Variable-Length Feedback Codes Over Known and Unknown Channels With Non-Vanishing Error Probabilities

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> remains a constant as the average decoding time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> approaches infinity. Our second-order achievability bound is an improvement of Polyanskiy et al.’s (2011) achievability bound. We also develop a universal VLF code that does not rely on the knowledge of the underlying channel parameters. Our universal VLF code employs the empirical mutual information as its decoding metric and universalizes the code by Polyanskiy et al. (2011). We 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.