# Coarsely Quantized Decoding and Construction of Polar Codes Using the Information Bottleneck Method
## 6 Numerical Results
### Information Bottleneck Decoders
The evidence for the second part of this statement is provided by the results for \(N=128\) in the same  
Figure 21: Block error rate of conventional (SCL) and a 4-bit information bottleneck (IB-SCL) decoders for \(N=1024,256\) and \(128\), rate 0.5, 16-bit CRC design \(E_{b}/N_{0}=3\) dB. (**a**) \(N_{L}=8\), (**b**) \(N_{L}=32\). Figure 14:0 Effect of using the approximation path metric update rule of Equation (10) on the block error rate of a 4-bit information bottleneck decoder with \(N_{L}=2,8\) or 32, 16-bit CRC, \(N=128\), rate 0.5, and design \(E_{b}/N_{0}=3\) dB. figure. Although the information sets generated for design \(E_{b}/N_{0}=3\) and 4 dB are the same in the case of \(N=128\), the decoder designed for \(E_{b}/N_{0}=4\) dB shows a degradation of 0.1 dB. Hence, the design \(E_{b}/N_{0}\) of an information bottleneck decoder needs to be carefully chosen. We have observed that the sensitivity of information bottleneck decoders to design \(E_{b}/N_{0}\) depends on codeword length \(N\), code rate, and decoder setting, i.e., with or without the outer CRC. Figure 21a compares the block error rate of the conventional and information bottleneck SCL decoder for a list size of \(N_{L}=8\) and various codeword lengths \(N\). The information bottleneck decoder shows a degradation of approximately \(0.16,0.25\), and 0.39 dB at a block error rate of \(10^{-3}\) for codeword lengths of \(N=128,256\), and 1024, respectively. Figure 21b shows the same comparison for a list size of \(N_{L}=32\), where a gap of \(0.13,0.22\), and 0.33 dB is observed at a block error rate of \(10^{-3}\) for codeword lengths of \(N=128,256\), and 1024, respectively, between the block error rate of the information bottleneck decoder and the conventional scheme. These results suggest that the information bottleneck decoders show a larger performance loss for larger codeword lengths. The exact computation of the path metric in an SCL decoding scheme according to Equation (9) involves logarithm and exponential functions which are not hardware-friendly. Therefore, the approximation of Equation (10) is preferred for hardware implementation which causes a negligible effect on the error rate of the conventional SCL decoder [27]. Figure 22 shows the effect of using the approximate path metric rule in an information bottleneck decoder for \(N=128\) and different list sizes. It can be seen that although the use of the approximate path metric update does not affect the error correction performance of the decoder for a list size \(N_{L}=2\), it does cause a minor performance degradation for larger list sizes. The performance loss caused by the use of the approximate update rule seems to increase with the list size. In the following, results on the aligned information bottleneck decoder of Section 5 are presented. First, Figure 23 compares the information bottleneck decoder of Section 4 with its aligned version for \(N=128\) and \(|\mathcal{T}|=16\). An alignment cardinality of \(|\mathcal{T}^{*}|=16\) has been used for the translation table alignment in the aligned decoder. It can be seen that the translation table alignment causes a minor degradation of approximately 0.1 dB for large list size of \(N_{L}=32\), but the degradation vanishes as the list size becomes smaller. Figure 24 shows that the degradation caused by the translation table alignment can effectively be eliminated if a larger alignment cardinality is used, e.g., \(|\mathcal{T}^{*}|=32\) or 128. Figure 14:1 Block error rate of a 4-bit aligned information bottleneck SCL decoder with alignment cardinality \(|\mathcal{T}^{*}|=16\), 16-bit CRC, \(N=128\), \(N_{L}=2,8\) or 32. The aligned information bottleneck decoder offers several benefits over the information bottleneck decoder of [26]. Firstly, the aligned decoder requires only a single translation table for converting the integer cluster labels into LLR values, regardless of the codeword length \(N\). Recall that the information bottleneck decoder without translation table alignment requires \(N\) distinct translation tables.