Feedback Insertion-Deletion Codes

In this paper, a new problem of transmitting information over the adversarial insertion-deletion channel with feedback is introduced. Suppose that we can transmit $n$ binary symbols one-by-one over the 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 which is able to transmit 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. For the substitution channel with random errors, an optimal algorithm with a nice intuition behind it was presented in 1963 by Horstein. However, the analysis of this algorithm for the adversarial substitution channel is quite complicated and was completed only 13 years later by Zigangirov. We revisit Zigangirov's result and present a more elaborate version of his proof.