  In this paper, a new problem of transmitting information over the adversarial
insertion-deletion channel with feedback is introduced. Suppose that the
encoder transmits $n$ binary symbols one-by-one over a channel, in which some
symbols can be deleted and some additional symbols can be inserted. After each
transmission, the encoder is notified about the insertions or deletions that
have occurred within the previous transmission and the encoding strategy can be
adapted accordingly. The goal is to design an encoder that is able to transmit
error-free as much information as possible under the assumption that the total
number of deletions and insertions is limited by $\tau n$, $0<\tau<1$. We show
how this problem can be reduced to the problem of transmitting messages over
the substitution channel. Thereby, the maximal asymptotic rate of feedback
insertion-deletion codes is completely established. The maximal asymptotic rate
for the adversarial substitution channel has been partially determined by
Berlekamp and later finished by Zigangirov. However, the analysis of the lower
bound by Zigangirov is quite complicated. We revisit Zigangirov's result and
present a more elaborate version of his proof.
