# Bit-Level Construction for Multiplicative-Repetition-Based Non-Binary Polar Codes +
## 3 Bit-Level Construction for Non-Binary Polar Codes
### Capacity-Based Bit-Level Construction
\[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]1 The code design procedure of MR-based non-binary polar codes with high-order modulation and probabilistic shaping. binary phase shift keying (BPSK) modulation is used over the AWGN channel. For comparison, the performance of the original Monte-Carlo construction [8], binary polar codes, and the construction scheme in [12] are also provided. Note that the binary polar codes were constructed based on the 5G construction [4]. The design SNR of all constructions was 2.0 dB, the CRC length and the list size were both set to 8. At the decoder, the symbol-level decoding algorithm was used for the non-binary polar codes. Therefore, the decoding complexity of the proposed construction scheme was the same as that of the symbol-level construction scheme. Figure 8 depicts the performance comparison between the symbol-level construction and the bit-level construction when \(N=32\) and \(R=1/4\) over GF(16). The SC decoding and CA-SCL decoding algorithms were used. When using the SC decoder, the bit-level MC construction and the bit-level CD construction proposed in this work exhibited the same performance, which was 0.2 dB better than the symbol-level MC construction. When using the CA-SCL decoder, the bit-level CD construction also achieved the best BLER performance, outperforming the MC method by 0.2 dB. Since the quantization level \(\mu=32\), the performance loss of the CD scheme increased with the code length and rate. With regard to Figure 9, when \(N=64,R=1/4\) over GF(16), and the SC decoding algorithm was used, the symbol-level construction scheme had a similar BLER performance compared with the binary polar code, and the bit-level construction scheme performed 0.5 dB better than the above two schemes at BLER \(=1\times 10^{-5}\). The proposed scheme and the exhaustive search scheme had the same BLER performance, but the proposed scheme had a lower complexity. When \(N=128\), the proposed bit-level construction scheme and symbol-level construction scheme had the same BLER performance, which was 0.75 dB better than that of the binary polar codes. Figure 9 also shows the performance comparison using the CA-SCL decoding algorithm. Since the CRC bits occupied the information sub-channels, the error probability upper bound of the proposed construction scheme increased. When \(N=64\), the bit-level

\[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]2 Performance comparison among the bit-level construction and symbol-level construction. construction scheme achieved a performance gain of 0.25 dB and 0.15 dB compared with symbol-level construction and binary polar codes. When \(N=128\), the bit-level construction scheme also achieved a gain of 0.15 dB compared with the symbol-level construction scheme, which was 0.25 dB better than the binary polar code in the high SNR region. In Figure 10, when \(N=64,R=1/2\), with the symbol-level construction scheme, the bit-level construction and the exhaustive search scheme had the same BLER performance under the SC decoding algorithm, which was 0.7 dB better than the binary polar code. Using the CA-SCL decoding with the CRC length of 24, the proposed scheme had the same BLER performance as the binary polar code, and outperformed the symbol-level construction by 0.2 dB at BLER = \(10^{-4}\). Figure 11 shows the performance comparison between the proposed construction scheme and the scheme in [12], where \(N=128,R=1/2\), and the list size was 16. The proposed scheme was 0.1 dB better than the scheme in [12] and 0.2 dB better than the symbol-level construction scheme and the binary polar code at BLER = \(2\times 10^{-4}\). Since the input symbols in our scheme satisfy \(v_{j}\in\{0,1,\ldots,2^{i}-1\}\), the decoding search space was reduced compared with that in [12]. We now evaluate the code performance under 16-QAM modulation.