2 Search Results for "Lebedev, Vladimir"

Document
DOI: 10.4230/LIPIcs.ITCS.2025.22
Round-Vs-Resilience Tradeoffs for Binary Feedback Channels

Authors: Mark Braverman, Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
In a celebrated result from the 60’s, Berlekamp showed that feedback can be used to increase the maximum fraction of adversarial noise that can be tolerated by binary error correcting codes from 1/4 to 1/3. However, his result relies on the assumption that feedback is "continuous", i.e., after every utilization of the channel, the sender gets the symbol received by the receiver. While this assumption is natural in some settings, in other settings it may be unreasonable or too costly to maintain. In this work, we initiate the study of round-restricted feedback channels, where the number r of feedback rounds is possibly much smaller than the number of utilizations of the channel. Error correcting codes for such channels are protocols where the sender can ask for feedback at most r times, and, upon a feedback request, it obtains all the symbols received since its last feedback request. We design such error correcting protocols for both the adversarial binary erasure channel and for the adversarial binary corruption (bit flip) channel. For the erasure channel, we give an exact characterization of the round-vs-resilience tradeoff by designing a (constant rate) protocol with r feedback rounds, for every r, and proving that its noise resilience is optimal. Designing such error correcting protocols for the corruption channel is substantially more involved. We show that obtaining the optimal resilience, even with one feedback round (r = 1), requires settling (proving or disproving) a new, seemingly unrelated, "clean" combinatorial conjecture, about the maximum cut in weighted graphs versus the "imbalance" of an average cut. Specifically, we prove an upper bound on the optimal resilience (impossibility result), and show that the existence of a matching lower bound (a protocol) is equivalent to the correctness of our conjecture.
Cite as

Mark Braverman, Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang. Round-Vs-Resilience Tradeoffs for Binary Feedback Channels. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2025.22,
  author =	{Braverman, Mark and Efremenko, Klim and Kol, Gillat and Saxena, Raghuvansh R. and Zhang, Zhijun},
  title =	{{Round-Vs-Resilience Tradeoffs for Binary Feedback Channels}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{22:1--22:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.22},
  URN =		{urn:nbn:de:0030-drops-226506},
  doi =		{10.4230/LIPIcs.ITCS.2025.22},
  annote =	{Keywords: Round-restricted feedback channel, error correcting code, noise resilience}
}
Document
DOI: 10.4230/DagSemProc.06201.4
Non--binary error correcting codes with noiseless feedback, localized errors, or both

Authors: Rudolf Ahlswede, Christian Deppe, and Vladimir Lebedev

Published in: Dagstuhl Seminar Proceedings, Volume 6201, Combinatorial and Algorithmic Foundations of Pattern and Association Discovery (2006)

Abstract
We investigate non--binary error correcting codes with noiseless feedback, localized errors, or both. It turns out that the Hamming bound is a central concept. For block codes with feedback we present here a coding scheme based on an idea of erasions, which we call the {\bf rubber method}. It gives an optimal rate for big error correcting fraction $\tau$ ($>{1\over q}$) and infinitely many points on the Hamming bound for small $\tau$. We also consider variable length codes with all lengths bounded from above by $n$ and the end of a word carries the symbol $\Box$ and is thus recognizable by the decoder. For both, the $\Box$-model with feedback and the $\Box$-model with localized errors, the Hamming bound is the exact capacity curve for $\tau <1/2.$ Somewhat surprisingly, whereas with feedback the capacity curve coincides with the Hamming bound also for $1/2\leq \tau \leq 1$, in this range for localized errors the capacity curve equals 0. Also we give constructions for the models with both, feedback and localized errors.
Cite as

Rudolf Ahlswede, Christian Deppe, and Vladimir Lebedev. Non--binary error correcting codes with noiseless feedback, localized errors, or both. In Combinatorial and Algorithmic Foundations of Pattern and Association Discovery. Dagstuhl Seminar Proceedings, Volume 6201, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{ahlswede_et_al:DagSemProc.06201.4,
  author =	{Ahlswede, Rudolf and Deppe, Christian and Lebedev, Vladimir},
  title =	{{Non--binary error correcting codes with noiseless feedback, localized errors, or both}},
  booktitle =	{Combinatorial and Algorithmic Foundations of Pattern and Association Discovery},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6201},
  editor =	{Rudolf Ahlswede and Alberto Apostolico and Vladimir I. Levenshtein},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06201.4},
  URN =		{urn:nbn:de:0030-drops-7849},
  doi =		{10.4230/DagSemProc.06201.4},
  annote =	{Keywords: Error-correcting codes, localized errors, feedback, variable length codes}
}
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