# Coarsely Quantized Decoding and Construction of Polar Codes Using the Information Bottleneck Method
## 1 Introduction
The error rate of a 4-bit information bottleneck SCL decoder is only slightly worse than that of a double-precision floating-point SCL decoder. The lookup tables used in the information bottleneck SCL decoders are categorized into _decoding_ and _translation_ tables. The decoding tables are used for the successive cancellation part of the decoder, while the translation tables are used to translate the integer valued messages into LLRs for path metric updates. For a codeword length \(N\), \(2N-2\) distinct decoding tables and \(N\) distinct translation tables are used [26]. Thus, the space complexity of an information bottleneck SCL decoder increases with the codeword length \(N\). Quantized SC and SCL decoders have been studied in, e.g., [27, 28, 29, 30, 31]. There are two major differences between these quantization approaches from the literature and the one proposed in this paper. Firstly, unlike the aforementioned quantization approaches, the quantization performed using the information bottleneck method is nonuniform in nature. Secondly, the quantization in [27, 28, 29, 30, 31] assigns a numerical value, e.g., reconstructed from the quantization labels, to each quantized LLR, which is then used in the relevant arithmetic operation. By contrast, the information bottleneck SCL decoder works entirely with the integer valued quantization labels and translates them into LLRs only when required, i.e., for path metric update. In addition to addressing the trade-off between quantization bitwidth and rate loss of an SC decoder, the authors in [28] proposed a very coarse 3-level, i.e., 2-bit, quantization scheme. Moreover, it was shown that the performance loss caused by the quantization is more pronounced for low code rates. The 3-level quantization scheme of [28] was deployed in short block length polar codes in [31], where it leads to significant degradation in the error rate. However, the author proposed improvements to the SCL decoder which drastically reduce this degradation. In this paper, we present the polar code construction using the information bottleneck method [25] in detail, along with its comparison to the closely related construction method of [PERSON]. Afterwards, the information bottleneck decoders of [26] are reviewed accompanied by detailed insights into their working. The huge number of lookup tables used in the information bottleneck decoders of [26] can be viewed as a space complexity cost for their computational simplicity. We demonstrate that using the _message alignment_ technique of [7], the required number of tables and, hence, the space complexity of the information, bottleneck SCL decoders can be reduced. The rest of the paper is structured as follows. After a brief recap of polar codes and the information bottleneck method in Section 2, polar code constructing using the information bottleneck method and [PERSON]'s method is discussed in Section 3. Section 4 shows how the results of Section 3 are used for decoding. Section 5 presents our latest work on the subject: space complexity reduction of the information bottleneck SCL decoder. Section 6 presents simulation results before concluding the paper in Section 7.