This paper focuses on an improved Gaussian approximation (GA) based construction of polar codes with successive cancellation (SC) decoding over an additive white Gaussian noise (AWGN) channel. Arıkan proved that polar codes with low-complexity SC decoding can approach the channel capacity of an arbitrary symmetric binary-input discrete memoryless channel, provided that the code length is chosen large enough. Nevertheless, how to construct such codes over an AWGN channel with low computational effort has been an open problem. Compared to density evolution, the GA is known as a low complexity yet powerful technique that traces the evolution of the mean log likelihood ratio (LLR) value by iterating a nonlinear function. Therefore, its high-precision numerical evaluation is critical as the code length increases. In this work, by analyzing the asymptotic behavior of this nonlinear function, we propose an improved GA approach that makes an accurate trace of mean LLR evolution feasible. With this improved GA, through numerical analysis and simulations with code lengths up to $N=2^{18}$ , we explicitly demonstrate that various code-rate polar codes with long codeword and capacity approaching behavior can be easily designed.

中文翻译：

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基于改进高斯近似的长码字逐次对消译码容量逼近极性码

本文重点介绍在加性高斯白噪声 (AWGN) 信道上具有连续消除 (SC) 解码的极性码的改进的基于高斯近似 (GA) 的构造。Arıkan 证明了具有低复杂度 SC 解码的极性码可以接近任意对称二进制输入离散无记忆信道的信道容量，前提是选择的码长足够大。然而，如何以低计算量在 AWGN 信道上构建这样的代码一直是一个悬而未决的问题。与密度演化相比，遗传算法被认为是一种低复杂性但功能强大的技术，它通过迭代非线性函数来跟踪平均对数似然比 (LLR) 值的演化。因此，随着代码长度的增加，其高精度数值评估至关重要。在这项工作中，通过分析这种非线性函数的渐近行为，我们提出了一种改进的 GA 方法，使平均 LLR 演化的准确轨迹变得可行。使用这种改进的 GA，通过代码长度高达 $N=2^{18}$ 的数值分析和模拟，我们明确证明可以轻松设计具有长码字和容量逼近行为的各种码率极性码。