# 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
Two codes of length \(N=512\) over GF(16) with code rates \(1/4\) and \(1/2\), respectively, were evaluated. Assume that the CA-SCL decoding algorithm was used. The performance of both non-binary polar codes and binary polar codes with bit interleaved coded modulation (BICM) is shown in \[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]3 When \(R=1/4\), the proposed scheme achieved performance gains of 0.25 and 1 dB compared with the symbol-level construction and binary polar codes. When \(R=1/2\), our scheme had performance gains of 0.2 dB and 0.5 dB, respectively. \[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]4 Performance comparison among the proposed scheme, conventional scheme, and binary polar codes, where \(N=128,64\), \(R=1/4\) over GF(16), and the SC and CA-SCL decoding algorithms were used. \[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]5 Performance comparison among the proposed scheme, conventional scheme, and binary polar codes, and scheme in [12] where \(N=128\), \(R=1/2\) over GF(16) and the CA-SCL decoding algorithm was used. \[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]6 Performance comparison among the proposed scheme, conventional scheme, and binary polar codes, where \(N=64\), \(R=1/2\) over GF(16), and the SC and CA-SCL decoding algorithms were used. The performance of the probabilistic shaped MR non-binary polar codes is shown in \[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]7 We compared the shaping gain and coding gain of the bit-level construction and the symbol-level construction schemes. When \(N=512\), \(R=1/2\), both bit-level construction and symbol construction had 0.1 dB shaping gain at \(\text{BLER}=1\times 10^{-3}\), while the coding gain was 0.2 dB. When \(N=512\), \(R=3/4\), both the bit-level construction and symbol-level construction had a 0.25 dB shaping gain at \(\text{BLER}=1\times 10^{-3}\), while the coding gain was 0.1 dB. In a word, the bit-level construction had a significant shaping gain with a high coding rate. \[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]8 Performance with 16-QAM over the AWGN channel. \[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]9 Probabilistic shaping performance with 16-QAM over the AWGN channel. A coded modulation performance comparison between BICM and hybrid MLC-BICM with 64-QAM modulation is shown in \[P_{b}(s)=\frac{P_{e,i+1}(j)-P_{e,i}(j)}{1-P_{e,i}(j)} \tag{29}\]0 For the BICM system, the code length \(N=256,R=5/6\), the crc length was 12, and the list size was 8. For the hybrid MLC and BICM system, the signal constellation was partitioned into subsets of size 4, which meant that 512 out of 1280 information bits were uncoded and the rest of the information bits were protected using a non-binary (binary) polar code with a rate of 3/4. The simulation results show that the hybrid MLC-BICM scheme was about 0.4 dB better than the BICM scheme at a bit error rate (BER) \(=1\times 10^{-4}\). Meanwhile, for the BICM system, the bit-level construction achieved a 0.25 dB gain at BER \(=1\times 10^{-4}\). ## 6 Conclusions

This paper presents an improved bit-level construction for MR-based non-binary polar codes, incorporating both CD and MC methods. The number of transmitted bits is determined by evaluating the error probability of each synthesized channel and the lower bound of its capacity, thereby minimizing the symbol error probability.