# Coarsely Quantized Decoding and Construction of Polar Codes Using the Information Bottleneck Method
## 6 Numerical Results
### Information Bottleneck Decoders
Figure 18 compares the block error rate of information bottleneck and conventional SCL decoders for \(N=128\) and \(N_{L}=32\)  
Figure 16: Block error rate of double-precision floating-point SCL decoder with channel quantizers of different resolutions designed using the information bottleneck method. \(N_{L}=32\) without the outer CRC code, \(N=128\), code rate 0.5, and design \(E_{b}/N_{0}=3\) dB. Figure 15: Block error rate of double-precision floating-point SCL decoder with channel quantizers of different resolutions designed using the information bottleneck method. \(N_{L}=32\), \(N_{crc}=16\), \(N=1024\), code rate 0.5, and design \(E_{b}/N_{0}=3\) dB. without the outer CRC code (cf. Figure 16). It can be seen that the information bottleneck decoders with \(|\mathcal{T}|=16,8\) and \(4\) show \(0.2,0.55\) and \(1.9\) dB degradation, respectively, at a block error rate of \(10^{-3}\). Compared to the information bottleneck decoder with 2-bit resolution, the 3-level quantized decoder of [31] exhibits a degradation of \(2.6\) dB for similar decoder parameters (Figure 3.4 in [31]). However, the enhancements proposed in [31] drastically improve the performance of the 3-level quantized SCL decoder and limit the degradation caused by the very coarse quantization to well below \(1.5\) dB. For the case of information bottleneck decoders, as presented in this paper, the performance degradation can only be reduced using higher bitwidths, e.g., \(|\mathcal{T}|=16\). Figure 17: Block error rate comparison between conventional decoder (SCL) and information bottleneck decoders (IB-SCL) constructed for \(|\mathcal{T}|=8,16\) or \(32\). \(N=1024\), \(N_{L}=32\), \(16\)-bit CRC, code rate \(0.5\), and design \(E_{b}/N_{0}=3\) dB. Figure 18: Block error rate comparison between conventional decoder (SCL) and information bottleneck decoders (IB-SCL) constructed for \(|\mathcal{T}|=4,8\) or \(16\). \(N=128\), \(N_{L}=32\), no CRC, code rate \(0.5\), and design \(E_{b}/N_{0}=3\) dB. Figure 19: Block error rate comparison between conventional decoder (SCL) and information bottleneck decoders (IB-SCL) constructed for \(|\mathcal{T}|=4,8\) or \(16\). \(N=256\), \(N_{L}=32\), no CRC, code rate \(37/256\), and design \(E_{b}/N_{0}=4\) dB. Figure 20: Effect of design \(E_{b}/N_{0}\) in the mismatched use of the information bottleneck decoders. \(N=128\) or \(1024\), \(|\mathcal{T}|=16\), \(N=128\), \(N_{L}=32\), \(N_{cc}=16\), and code rate \(0.5\). Figure 19 shows the block error rate of an information bottleneck SCL decoder for a low code rate of \(\approx\)0.145 with \(N=256\), \(N_{L}=32\), no CRC, and design \(E_{b}/N_{0}=4\) dB. The information bottleneck decoders with \(|\mathcal{T}|=16,8\), and 4 show a degradation of \(0.3,0.85\), and 2.9 dB, respectively, at a block error rate of \(10^{-3}\). The degradation due to quantization in the information bottleneck decoders is starker (cf. Figure 18) at low code rates in accordance with the theoretical predictions of [28]. In the remainder of this document, the investigations are restricted to information bottleneck decoders having a 4-bit resolution, i.e., \(|\mathcal{T}|=16\). Figure 20 shows the effect of the choice of design \(E_{b}/N_{0}\) for generating information bottleneck decoders that are used mismatched to the operating \(E_{b}/N_{0}\) range. The decoders in the figure are used in the CRC-aided setting with \(N_{\sigma\epsilon}=16\), \(N_{L}=32\), code rate of 0.5, and design \(E_{b}/N_{0}=3\) or 4 dB. For \(N=1024\), the information bottleneck decoder constructed for design \(E_{b}/N_{0}=4\) dB shows a degradation of 0.15 dB compared to the one designed for \(E_{b}/N_{0}=3\) dB at a block error rate of \(10^{-3}\). This degradation is partly due to the difference in the information sets obtained for the two design \(E_{b}/N_{0}\) values and partly due to the decoder itself.