  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.
