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List Decoding Bounds for Binary Codes with Noiseless Feedback

Authors Meghal Gupta , Rachel Yun Zhang

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  • Mathematics of computing → Coding theory
Keywords
  • error-correcting codes
  • feedback
  • list decoding

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Abstract

In an error-correcting code, a sender encodes a message x ∈ {0, 1}^k such that it is still decodable by a receiver on the other end of a noisy channel. In the setting of error-correcting codes with feedback, after sending each bit, the sender learns what was received at the other end and can tailor future messages accordingly. 
While the unique decoding radius of feedback codes has long been known to be 1/3, the list decoding capabilities of feedback codes is not well understood. In this paper, we provide the first nontrivial bounds on the list decoding radius of feedback codes for lists of size 𝓁. For 𝓁 = 2, we fully determine the 2-list decoding radius to be 3/7. For larger values of 𝓁, we show an upper bound of 1/2 - 1/{2^(𝓁+2) - 2}, and show that the same techniques for the 𝓁 = 2 case cannot match this upper bound in general.

Cite As Get BibTex

Meghal Gupta and Rachel Yun Zhang. List Decoding Bounds for Binary Codes with Noiseless Feedback. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 60:1-60:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.60

Author Details

Meghal Gupta
  • University of California Berkeley, CA, USA
Rachel Yun Zhang
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Gupta, Meghal: Supported by a NSF GFRP Fellowship.
  • Zhang, Rachel Yun: Supported by NSF Graduate Research Fellowship 2141064. Supported in part by DARPA under Agreement No. HR00112020023 and by an NSF grant CNS-2154149.

