:if Performance loss at target FER/BER \(<\Delta\)then
12:break
13:endif
14:endfor
15:return\(S_{\text{Rate-1}}\), \(k_{c}\)
16:endprocedure
```

**Algorithm 2** Iterative threshold approximation. The computational effort for each simulation depends on the choice of the target FER/BER. However, the maximum number of needed simulation runs is \(L\) for the estimation of \(S_{\text{Rate-1}}\) and \(L+1\) for the estimation of \(k_{c}\). In Figure 6b-d, \(m_{l,j}\) is shown resulting from ExPOS-based candidate generation.