The min-sum approximation is widely used in the decoding of polar codes. Although it is a numerical approximation, hardly any penalties are incurred in practice. We give a theoretical justification for this. We consider the common case of a binary-input, memoryless, and symmetric channel, decoded using successive cancellation and the min-sum approximation. Under mild assumptions, we show the following. For the finite length case, we show how to exactly calculate the error probabilities of all synthetic (bit) channels in time $O(N^{1.585})$, where $N$ is the codeword length. This implies a code construction algorithm with the above complexity. For the asymptotic case, we develop two rate thresholds, denoted $R_{\mathrm{L}} = R_{\mathrm{L}}(\lambda)$ and $R_{\mathrm{U}} =R_{\mathrm{U}}(\lambda)$, where $\lambda(\cdot)$ is the labeler of the channel outputs (essentially, a quantizer). For any $0 < \beta < \frac{1}{2}$ and any code rate $R < R_{\mathrm{L}}$, there exists a family of polar codes with growing lengths such that their rates are at least $R$ and their error probabilities are at most $2^{-N^\beta}$. That is, strong polarization continues to hold under the min-sum approximation. Conversely, for code rates exceeding $R_{\mathrm{U}}$, the error probability approaches $1$ as the code-length increases, irrespective of which bits are frozen. We show that $0 < R_{\mathrm{L}} \leq R_{\mathrm{U}} \leq C$, where $C$ is the channel capacity. The last inequality is often strict, in which case the ramification of using the min-sum approximation is that we can no longer achieve capacity.

翻译：Min-Sum近似在极化码译码中被广泛使用。尽管这是一种数值近似，但在实践中几乎不会带来性能损失。本文为此提供了理论依据。我们考虑二进制输入、无记忆且对称的信道，采用连续消除译码与Min-Sum近似的常见场景。在温和假设下，我们证明了以下结论：对于有限码长情形，我们展示了如何在$O(N^{1.585})$时间内精确计算所有合成（比特）信道的错误概率，其中$N$为码字长度。这导出了具有上述复杂度的码构造算法。对于渐近情形，我们建立了两个速率阈值，记为$R_{\mathrm{L}} = R_{\mathrm{L}}(\lambda)$与$R_{\mathrm{U}} =R_{\mathrm{U}}(\lambda)$，其中$\lambda(\cdot)$为信道输出的标记器（本质上是一种量化器）。对于任意$0 < \beta < \frac{1}{2}$与任意码率$R < R_{\mathrm{L}}$，存在码长递增的极化码族，其码率至少为$R$且错误概率至多为$2^{-N^\beta}$。这表明在Min-Sum近似下强极化现象仍然成立。反之，对于超过$R_{\mathrm{U}}$的码率，无论冻结位如何选取，错误概率均随码长增加趋近于$1$。我们证明了$0 < R_{\mathrm{L}} \leq R_{\mathrm{U}} \leq C$，其中$C$为信道容量。最后一个不等式通常为严格不等式，此时使用Min-Sum近似的结果是我们无法再达到信道容量。