# 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
The codewords are generated by \(c_{1}^{N}=v_{1}^{N}\mathbf{G}_{N}\). A signal mapper \(\pi\) maps the polar coded symbols \(c_{i},1\leq i\leq N\) to modulated symbols \(x_{i}=\pi(c_{i})\in\mathcal{X}\). We define a symbol set \(\mathcal{S},\mathcal{A}\cup\mathcal{F}\cup\mathcal{S}=\{1,2,\ldots,N\}\). By reformulating (8) as \(v_{1}^{N}\mathbf{G}_{N}+\pi^{-1}(x_{1}^{N})=\mathbf{0}\), we can regard this as a polar codeword \(c_{1}^{N}\) transmitted over a \(2^{q}\)-ary symmetric channel with transition probability \(P_{X}(x)\), where \(\pi^{-1}(x_{1}^{N})\) is the noise sequence and the channel output is the all-zero vector. Therefore, we can use

Figure 5: System model of shaping non-binary polar codes with hybrid MLC and BICM scheme. symbol-level or bit-level construction to obtain the shaping set \(\mathcal{S}\) and use the SC decoder to obtain the shaping symbol value. Note that \(|\mathcal{S}|=\lceil N(1-H_{2^{q}}(P_{X}(x)))\rceil\) symbols, where \(H(P_{X}(x))=\sum_{x\in\mathcal{X}}-P_{X}(x)\log_{2^{q}}(P_{X}(x))\). At the decoder, the initial probabilities are generated based on the channel output \(y_{1}^{N}\) as

\[W(x_{i}\mid y_{i})\propto W_{\text{AWGN}}(y_{i}\mid x_{i})P_{X}(x_{i}), \tag{33}\]

where \(W_{\text{AWGN}}(y_{i}|x_{i})\) is the transition probability over the AWGN channel. For a hybrid MLC and BICM coded modulation framework, since the LLBs are without channel coding, probabilistic shaping can only be achieved at the HLBs. Based on the results of (32), we can calculate the probability of each subset. However, the probability distribution of each subset is close to a uniform distribution. ### Code Design

In this part, we introduce the code design of the MR-based non-binary polar codes with high-order modulation and probabilistic shaping in detail, as illustrated in Figure 7.

For the symbol-level shaping method, we first calculate the number of shaping symbols \(|\mathcal{S}|\) according to the optimal distribution \(P_{X}(x)\). Then, we use the symbol-level construction over the \(2^{q}\)-ary symmetric channel to obtain the shaping symbol set \(\mathcal{S}\). Note that both methods mentioned above can be used. Then, for the AWGN channel, we calculate the capacities or error probabilities for each synthesized channel and assign the best synthesized channels (excluding the ones in \(\mathcal{S}\)) to \(\mathcal{A}\). The remaining synthesized channels are assigned to \(\mathcal{F}\). Once the symbol sets \(\mathcal{S},\mathcal{A},\mathcal{F}\) are known, we can use the SC decoder to calculate the values of the shaping symbols. For this purpose, we regard \(v_{j},j\in\mathcal{A}\cup\mathcal{F}\) as frozen symbols and \(P_{X}(x)\) as the channel observation of an all-zero vector. For the bit-level shaping method, we define a shaping bit set \(\mathcal{S}_{\text{bit}}\) and a frozen bit set \(\mathcal{F}_{\text{bit}}\), \(\mathcal{S}_{\text{bit}}\cup\mathcal{B}\cup\mathcal{F}_{\text{bit}}=\{1,2, \ldots,Nq\}\). The number of shaping bits can be calculated by \(|\mathcal{S}_{\text{bit}}|=\lceil N(q-H_{2}(P_{X}(x)))\rceil\). The shaping bit set \(\mathcal{S}_{\text{bit}}\), the information bit set \(\mathcal{B}\), and the frozen bit set \(\mathcal{F}_{\text{bit}}\) can be obtained through the bit-level construction. Due to the bit-level construction, a synthetic channel may contain both information bits and shaping bits simultaneously, which is the main difference compared with the symbol-level shaping method. ## 5 Simulation Results

In this section, the performance of the proposed construction scheme based on the CD method and MC method for MR-based non-binary polar codes is evaluated. Assume that

\[P_{e}(N,K,\mathcal{A}_{1})=P(\mathcal{E})\leq\sum_{j\in\mathcal{A}_{1}}P( \mathcal{E}_{j})\approx\sum_{j\in\mathcal{A}_{1}}P_{e,I(j)}(j), \tag{28}\]0 System model of shaping non-binary polar codes with BICM scheme.