Document Open Access Logo

On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors

Authors Alexander R. Block , Jeremiah Blocki , Kuan Cheng , Elena Grigorescu , Xin Li , Yu Zheng, Minshen Zhu

Thumbnail PDF


  • Filesize: 0.86 MB
  • 25 pages

Document Identifiers

Author Details

Alexander R. Block
  • University of Maryland, College Park, MD, USA
  • Georgetown University, Washington, D.C., USA
Jeremiah Blocki
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Kuan Cheng
  • Center on Frontiers of Computing Studies, Peking University, China
Elena Grigorescu
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Xin Li
  • Department of Computer Science, Johns Hopkins University, Baltimore, MD, USA
Yu Zheng
  • Meta Platforms, Inc., Bellevue, WA, USA
Minshen Zhu
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA


We are indebted to some anonymous reviewers who helped us improve the presentation of the paper.

Cite AsGet BibTex

Alexander R. Block, Jeremiah Blocki, Kuan Cheng, Elena Grigorescu, Xin Li, Yu Zheng, and Minshen Zhu. On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 14:1-14:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Locally Decodable Codes (LDCs) are error-correcting codes C:Σⁿ → Σ^m, encoding messages in Σⁿ to codewords in Σ^m, with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length m that is super-polynomial in n, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) show how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting, and introduce their variants in the insertion and deletion (Insdel) error setting. Standard LDCs for Insdel errors were first studied by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), and are further motivated by recent advances in DNA random access bio-technologies. Our first result is an exponential lower bound on the length of Hamming RLDCs making 2 queries (even adaptively), over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a "phase-transition"-type behavior on the codeword length for some constant-query complexity. We achieve these lower bounds via a transformation of RLDCs to standard Hamming LDCs, using a careful analysis of restrictions of message bits that fix codeword bits. We further define two variants of RLDCs in the Insdel-error setting, a weak and a strong version. On the one hand, we construct weak Insdel RLDCs with almost linear codeword length and constant query complexity, matching the parameters of the Hamming variants. On the other hand, we prove exponential lower bounds for strong Insdel RLDCs. These results demonstrate that, while these variants are equivalent in the Hamming setting, they are significantly different in the insdel setting. Our results also prove a strict separation between Hamming RLDCs and Insdel RLDCs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Mathematics of computing → Coding theory
  • Theory of computation → Lower bounds and information complexity
  • Relaxed Locally Decodable Codes
  • Hamming Errors
  • Insdel Errors
  • Lower Bounds


