Why block length and delay behave differently if feedback is present

TL;DR

This paper demonstrates that feedback significantly improves error exponents in fixed-delay communication systems, introduces the uncertainty-focusing bound, and shows it is achievable in various scenarios, challenging previous conjectures.

Contribution

The paper introduces the uncertainty-focusing bound for fixed-delay systems with feedback and proves its achievability across multiple channel types, providing new insights into error exponent behavior.

Findings

Feedback enhances error exponents at fixed delay.

The uncertainty-focusing bound is asymptotically achievable with feedback.

High-rate systems can surpass the sphere-packing bound, countering prior conjectures.

Abstract

For output-symmetric DMCs at even moderately high rates, fixed-block-length communication systems show no improvements in their error exponents with feedback. In this paper, we study systems with fixed end-to-end delay and show that feedback generally provides dramatic gains in the error exponents. A new upper bound (the uncertainty-focusing bound) is given on the probability of symbol error in a fixed-delay communication system with feedback. This bound turns out to have a similar form to Viterbi's bound used for the block error probability of convolutional codes as a function of the fixed constraint length. The uncertainty-focusing bound is shown to be asymptotically achievable with noiseless feedback for erasure channels as well as any output-symmetric DMC that has strictly positive zero-error capacity. Furthermore, it can be achieved in a delay-universal (anytime) fashion even if the feedback itself is delayed by a small amount. Finally, it is shown that for end-to-end delay, it is generally possible at high rates to beat the sphere-packing bound for general DMCs -- thereby providing a counterexample to a conjecture of Pinsker.