# Coarsely Quantized Decoding and Construction of Polar Codes Using the Information Bottleneck Method
## 6 Numerical Results
### Information Bottleneck Decoders  
In this section, the simulation results on the error correction performance of the coarsely quantized information bottleneck decoders are discussed. In all the simulations in the following discussion, an AWGN channel with BPSK modulation is assumed. The information bottleneck decoders and the information bottleneck quantizer have the same resolution, i.e., \(|\mathcal{Y}|=|\mathcal{T}|\). The conventional SCL decoders use double-precision, floating-point arithmetic, i.e., they assume virtually no quantization and have a 64-bit resolution. The CRC-aided settings are used where a value for \(N_{\textit{crc}}\) is specified. It should be pointed out that we did not attempt to optimize the design \(E_{b}/N_{0}\) over the error rate of the information bottleneck decoders in this work. The design \(E_{b}/N_{0}\) is selected as mentioned in Section 6.1 to obtain the information set \(\mathcal{A}\) of a polar code. Then, the error correction performance of a conventional SCL (SC) decoder is compared to that of an information bottleneck SCL (SC) decoder generated for that design \(E_{b}/N_{0}\). First, Figures 15 and 16 show the block error rate of double-precision floating-point SCL decoder when used with channel quantizers of various bitwidths designed using the information bottleneck method. Figure 15 shows the results for \(N=1024\), \(N_{\textit{crc}}=16\), list size \(N_{L}=32\), and quantizer cardinalities \(|\mathcal{Y}|=4,8,16\) or \(32\). It can be seen that a 5-bit quantizer causes virtually no error rate degradation, whereas the 2-bit quantizer causes \(\approx 0.6\) dB degradation at a block error rate of \(10^{-3}\). Figure 16 shows the results of the same experiment repeated for \(N=128\) and \(N_{L}=32\) but without the outer CRC code. The CRC is dropped for the sake of fair comparison with the results of [31] since the author did not use the CRC-aided setting in his experiments. Especially, the results for 2-bit resolution are comparable to those of [31]. Firstly, we see that a channel quantizer with \(|\mathcal{Y}|=16\) shows negligible degradation for the shorter codeword length. Secondly, the 2-bit quantizer causes approximately 0.55 dB degradation, which is slightly less than the 0.8 dB degradation reported for the 3-level, i.e., 2-bit, quantizer in Figure 3.1 in [31]. It is, however, pointed out that the code construction methods used in [31] are different than the one used in this document. Based on Figures 15 and 16, \(|\mathcal{Y}|=16\) or \(32\) seems a good choice to minimize the effect of channel quantization on the decoder performance. Figure 14: Block error rate of polar codes constructed using Gaussian approximation, Tal and Vardy’s method, and the information bottleneck method using a conventional successive cancellation (SC) and successive cancellation list (SCL) decoder with \(N_{L}=8,32\), 16-bit cyclic redundancy check (CRC), \(N=1024\), rate 0.5, and design \(E_{b}/N_{0}=3\) dB. Figures 17-22 present simulation results for the information bottleneck decoders without the message alignment. Figure 17 illustrates the trade-off between the space complexity and the performance of the information bottleneck SCL decoders where the quantized decoders for different choices of \(|\mathcal{T}|\) are compared to a conventional SCL decoder for \(N=1024\), list size \(N_{L}=32\), and \(N_{crc}=16\). As expected, the information bottleneck decoder having the lowest resolution, i.e., \(|\mathcal{T}|=8\) shows the largest degradation of \(\approx\)0.65 dB where the channel quantizer contributes 0.25 dB (as in Figure 15). The performance degradation of the quantized decoding can be reduced to \(\approx\)0.33 dB and \(\approx\)0.2 dB using decoders constructed for \(|\mathcal{T}|=16\) and \(|\mathcal{T}|=32\), respectively.