中小偏差区间下信道编码的三阶分析
Third-order Analysis of Channel Coding in the Small-to-Moderate Deviations Regime
AI总结：
本文研究了高斯信道与非奇异离散无记忆信道在中小偏差区间下的三阶编码特性，引入信道偏度这一新量，改进了现有渐近展开界，并在多种实际参数下实现了最精确的近似。
摘要：
本文研究了非奇异离散无记忆信道和带最大功率约束的高斯信道的三阶特性。我们展开式中的三阶项采用了一个新量，这里称为信道偏度，随着错误概率的降低，它对近似精度的影响更为显著。对于高斯信道，在中心极限定理（CLT）区间内评估Shannon 1959年随机编码界和Vazquez-Vilar 2021年元反演界，能够实现对信道偏度的精确计算。对于离散无记忆信道，本工作将Moulin 2017年关于非奇异信道最大可达消息集大小渐近展开的界从CLT区间推广至包含中等偏差（MD）区间，从而改进了Altuğ和Wagner 2014年的MD结果。以一个二元对称信道为例，对于大多数实际重要的 $(n, ε)$ 对，包括 $n \in [100, 500]$ 和 $ε\in [10^{-10}, 10^{-1}]$，包含到信道偏度的近似在文献中的多种展开式中是最准确的。本文还推导了MD区间内二元假设检验第二类错误指数的三阶项；所得的三阶项与信道偏度相似。
英文摘要：
This paper studies the third-order characteristic of nonsingular discrete memoryless channels and the Gaussian channel with a maximal-power constraint. The third-order term in our expansions employs a new quantity here called the channel skewness, which affects the approximation accuracy more significantly as the error probability decreases. For the Gaussian channel, evaluating Shannon's 1959 random coding bound and Vazquez-Vilar's 2021 meta-converse bound in the central limit theorem (CLT) regime enables exact computation of the channel skewness. For discrete memoryless channels, this work generalizes Moulin's 2017 bounds on the asymptotic expansion of the maximum achievable message set size for nonsingular channels from the CLT regime to include the moderate deviations (MD) regime, thereby refining Altuğ and Wagner's 2014 MD result. For an example binary symmetric channel and most practically important $(n, ε)$ pairs, including $n \in [100, 500]$ and $ε\in [10^{-10}, 10^{-1}]$, an approximation up to the channel skewness is the most accurate among several expansions in the literature. A derivation of the third-order term in the type-II error exponent of binary hypothesis testing in the MD regime is also included; the resulting third-order term is similar to the channel skewness.
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