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 $n_1, n_2, \dots, n_K$ rather than allowing decoding at any time $n = 0, 1, 2, \dots$. 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 $\ln M \approx \frac{N C(P)}{1-\epsilon} - \sqrt{N \ln_{(K-1)}(N) \frac{V(P)}{1-\epsilon}}$, 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.