  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.
