# 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
Note that for some \(s_{1}>s_{2},s_{1}=(j-1)q+i_{1}+1,s_{2}=(j-1)q+i_{2}+1,0\leq i_{2}<i_{1}<q\), there are \(P_{b}(s_{1})\leq P_{T}<P_{b}(s_{2})\), where \(P_{T}\) is the \(Kq\)-th smallest probability. This means that \(s_{1}\in\mathcal{B},s_{2}\
otin\mathcal{B}\), which means the construction result does not satisfy the fixed carry pattern. Therefore, we must ensure that, for this case, \(s_{2}\) is selected earlier than \(s_{1}\) when we choose the \(Kq\) smallest \(P_{b}(s)\), which is different from the construction scheme in [12; 14]. The overall procedure of the error-probability-based bit-level construction algorithm is shown in Algorithm 5. Figure 4 shows the error probability of each decoding bit under the SC decoder, where \(N=64\) symbols, \(K=32\) symbols over GF(16), and the \(E_{b}/N_{0}=2.0\) dB. Construction for MR-Based Non-Binary Polar Codes with High-Order Modulation and Probabilistic Shaping

To achieve high spectral efficiency, MR-based non-binary polar codes combined with high-order modulation are considered. It is well known that the optimal input distribution for AWGN channels is Gaussian. As the spectral efficiency increases, the performance loss caused by the use of uniform signaling can be up to 1.53 dB [23; 24]. Probabilistic

Figure 4: The error probability of each decoding bit under the SC decoder. shaping is a technique developed to mitigate this loss. In this section, we introduce the code construction for MR-based non-binary polar codes with BICM and probabilistic shaping. ### Hybrid MLC and BICM Coded Modulation Framework

[PERSON] et al. proposed a hybrid MLC and BICM coded modulation framework based on Ungerboeck set partitioning, to meet requirements for higher spectral efficiency and higher throughput in future 6G communications in [28]. Figure 5 shows the hybrid MLC-BICM coded modulation system model combined with non-binary polar codes. MLC incorporates Ungerboeck set partitioning, which divides the signal constellation \(\mathcal{X}\) into \(2^{q-1}\) subsets through \(q-1\) partitioning steps. Let \(d_{j},(j=0,1,\ldots,q-1)\) denote the minimum square Euclidean distances of the \(j\)-th level subsets, where the average energy of the original constellation is normalized to \(1\). The \(j\)-th level bit error rate (BER) of the intra-subset can be calculated by

\[P(d_{j})\approx\frac{1}{2}\operatorname{erfc}\left(\sqrt{\frac{E_{s}d_{j}}{4N_ {0}}}\right)=\frac{1}{2}\operatorname{erfc}\left(\sqrt{\frac{\rho E_{b}d_{j}}{4 N_{0}}}\right), \tag{31}\]

where \(E_{s}\) is the average energy of the constellation \(\mathcal{X}\), \(E_{b}\) is the average energy of each information bit [30], and \(\rho\) is the corresponding spectral efficiency (SE) in bits/two-dimensional (2D) symbol. Therefore, at a certain level of the subset partitioning chain, the signals within the subset can be detected by a hard decision with high reliability, because the BER is reliable enough, such as a BER \(\leq 10^{-5}\). We divide the index bits of the transmitted signal into two groups: the high-level bits (HLBs) used to index the subset, and the low-level bits (LLBs) used to index the signal in the subset. To achieve better performance, we apply Gray labeling for both LLBs and HLBs. ### Probabilistic Shaping for MR-Based Non-Binary Polar Codes

Figure 6 depicts a block diagram of shaping non-binary polar codes with the BICM scheme. Suppose that a two-dimensional signal constellation \(\mathcal{X}\) of size \(|\mathcal{X}|=2^{q}\) is used. In other words, the field matches the modulation order. In order to ensure that the transmitted symbols follow a carefully optimized distribution, such as the Maxwell-Boltzmann distribution

\[P_{X}(x)=\frac{\exp\left(-\lambda\|x\|^{2}\right)}{\sum_{\mathcal{X}\in \mathcal{X}}\exp(-\lambda\|x\|^{2})}, \tag{32}\]

where \(\lambda\geq 0\), the information sequence \(\mathbf{d}=d_{1},d_{2},\ldots,d_{K}\) needs to contain some shaping symbols. After inserting the shaping symbols and frozen symbols, the input sequence \(\mathbf{v}=v_{1},v_{2},\ldots,v_{N}\) can be obtained.