# Coarsely Quantized Decoding and Construction of Polar Codes Using the Information Bottleneck Method
## 6 Numerical Results
### Code Construction  
Figures 12 and 13 present the so-called _frozen charts_, a compact visualization of information sets from [38], for polar codes constructed using [PERSON] and Vardy's method and the information bottleneck method, as well as the Gaussian approximation [24]. A frozen chart is a grid of squares where each square denotes a bit channel, starting from top left and reading columnwise. A colored square denotes a frozen bit position, while a white square represents an information bit position. The bit channels in the frozen charts are sorted with respect to the descending order of their error probabilities obtained from [PERSON] and Vardy's method with \(\left|\mathcal{T}\right|=512\) such that the top left square represents the least reliable bit channel, while the bottom right square represents the most reliable bit channel. Each polar code is constructed for rate 0.5 such that \(K=N/2\) worst bit channels are frozen. The design \(E_{b}/N_{0}\) for any polar code presented in this paper is selected in the following way. First, the block error probability of the code with target code rate and codeword length \(N\) is computed using Equation (3) from [39] for various candidate \(E_{b}/N_{0}\) values. Then, the candidate value which achieves the block error probability of \(10^{-3}\) at the smallest \(E_{b}/N_{0}\) is selected as the design \(E_{b}/N_{0}\) for the polar code. Figure 12: Frozen charts for half-rate polar code with \(N=128\) and design \(E_{b}/N_{0}=3\) dB, constructed using (**a**) [PERSON]’s method with \(\left|\mathcal{T}\right|=512\), (**b**) [PERSON]’s method with \(\left|\mathcal{T}\right|=16\), (**c**) information bottleneck method with \(\left|\mathcal{T}\right|=16\), (**d**) information bottleneck method with \(\left|\mathcal{T}\right|=32\), and (**e**) Gaussian approximation. Figure 12 shows the frozen charts obtained for \(N=128\). It is seen that the codes obtained from all of the construction methods are the same. Especially, [PERSON] and Vardy polar codes for \(|\mathcal{T}|=512\) and \(|\mathcal{T}|=16\) are the same. This result suggests that although the bounds on the error probability of the bit channels from [PERSON]'s method might not be tight with \(|\mathcal{T}|=16\), the resulting information set is the same as the one obtained for \(|\mathcal{T}|=512\). The frozen charts of the codes constructed using the information bottleneck method for \(|\mathcal{T}|=16\) and \(|\mathcal{T}|=32\) are also the same. Hence, for a given code rate and design \(E_{b}/N_{0}\), all the methods compared here produce the same information sets, regardless of the choice of \(|\mathcal{T}|\). Figure 13 shows the frozen charts obtained for \(N=1024\). It is seen that the codes obtained from various construction methods are slightly different for the larger codeword length. [PERSON] and [PERSON] polar codes for \(|\mathcal{T}|=512\) and \(|\mathcal{T}|=16\) differ in a few positions (cf. Figure 13a,b). The same is true for codes constructed with the information bottleneck method (cf. Figure 13c,d). Interestingly, the codes constructed using [PERSON] and [PERSON]'s method with \(|\mathcal{T}|=512\) in Figure 13a are exactly the same as the ones constructed using the Gaussian approximation (cf. Figure 13e) and differ from the frozen chart of the information bottleneck polar code of Figure 13d in only two positions. Hence, the codes constructed for coarsely quantized decoders do not differ significantly from those designed for high-resolution decoding. The important question here is how much such small differences affect the error correction performance of the polar code. Figure 14 shows the block error rate of using the information sets of Figure 13. It is obvious from Figure 14 that the minor differences in the information sets obtained from various construction methods and different quantization parameters \(|\mathcal{T}|\) have a negligible effect on the block error rates of polar code under SC decoding and CRC-aided SCL decoding with different list sizes.