## 2 Background
### Fast-SSCL Decoding and Rate-1 Nodes  
Similar to the PFT optimization for the SC algorithm, optimized Rate-0, Rate-1, REP and SPC nodes also exist for the SCL algorithm [18; 19; 10; 11; 20]. The calculation of PMs and the candidate generation, i.e., proper path splitting, in the optimized nodes are the main tasks in adapting the PFT pruning techniques to SCL decoding. State-of-the-art Fast-SSCL decoding was introduced in [11] with the focus on the decoding of Rate-1 nodes. Instead of considering all \(L\cdot 2^{N_{v}}\) possible candidates in a Rate-1 node of length \(N_{v}\) and list size \(L\), the authors of [11] proved that the number of path splits needed to avoid any error-correction performance degradation is  
\[P=\min(N_{v},L-1). \tag{4}\]  
The number of all \(L\cdot 2^{P}\) possibly considered candidates is reduced by (4), when \(L-1<N_{v}\). However, the Rate-1 node decoding algorithm in [11] is derived from list sphere decoding [17] and involves a sorting step after each bit estimation where the decoding path is split. Consequently, for each of the \(P\) path splits, the number of paths doubles, while only the \(L\) most probable ones are considered after \(2L\) to \(L\) sorting in the following splitting step. Thus, the number of visited paths in a Rate-1 node of [11] is  
\[L_{\text{out,s}}=L\cdot(P+1). \tag{5}\]  
In [12], a parallel decoding implementation of a Rate-1 node is used. The corresponding architecture is shown in Figure 2. To achieve low latency and high throughput, all the relevant candidate partial sums \(\boldsymbol{\beta}^{l,j}\) according to (4) are estimated in parallel. The \(2^{P}\) possible candidates originating from input path \(l\in[0,L-1]\) are indexed by \(j\in[0,2^{P}-1]\). For the parallel processing, \(2^{P}-1\) splitting patterns for each of the \(L\) input paths are determined in the _Candidate Generator_. These patterns are used to generate the candidate partial sumsby applying the bitwise XOR operation with the HDD estimates \(\mathbf{\beta}^{l,0}\) of the dedicated LLR vectors \(\mathbf{\alpha}^{v,l}\). Consequently,  
\[L_{\text{out,p}}=L\cdot 2^{P} \tag{6}\]  
candidates are considered and need to be sorted to extract the \(L\) most probable partial sums \(\mathbf{\beta}^{v,l}\).  
The notation of PMs is as follows: PM\({}_{l,j}\) denotes the PM of the candidate path with index \(j\) originating from input path \(l\). To estimate PM\({}_{l,j}\), Equation (3) is restated as  
\[\text{PM}_{l,j}=\text{PM}_{l}+\sum_{\begin{subarray}{c}i=0\\ \beta_{i}^{l,j}\
eq\beta_{i}^{l,0}\end{subarray}}^{N_{p}-1}\Big{|}\mathbf{\alpha} _{i}^{v,l}\Big{|}. \tag{7}\]  
Note that the PM of an HDD estimate is always equal to the input PM and written as PM\({}_{l,0}\). PM\({}_{l}\) with a single index is used to denote the \(l\)-th PM in a sorted list of PMs.  
Due to the exponential increase of the number of candidate paths with \(N_{v}\) for \(N_{v}>L\), in [12], a restriction on the maximal node size of Rate-1 nodes \(N_{v,Rate\text{-}1_{max}}=4\) was set for \(L>4\). This limits tree pruning to layer \(s=2\) for Rate-1 nodes and thus the optimization of the PFT. This especially affects the rightmost Rate-1 nodes, which tend to be large due to polarization being predominant at high bit channels' indices.  
Figure 2: Parallel Rate-1 node architecture.  
Concerning the rightmost Rate-1 nodes (e.g., node \(v=6\) in Figure 1), it is important to recollect their relation to CRC. Without the presence of an CRC, the rightmost Rate-1 nodes only need to proceed with the most probable path [7]. In contrast, using a CRC requires the maintenance of the list of \(L\) paths to preserve error-correction performance compared to CA-SCL decoding.