# Bit-Level Construction for Multiplicative-Repetition-Based Non-Binary Polar Codes +
## 1 Introduction  
Binary polar codes, proposed by [PERSON], can achieve symmetric capacity of binary-input discrete memoryless channels (BI-DMCs) [1]. However, the performance of polar codes at finite block lengths is unsatisfactory with the successive cancellation (SC) decoding algorithm. In order to improve the performance of polar codes in the finite length regime, the cyclic redundancy check (CRC)-aided successive cancellation list (CA-SCL) decoding algorithm was proposed in [2, 3]. In practice, the CRC-polar concatenation scheme has shown excellent performance at short to medium code lengths with CA-SCL decoding compared with low-density-parity-check (LDPC) codes and Turbo codes. Due to their superior performance, CRC-polar codes have been selected as the coding standard for the control channel in the fifth generation (5G) wireless communication system [4]. It was proved in [5; 6] that non-binary polar codes with arbitrary finite input-alphabet sizes can achieve symmetric capacity. It was also shown that all discrete memoryless channels (DMCs) can be polarized by randomized constructions [5]. Subsequently, [PERSON] and [PERSON] [7] proposed non-binary polar codes based on an \(l\times l\) matrix of Reed-Solomon (RS) codes over GF(\(p\)), where \(p\) is the power of a prime. They also proved that the exponent of their non-binary polar codes is \(\log(l! )/(l\log(l))\) for all \(l\leq p\), and this exponent can become arbitrarily close to 1 as \(l\) increases. In 2016, [PERSON] et al. [8] introduced four-dimensional polar codes based on the RS kernel of GF(4) (RS4). Simulation results demonstrated that non-binary polar codes based on the RS kernel outperformed their binary counterparts. Moreover, binary and RS4-based non-binary kernels were mixed in [9] to reduce the decoding complexity of RS4-based polar codes. However, decoder implementations are not easily generalizable across different field sizes. To overcome this limitation, a novel multiplicative repetition (MR)-based non-binary code was proposed in [10; 11; 12; 13; 14]. The conventional decoding algorithm for binary polar codes can be easily extended to this class of non-binary polar codes. In [12], [PERSON] and [PERSON] introduced the concept of two-stage polarization to construct non-binary polar codes, which exhibited comparable performance with RS4-based non-binary polar codes. In [13], [PERSON] and [PERSON] proposed a kernel selection method to maximize the polarization effect, where the kernel had a fixed multiplicative factor. For binary polar codes over a binary-input additive white Gaussian noise (BI-AWGN) channel, several construction algorithms exist, including the Monte-Carlo (MC) algorithm [1], Gaussian approximation (GA) algorithm [15], polarized weight (PW) algorithm [16], and others [17; 18]. For MR-based non-binary polar codes, the Monte-Carlo construction and channel degradation (CD) construction [19] are mainly used. In 2022, [PERSON] [20] proposed a new construction algorithm for non-binary I-polar codes based on the polarization effect and the block error rate (BLER) upper bound. To achieve high spectral efficiency, non-binary polar codes combined with multilevel coding (MLC) and bit-interleaved coded modulation (BICM) were proposed in [11; 20; 21; 22]. However, uniformly distributed symbols were considered in these schemes, resulting in a shaping loss up to 1.53 dB in AWGN channels [23; 24]. A classical scheme to mitigate this loss is combining probabilistic shaping technique with high-order modulation [25; 26]. In 2023, a method for constructing RS-kernel-based non-binary polar codes with shaping was proposed in [27]. This work aimed to improve the performance of MR-based non-binary polar codes at finite block lengths. For polar codes over GF(\(2^{q}\)), the capacity of many synthesized channels ranges from 0 to \(q\) bits per channel used at finite block lengths. A single symbol in a synthesized channel may not carry \(q\) bits, but can carry \(i\) bits, where \(1\leq i<q\).