A new problem of transmitting information over the adversarial insertion-deletion channel with feedback is introduced. Assume 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 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 completed 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.

引入了在具有反馈的对抗性插入-删除通道上传输信息的新问题。假设编码器通过一个信道一个一个地传输\(n\)个二进制符号，其中一些符号可以被删除并且一些附加符号可以被插入。每次传输后，编码器都会收到有关前一次传输中发生的插入或删除的通知，并且可以相应地调整编码策略。目标是在假设删除和插入的总数受\(\tau n\) , \(0<\tau<1 \). 我们展示了如何将这个问题简化为通过替代通道传输消息的问题。从而，完全建立了反馈插入-删除码的最大渐近率。对抗性替代通道的最大渐近率部分由 Berlekamp 确定，后来由 Zigangirov 完成。但是，Zigangirov 对下界的分析相当复杂。我们重新审视 Zigangirov 的结果，并展示了他的证明的更详细版本。