# All-Integer Quantization for Low-Complexity Min-Sum Successive Cancellation Polar Decoder
## 1 Introduction  
Polar codes, exploiting channel polarization, were proven by [PERSON] [1] to achieve channel capacity on symmetric binary-input discrete memoryless channels (B-DMCs) when code length \(N\) becomes large. For finite block lengths, SC decoding was first introduced for polar codes. To simplify the hardware implementation, an approximation method for SC decoding was explored in [2]. Applying the min-sum approximation (MSA) forms the MS-SC decoder, which has slightly lower performance than that of the original SC decoder. To significantly improve the decoding performance, decoding algorithms such as Successive Cancellation List (SCL) decoding and Cyclic Redundancy Check (CRC)-aided SCL decoding were proposed in [3]. Moreover, the computational complexity of SCL decoding grows linearly with the list size. Due to their superior error correction capabilities, polar codes have been selected for use in 5G enhanced mobile broadband (eMBB) scenarios, particularly in the control channel [4]. Additionally, applications such as speech communication [5], decoders based on field-programmable gate arrays (FPGAs) [6], and exploration for future 6G communication [7] highlight the diverse potential of polar codes. In theory, decoding algorithms rely on infinite-precision calculations, but practical implementations, such as those discussed in [8], do not require such high precision. An approximation technique, known as quantization, is important for practical channel coding schemes, including polar, turbo, and low-density parity-check (LDPC) codes, as it significantly reduces computational complexity and hardware resource usage. However, reducedprecision can degrade the error correction performance. Therefore, quantization is a key challenge in improving polar decoding while balancing complexity and performance. Reducing the number of quantization levels is an alternative approach to achieve a balance between computational complexity and decoding performance in SC decoding. A nonuniform quantized SC decoding method was proposed in [9] to reduce complexity without a significant degradation in the error-correction performance, demonstrating only a 0.1-0.2 dB performance loss at a frame error rate (FER) of \(10^{-3}\) when using 2-, 3-, and 4-bit precision levels compared to floating-point SC decoding. The minimum distortion quantizers proposed in [10] achieved an FER close to floating-point decoding with 5-bit nonuniform quantization. However, this method still results in a 0.1 dB loss at an FER of \(10^{-4}\). An alternative approach, as proposed in [11], combines a 1-bit quantizer with an improved polar belief propagation (BP) decoder, which applies deep learning techniques to enhance the decoding process through a neural network decoder. This approach achieves a performance gain of 2 dB compared to the 1-bit quantized BP decoder. To further minimize the computational effort, the Information Bottleneck method was applied to LDPC decoding, where quantizers mapped log-likelihood ratios (LLRs) to unsigned integers, leading to decoders that outperformed traditional min-sum decoders in terms of error performance and computational efficiency [12]. Similarly, in [13], lookup tables were utilized in a discrete SCL decoding algorithm, achieving a performance comparable to a floating-point SCL decoder with a list size of 32, proving the feasibility of integer-based decoding for practical applications. Additionally, ref. [14] applied a 4-bit all-integer quantization to an MSA-based LDPC decoder. Their proposed method achieved performance close to that of its floating-point counterpart, highlighting the potential of integer quantization in modern channel coding techniques. High-precision floating-point calculation is often computationally intensive and less suitable for hardware-constrained environments such as resource-constrained Internet of Things (IoT) devices [15], wireless sensor networks [16], and NAND flash memory [17].