Variable-length coding with feedback in the non-asymptotic regime

Without feedback, the backoff from capacity due to non-asymptotic block length can be quite substantial for block lengths and error probabilities of interest in many practical applications. In this paper, novel achievability bounds are used to demonstrate that in the non-asymptotic regime, the maximal achievable rate improves dramatically thanks to variable-length coding with feedback. For example, for the binary symmetric channel with capacity 1/2 the blocklength required to achieve 90% of the capacity is smaller than 200, compared to at least 3100 for the best fixed-blocklength, non-feedback code. Virtually all the advantages of noiseless feedback are shown to be achievable with decision-feedback only. It is demonstrated that the non-asymptotic behavior of the fundamental limit depends crucially on the particular model chosen for the “end-of-packet” control signal.