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

Recep Can Yavas, Victoria Kostina, Michelle Effros

Abstract: 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K &lt; \infty$</tex> decoding times <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n_{1}, n_{2}, \ldots, n_{K}$</tex> rather than allowing decoding at any time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n=0,1,2, \ldots$</tex> . We consider stop-feedback, which is one-bit feedback transmitted from the receiver to the transmitter at times <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n_{1}, n_{2},\ldots$</tex> only to inform her whether to stop. We prove an achievability bound for VLF codes with the asymptotic approximation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ln M \approx \frac{NC(P)}{1-\epsilon}-\sqrt{N\ \ln_{(K-1)}(N)\frac{V(P)}{1-\epsilon}}$</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ln(K)(\cdot)$</tex> denotes the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> -fold nested logarithm function, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> is the average decoding time, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C(P)$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$V(P)$</tex> are the capacity and dispersion of the Gaussian channel, respectively. Our achievability bound evaluates a non-asymptotic bound and optimizes the decoding times <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n_{1}, \ldots, n_{K}$</tex> within our code architecture.