We study variable-length feedback (VLF) codes with noiseless feedback for discrete memoryless channels. We present a novel non-asymptotic bound, which analyzes the average error probability and average decoding time of our modified Yamamoto--Itoh scheme. We then optimize the parameters of our code in the asymptotic regime where the average error probability $\epsilon$ remains a constant as the average decoding time $N$ approaches infinity. Our second-order achievability bound is an improvement of Polyanskiy et al.'s (2011) achievability bound. We also universalize our code by employing the empirical mutual information in our decoding metric and derive a second-order achievability bound for universal VLF codes. Our results for both VLF and universal VLF codes are extended to the additive white Gaussian noise channel with an average power constraint. The former yields an improvement over Truong and Tan's (2017) achievability bound. The proof of our results for universal VLF codes uses a refined version of the method of types and an asymptotic expansion from the nonlinear renewal theory literature.

翻译：我们针对离散无记忆信道研究具有无噪声反馈的可变长度反馈（VLF）码。提出了一种新颖的非渐近界，该界分析了改进的Yamamoto-Itoh方案的平均错误概率和平均解码时间。随后，在平均错误概率$\epsilon$随平均解码时间$N$趋于无穷而保持常数的渐近区域中，我们优化了编码参数。我们的二阶可达性界是Polyanskiy等人（2011）可达性界的改进。此外，通过在解码度量中利用经验互信息，我们实现了编码的普适化，并推导了通用VLF码的二阶可达性界。针对VLF码和通用VLF码的结论均被推广至带有平均功率约束的加性高斯白噪声信道。前者相比Truong和Tan（2017）的可达性界有所改进。通用VLF码结论的证明采用了精细化的类型方法以及非线性更新理论文献中的渐近展开。