Linear Relaxed Locally Decodable and Correctable Codes Do Not Need Adaptivity and Two-Sided Error

Author Guy Goldberg

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Guy Goldberg
  • Weizmann Institute of Science, Rehovot, Israel


We would like to thank Irit Dinur for her guidance and encouragement. We would also like to thank Oded Goldreich for insightful discussions, and to Yotam Dikstein for his helpful comments.

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Guy Goldberg. Linear Relaxed Locally Decodable and Correctable Codes Do Not Need Adaptivity and Two-Sided Error. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 74:1-74:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Relaxed locally decodable codes (RLDCs) are error-correcting codes in which individual bits of the message can be recovered by querying only a few bits from a noisy codeword. For uncorrupted codewords, and for every bit, the decoder must decode the bit correctly with high probability. However, for a noisy codeword, a relaxed local decoder is allowed to output a "rejection" symbol, indicating that the decoding failed. We study the power of adaptivity and two-sided error for RLDCs. Our main result is that if the underlying code is linear, adaptivity and two-sided error do not give any power to relaxed local decoding. We construct a reduction from adaptive, two-sided error relaxed local decoders to non-adaptive, one-sided error ones. That is, the reduction produces a relaxed local decoder that never errs or rejects if its input is a valid codeword and makes queries based on its internal randomness (and the requested index to decode), independently of the input. The reduction essentially maintains the query complexity, requiring at most one additional query. For any input, the decoder’s error probability increases at most two-fold. Furthermore, assuming the underlying code is in systematic form, where the original message is embedded as the first bits of its encoding, the reduction also conserves both the code itself and its rate and distance properties We base the reduction on our new notion of additive promise problems. A promise problem is additive if the sum of any two YES-instances is a YES-instance and the sum of any NO-instance and a YES-instance is a NO-instance. This novel framework captures both linear RLDCs and property testing (of linear properties), despite their significant differences. We prove that in general, algorithms for any additive promise problem do not gain power from adaptivity or two-sided error, and obtain the result for RLDCs as a special case. The result also holds for relaxed locally correctable codes (RLCCs), where a codeword bit should be recovered. As an application, we improve the best known lower bound for linear adaptive RLDCs. Specifically, we prove that such codes require block length of n ≥ k^{1+Ω(1/q²)}, where k denotes the message length and q denotes the number of queries.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Locally decodable codes
  • Relaxed locally correctable codes
  • Relaxed locally decodable codes


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  1. Omar Alrabiah, Venkatesan Guruswami, Pravesh Kothari, and Peter Manohar. A near-cubic lower bound for 3-query locally decodable codes from semirandom CSP refutation. Electron. Colloquium Comput. Complex., TR22-101, 2022. Google Scholar
  2. Vahid R. Asadi and Igor Shinkar. Relaxed locally correctable codes with improved parameters. In ICALP, volume 198 of LIPIcs, pages 18:1-18:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  3. Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil P. Vadhan. Robust pcps of proximity, shorter pcps, and applications to coding. SIAM J. Comput., 36(4):889-974, 2006. URL:
  4. Eli Ben-Sasson, Prahladh Harsha, and Sofya Raskhodnikova. Some 3cnf properties are hard to test. In STOC, pages 345-354. ACM, 2003. Google Scholar
  5. Manuel Blum and Sampath Kannan. Designing programs that check their work. J. ACM, 42(1):269-291, 1995. Google Scholar
  6. Alessandro Chiesa, Tom Gur, and Igor Shinkar. Relaxed locally correctable codes with nearly-linear block length and constant query complexity. In SODA, pages 1395-1411. SIAM, 2020. Google Scholar
  7. Gil Cohen and Tal Yankovitz. Relaxed locally decodable and correctable codes: Beyond tensoring. Electron. Colloquium Comput. Complex., TR22-045, 2022. URL:
  8. Gil Cohen and Tal Yankovitz. Asymptotically-good RLCCs with (log n)^(2+o(1)) queries. Electron. Colloquium Comput. Complex., TR23-110, 2023. Google Scholar
  9. Marcel de Sena Dall'Agnol, Tom Gur, and Oded Lachish. A structural theorem for local algorithms with applications to coding, testing, and privacy. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 1651-1665. SIAM, 2021. URL:
  10. Oded Goldreich. On the lower bound on the length of relaxed locally decodable codes. Electron. Colloquium Comput. Complex., TR23-064, 2023. Google Scholar
  11. Oded Goldreich, Tom Gur, and Ilan Komargodski. Strong locally testable codes with relaxed local decoders. In CCC, volume 33 of LIPIcs, pages 1-41. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. Google Scholar
  12. Oded Goldreich and Madhu Sudan. Locally testable codes and pcps of almost-linear length. J. ACM, 53(4):558-655, 2006. Google Scholar
  13. Tom Gur and Oded Lachish. On the power of relaxed local decoding algorithms. In SODA, pages 1377-1394. SIAM, 2020. Google Scholar
  14. Tom Gur, Govind Ramnarayan, and Ron Rothblum. Relaxed locally correctable codes. Electron. Colloquium Comput. Complex., TR17-143, 2017. Google Scholar
  15. Tom Gur, Govind Ramnarayan, and Ron Rothblum. Relaxed locally correctable codes. Theory Comput., 16:1-68, 2020. URL:
  16. Richard Wesley Hamming. Error detecting and error correcting codes. Bell System Technical Journal, 29:147-160, 1950. Google Scholar
  17. Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In F. Frances Yao and Eugene M. Luks, editors, Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 80-86. ACM, 2000. URL:
  18. Iordanis Kerenidis and Ronald de Wolf. Exponential lower bound for 2-query locally decodable codes via a quantum argument. In STOC, pages 106-115. ACM, 2003. Google Scholar
  19. Swastik Kopparty and Shubhangi Saraf. Local testing and decoding of high-rate error-correcting codes. Electron. Colloquium Comput. Complex., TR17-126, 2017. Google Scholar
  20. Vinayak M. Kumar and Geoffrey Mon. Relaxed local correctability from local testing. CoRR, abs/2306.17035, 2023. Google Scholar
  21. Richard J. Lipton. Efficient checking of computations. In STACS, volume 415 of Lecture Notes in Computer Science, pages 207-215. Springer, 1990. Google Scholar
  22. Noga Ron-Zewi and Ron D. Rothblum. Local proofs approaching the witness length. IACR Cryptol. ePrint Arch., page 1062, 2019. Google Scholar
  23. C.E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37(1):10-21, 1949. URL:
  24. David P. Woodruff. New lower bounds for general locally decodable codes. Electron. Colloquium Comput. Complex., TR07-006, 2007. Google Scholar
  25. David P. Woodruff. A quadratic lower bound for three-query linear locally decodable codes over any field. J. Comput. Sci. Technol., 27(4):678-686, 2012. Google Scholar
  26. Sergey Yekhanin. Locally decodable codes. Found. Trends Theor. Comput. Sci., 6(3):139-255, 2012. Google Scholar