References

  1. Rudolf Ahlswede, Ferdinando Cicalese, Christian Deppe, and Ugo Vaccaro. Two batch search with lie cost. IEEE transactions on information theory, 55(4):1433-1439, 2009. URL: https://doi.org/10.1109/TIT.2009.2013014.
  2. Omar Alrabiah, Venkatesan Guruswami, and Ray Li. Randomly punctured reed-solomon codes achieve list-decoding capacity over linear-sized fields. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1458-1469, 2024. URL: https://doi.org/10.1145/3618260.3649634.
  3. Elwyn R. Berlekamp. Block coding with noiseless feedback. PhD thesis, Massachusetts Institute of Technology, 1964. Google Scholar
  4. Elwyn R. Berlekamp. Block coding for the binary symmetric channel with noiseless, delayless feedback. Error-correcting Codes, pages 61-88, 1968. Google Scholar
  5. Volodia M Blinovskiı. Bounds for codes in decoding by a list of finite length. Problemy Peredachi Informatsii, 22(1):11-25, 1986. Google Scholar
  6. Joshua Brakensiek, Sivakanth Gopi, and Visu Makam. Generic reed-solomon codes achieve list-decoding capacity. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 1488-1501, 2023. URL: https://doi.org/10.1145/3564246.3585128.
  7. Christian Deppe. Solution of ulam’s searching game with three lies or an optimal adaptive strategy for binary three-error-correcting codes. Discrete Mathematics, 224(1-3):79-98, 2000. URL: https://doi.org/10.1016/S0012-365X(00)00109-6.
  8. Christian Deppe. Coding with feedback and searching with lies. In Entropy, Search, Complexity, pages 27-70. Springer, 2007. Google Scholar
  9. Peter Elias. List decoding for noisy channels. Massachusetts Institute of Technology, 1957. Google Scholar
  10. Ran Gelles. Coding for Interactive Communication: A Survey. Foundations and Trends® in Theoretical Computer Science, 13:1-161, January 2017. URL: https://doi.org/10.1561/0400000079.
  11. Zeyu Guo and Zihan Zhang. Randomly punctured reed-solomon codes achieve the list decoding capacity over polynomial-size alphabets. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 164-176. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00019.
  12. Meghal Gupta, Venkatesan Guruswami, and Rachel Yun Zhang. Binary error-correcting codes with minimal noiseless feedback. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 1475-1487, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585126.
  13. Venkatesan Guruswami. List decoding of error correcting codes. PhD thesis, Massachusetts Institute of Technology, 2001. URL: https://api.semanticscholar.org/CorpusID:9164141.
  14. Venkatesan Guruswami. List decoding of binary codes-a brief survey of some recent results. In Coding and Cryptology: Second International Workshop, IWCC 2009, Zhangjiajie, China, June 1-5, 2009. Proceedings 2, pages 97-106. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-01877-0_10.
  15. Venkatesan Guruswami. Linear-algebraic list decoding of folded reed-solomon codes. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 77-85. IEEE, 2011. URL: https://doi.org/10.1109/CCC.2011.22.
  16. Venkatesan Guruswami et al. Algorithmic results in list decoding. Foundations and Trendsregistered in Theoretical Computer Science, 2(2):107-195, 2007. Google Scholar
  17. Venkatesan Guruswami, Ray Li, Jonathan Mosheiff, Nicolas Resch, Shashwat Silas, and Mary Wootters. Bounds for list-decoding and list-recovery of random linear codes. arXiv preprint arXiv:2004.13247, 2020. URL: https://arxiv.org/abs/2004.13247.
  18. Venkatesan Guruswami and Srivatsan Narayanan. Combinatorial limitations of average-radius list-decoding. IEEE Transactions on Information Theory, 60(10):5827-5842, 2014. URL: https://doi.org/10.1109/TIT.2014.2343224.
  19. Venkatesan Guruswami and Atri Rudra. Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy. IEEE Transactions on information theory, 54(1):135-150, 2008. URL: https://doi.org/10.1109/TIT.2007.911222.
  20. Wojciech Guzicki. Ulam’s searching game with two lies. Journal of Combinatorial Theory, Series A, 54(1):1-19, 1990. URL: https://doi.org/10.1016/0097-3165(90)90002-E.
  21. Bernhard Haeupler, Pritish Kamath, and Ameya Velingker. Communication with Partial Noiseless Feedback. In APPROX-RANDOM, 2015. Google Scholar
  22. S Muthukrishnan. On optimal strategies for searching in presence of errors. In Proceedings of the fifth annual ACM-SIAM symposium on discrete algorithms, pages 680-689, 1994. URL: http://dl.acm.org/citation.cfm?id=314464.314672.
  23. Alberto Negro, Giuseppe Parlati, and P Ritrovato. Optimal adaptive search: reliable and unreliable models. In Proc. 5th Italian Conf. on Theoretical Computer Science, pages 211-231. World Scientific, 1995. Google Scholar
  24. Andrzej Pelc. Solution of ulam’s problem on searching with a lie. Journal of Combinatorial Theory, Series A, 44(1):129-140, 1987. URL: https://doi.org/10.1016/0097-3165(87)90065-3.
  25. Andrzej Pelc. Searching games with errors—fifty years of coping with liars. Theoretical Computer Science, 270(1-2):71-109, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00303-6.
  26. Morris Plotkin. Binary codes with specified minimum distance. IRE Transactions on Information Theory, 6(4):445-450, 1960. URL: https://doi.org/10.1109/TIT.1960.1057584.
  27. A. Renyi. On a problem of information theory. In MTA Mat. Kut. Int. Kozl., volume 6B, pages 505-516, 1961. Google Scholar
  28. Nicolas Resch, Chen Yuan, and Yihan Zhang. Zero-rate thresholds and new capacity bounds for list-decoding and list-recovery. IEEE Transactions on Information Theory, 2024. Google Scholar
  29. Eric Ruzomberka, Yongkyu Jang, David J. Love, and H. Vincent Poor. The capacity of channels with o(1)-bit feedback. In 2023 IEEE International Symposium on Information Theory (ISIT), pages 2768-2773, 2023. URL: https://doi.org/10.1109/ISIT54713.2023.10206714.
  30. Ofer Shayevitz. On error correction with feedback under list decoding. In IEEE International Symposium on Information Theory, ISIT 2009, June 28 - July 3, 2009, Seoul, Korea, Proceedings, pages 1253-1257, August 2009. URL: https://doi.org/10.1109/ISIT.2009.5205965.
  31. Joel Spencer and Peter Winkler. Three Thresholds for a Liar. Combinatorics, Probability and Computing, 1(1):81-93, 1992. URL: https://doi.org/10.1017/S0963548300000080.
  32. Madhu Sudan. List decoding: Algorithms and applications. ACM SIGACT News, 31(1):16-27, 2000. URL: https://doi.org/10.1145/346048.346049.
  33. Stanislaw M Ulam. Adventures of a Mathematician. Univ of California Press, 1991. Google Scholar
  34. John M. Wozencraft. List decoding, volume 48. Massachusetts Institute of Technology, 1958. Google Scholar
  35. K.Sh. Zigangirov. Number of correctable errors for transmission over a binary symmetrical channel with feedback. Problems Inform. Transmission, 12:85-97, 1976. Google Scholar
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