# Coarsely Quantized Decoding and Construction of Polar Codes Using the Information Bottleneck Method
## 1 Introduction
Eventually, polar codes were included in the 5G standard for control channels in the eMBB scenario [19]. Although proposals for using other decoders, e.g., belief propagation for improving performance of polar codes appeared in the literature as early as [20], the CRC or parity check aided SCL decoder remains the practical decoding choice [21]. Polar codes exploit the _channel polarization_ phenomenon where the effective channels, known as the _bit channels_, experienced by the bits at the input of the polar encoder exhibit polarized behavior: The bit channels tend to be either extremely reliable or completely unreliable. The code uses the reliable bit channels to carry information bits, whereas the input of the unreliable bit channels is fixed to predetermined values. Hence, the order of the bit channels with respect to their reliabilities is determined in order to identify the most reliable ones, a process commonly referred to as _code construction_. This order of the bit channels is not universal and depends on the underlying physical channel. For channel models of practical importance, e.g., AWGN channels, the output alphabet of the bit channels expands exponentially in the codeword length, making exact and efficient computation of the bit channel reliabilities challenging. Therefore, approximate construction methods are used for such channel models. An important contribution regarding approximate construction methods was made in [22], where it was shown that the density evolution tool can also be used for the construction of polar codes. However, the precision requirements for an exact implementation of the method of [22] become impractical as the codeword length increases. The idea of [22] was extended by [PERSON] and [PERSON] in [23], where they provided an efficient implementation of the quantized density evolution for polar code construction. In order to keep track of the inaccuracy introduced by the quantization, the method of [23] determines an upper and a lower bound on the probability of error of each bit channel. Performing the density evolution under a Gaussian approximation provides another notable construction method [24] that tracks the reliability of each bit channel using a single scalar quantity instead of its transition probabilities, hence considerably reducing the computation complexity of density evolution. For a binary input AWGN channel, the sets of reliable bit channels of polar codes constructed with the Gaussian approximation and the Tal and Vardy method differ only in few elements. The idea of combining the information bottleneck method with density evolution [1] can also be utilized for polar code construction. However, unlike other construction methods, the benefits of this approach are twofold: Firstly, the discrete density evolution using the information bottleneck method compresses the output alphabet of the bit channels of a polar code to a small size, while minimizing the information loss due to the compression as much as possible. This compression facilitates computing the reliabilities of the bit channels in terms of their capacities, keeping track of the loss of the mutual information caused by the quantization [25]. Secondly, the intermediate steps of the discrete density evolution provide deterministic mappings between the quantized input and output messages for each node in the Tanner graph of the polar code. These deterministic mappings can replace the LLR computations of a conventional decoder, constituting an information bottleneck SC or SCL decoder [26]. Hence, a polar code and its tailored quantized decoder are obtained from the same process. Similar to the information bottleneck LDPC decoders [1, 7], all the operations in the information bottleneck SC decoder are table lookups of unsigned integers. However, the path metric computation in an information bottleneck SCL decoder is done similar to that in a conventional decoder as in [27].