Abstract
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.
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Acknowledgment
The authors are thankful to Zilin Jiang for the fruitful discussion on Zigangirov's proof.
Funding
The research of Georg Maringer was supported by the German Research Foundation (DFG) under Grant No. WA3907/4-1. The research of Nikita Polyanskii was supported in part by the German Research Foundation (DFG) under Grant no. WA3907/1-1. The research of Ilya Vorobyev was supported in part by the joint grant of the Russian Foundation for Basic Research (RFBR) and Japan Society for the Promotion of Science under Grant no. 20-51-50007, and by the RFBR under Grant no. 20-01-00559. The research of Lorenz Welter was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, grant agreement no. 801434.
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Translated from Problemy Peredachi Informatsii, 2021, Vol. 57, No. 3, pp. 17–47 https://doi.org/10.31857/S0555292321030025.
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Maringer, G., Polyanskii, N., Vorobyev, I. et al. Feedback Insertion-Deletion Codes. Probl Inf Transm 57, 212–240 (2021). https://doi.org/10.1134/S0032946021030029
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DOI: https://doi.org/10.1134/S0032946021030029