Variable-length Feedback Codes with Several Decoding Times for the Gaussian Channel

We investigate variable-length feedback (VLF) codes for the Gaussian point-to-point channel under maximal power, average error probability, and average decoding time constraints. Our proposed strategy chooses
$K < \infty$
decoding times
$n_{1}, n_{2}, \ldots, n_{K}$
rather than allowing decoding at any time
$n=0,1,2, \ldots$
. We consider stop-feedback, which is one-bit feedback transmitted from the receiver to the transmitter at times
$n_{1}, n_{2},\ldots$
only to inform her whether to stop. We prove an achievability bound for VLF codes with the asymptotic approximation
$\ln M \approx \frac{NC(P)}{1-\epsilon}-\sqrt{N\ \ln_{(K-1)}(N)\frac{V(P)}{1-\epsilon}}$
, where
$\ln(K)(\cdot)$
denotes the
$K$
-fold nested logarithm function,
$N$
is the average decoding time, and
$C(P)$
and
$V(P)$
are the capacity and dispersion of the Gaussian channel, respectively. Our achievability bound evaluates a non-asymptotic bound and optimizes the decoding times
$n_{1}, \ldots, n_{K}$
within our code architecture.