# Coarsely Quantized Decoding and Construction of Polar Codes Using the Information Bottleneck Method
## 6 Numerical Results
### Information Bottleneck Decoders
If a \(Q_{LLR}=6\) bit resolution is assumed for the LLR values in the translation tables, the information bottleneck decoder without the table alignment of Figure 23 will require \(N\cdot|\mathcal{T}|\cdot Q_{LLR}=12,\)288 bits to store the \(N\) translation tables. On the other hand, the aligned decoder of Figure 23 with \(|\mathcal{T}^{*}|=16\) will require only \(|\mathcal{T}^{*}|\cdot Q_{LLR}=96\) bits for the aligned translation table, thereby reducing the memory required for saving the translation tables by more than 99%. Hence, the aligned decoder is space-efficient in the sense that the space required for storing or implementing the translation table is far less and independent of the codeword length. Secondly, the dynamic range of LLR values stored in the single aligned translation table is significantly smaller than that for \(N\) distinct translation tables (cf. Figures 10 and 11). For the polar code of Figure 23, the magnitudes of LLR values stored in the \(N=128\) distinct translation tables ranged between 0 and 63. By contrast, the aligned translation table stores LLR magnitudes between 0 and 10, i.e., the LLR values in the aligned translation table require smaller bit resolution. The aligned translation table for \(|\mathcal{T}^{*}|=32\) or 128 (cf. Figure 24) will require 192 or 786 bits of memory, respectively, which is still a considerable reduction. However, recall that translation table alignment with \(|\mathcal{T}^{*}|>|\mathcal{T}|\) increases the bitwidth of the aligned decoding tables at the decision level, which increases their memory requirement. Unfortunately, this increase in the memory requirement of the aligned decoding tables outweighs the savings achieved from the translation table alignment. Finally, the fact that only \(|\mathcal{T}^{*}|\) distinct LLR values appear in the aligned information bottleneck SCL decoder can be exploited to circumvent the performance degradation caused be the use of the approximate path metric update rule. Since a single LLR value leads to a single path metric increment in Equation (9) for \(\hat{u}=0\), there exist only \(|\mathcal{T}^{*}|\) path metric increments when a path is extended with \(\hat{u}=0\) at any stage. Owing to the odd symmetry of the translation table (cf. Figure 11), the metric increments for \(\hat{u}=1\) are readily obtained from the increments for \(\hat{u}=0\), as shown in Figure 14:2 A path metric increment \(\Delta M\) in (9) for \(t^{*}=0,1,2,\ldots,15\), when the decoding path is extended with \(\hat{u}=0\), is equal to \(\Delta M\) for \(t^{*}=15,14,13,\ldots,0\), respectively, if the path is extended with \(\hat{u}=1\). The path metric update values in Figure 25 are computed according to the exact formulation of Equation (9). Therefore, the \(|\mathcal{T}^{*}|\) LLR values in the aligned translation table can  
Figure 14:3 Effect of alignment cardinality \(|\mathcal{T}^{*}|\) on the block error rate of a 4-bit aligned information bottleneck SCL decoder with \(N_{L}=8\) or 32, 16-bit CRC for \(N=128\), rate 0.5, and design \(E_{b}/N_{0}=3\) dB. be replaced by their respective path increment values, computed offline, for either value of \(u\), and used to translate cluster values \(t^{*}\in\{0,1,\ldots,|\mathcal{T}^{*}|-1\}\) directly into path increments for both the choices of \(u\in\{0,1\}\). As a result, the exact metric update in the aligned information bottleneck decoder is implementation-friendly, i.e., just an addition of real numbers, similar to the approximation of Equation (10). Perhaps the biggest question on the mind of the reader at this point is how much complexity reduction is achieved using an information bottleneck decoder. Unfortunately, the answer to this question is not straightforward and depends on how the proposed decoder is implemented. The focal point of this paper is to provide a new way to look at quantized polar decoders. Detailed implementation issues are subject to ongoing work.