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  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., 2022. URL:
  2. Alexandr Andoni, Thijs Laarhoven, Ilya P. Razenshteyn, and Erik Waingarten. Optimal hashing-based time-space trade-offs for approximate near neighbors. In SODA, pages 47-66, 2017. Google Scholar
  3. Vahid R. Asadi and Igor Shinkar. Relaxed locally correctable codes with improved parameters. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, volume 198 of LIPIcs, pages 18:1-18:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  4. László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. Checking computations in polylogarithmic time. In STOC, pages 21-31, 1991. Google Scholar
  5. James L. Banal, Tyson R. Shepherd, Joseph Berleant, Hellen Huang, Miguel Reyes, Cheri M. Ackerman, Paul C. Blainey, and Mark Bathe. Random access dna memory using boolean search in an archival file storage system. Nature Materials, 20:1272-1280, 2021. URL:
  6. Avraham Ben-Aroya, Oded Regev, and Ronald de Wolf. A hypercontractive inequality for matrix-valued functions with applications to quantum computing and ldcs. In FOCS, pages 477-486. IEEE Computer Society, 2008. Google Scholar
  7. 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. A preliminary version appeared in the Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC). Google Scholar
  8. Arnab Bhattacharyya, L. Sunil Chandran, and Suprovat Ghoshal. Combinatorial lower bounds for 3-query ldcs. In ITCS, volume 151 of LIPIcs, pages 85:1-85:8. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  9. Arnab Bhattacharyya, Zeev Dvir, Shubhangi Saraf, and Amir Shpilka. Tight lower bounds for linear 2-query lccs over finite fields. Comb., 36(1):1-36, 2016. Google Scholar
  10. Arnab Bhattacharyya and Sivakanth Gopi. Lower bounds for constant query affine-invariant lccs and ltcs. ACM Trans. Comput. Theory, 9(2):7:1-7:17, 2017. Google Scholar
  11. Arnab Bhattacharyya, Sivakanth Gopi, and Avishay Tal. Lower bounds for 2-query lccs over large alphabet. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 2017. Google Scholar
  12. Alexander R. Block and Jeremiah Blocki. Private and resource-bounded locally decodable codes for insertions and deletions. In 2021 IEEE International Symposium on Information Theory (ISIT), pages 1841-1846, 2021. URL:
  13. Alexander R. Block, Jeremiah Blocki, Elena Grigorescu, Shubhang Kulkarni, and Minshen Zhu. Locally decodable/correctable codes for insertions and deletions. In FSTTCS, volume 182 of LIPIcs, pages 16:1-16:17, 2020. Google Scholar
  14. Jeremiah Blocki, Kuan Cheng, Elena Grigorescu, Xin Li, Yu Zheng, and Minshen Zhu. Exponential lower bounds for locally decodable and correctable codes for insertions and deletions. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 739-750, 2022. URL:
  15. Jeremiah Blocki, Venkata Gandikota, Elena Grigorescu, and Samson Zhou. Relaxed locally correctable codes in computationally bounded channels. IEEE Transactions on Information Theory, 67(7):4338-4360, 2021. URL:
  16. Jeremiah Blocki, Shubhang Kulkarni, and Samson Zhou. On Locally Decodable Codes in Resource Bounded Channels. In Yael Tauman Kalai, Adam D. Smith, and Daniel Wichs, editors, 1st Conference on Information-Theoretic Cryptography (ITC 2020), volume 163, pages 16:1-16:23, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL:
  17. Manuel Blum and Sampath Kannan. Designing programs that check their work. J. ACM, 42(1):269-291, 1995. Google Scholar
  18. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci., 47(3):549-595, 1993. Google Scholar
  19. Joshua Brakensiek, Venkatesan Guruswami, and Samuel Zbarsky. Efficient low-redundancy codes for correcting multiple deletions. IEEE Trans. Inf. Theory, 64(5):3403-3410, 2018. Google Scholar
  20. Victor Chen, Elena Grigorescu, and Ronald de Wolf. Error-correcting data structures. SIAM J. Comput., 42(1):84-111, 2013. Google Scholar
  21. Kuan Cheng, Venkatesan Guruswami, Bernhard Haeupler, and Xin Li. Efficient linear and affine codes for correcting insertions/deletions. In SODA, pages 1-20. SIAM, 2021. Google Scholar
  22. Kuan Cheng, Bernhard Haeupler, Xin Li, Amirbehshad Shahrasbi, and Ke Wu. Synchronization strings: Highly efficient deterministic constructions over small alphabets. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2185-2204. SIAM, 2019. Google Scholar
  23. Kuan Cheng, Zhengzhong Jin, Xin Li, and Ke Wu. Deterministic document exchange protocols, and almost optimal binary codes for edit errors. In Mikkel Thorup, editor, FOCS, pages 200-211, 2018. Google Scholar
  24. Kuan Cheng, Zhengzhong Jin, Xin Li, and Ke Wu. Block edit errors with transpositions: Deterministic document exchange protocols and almost optimal binary codes. In ICALP, volume 132 of LIPIcs, pages 37:1-37:15, 2019. Google Scholar
  25. Kuan Cheng and Xin Li. Efficient document exchange and error correcting codes with asymmetric information. In SODA, pages 2424-2443. SIAM, 2021. Google Scholar
  26. Kuan Cheng, Xin Li, and Yu Zheng. Locally decodable codes with randomized encoding. CoRR, abs/2001.03692, 2020. URL:
  27. Alessandro Chiesa, Tom Gur, and Igor Shinkar. Relaxed locally correctable codes with nearly-linear block length and constant query complexity. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1395-1411. SIAM, 2020. Google Scholar
  28. Benny Chor, Eyal Kushilevitz, Oded Goldreich, and Madhu Sudan. Private information retrieval. J. ACM, 45(6):965-981, 1998. Google Scholar
  29. Gil Cohen and Tal Yankovitz. Relaxed locally decodable and correctable codes: Beyond tensoring. Electron. Colloquium Comput. Complex., TR22-045, 2022. URL:
  30. Marcel Dall'Agnol, Tom Gur, and Oded Lachish. A structural theorem for local algorithms with applications to coding, testing, and privacy. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1651-1665. SIAM, 2021. Google Scholar
  31. Yan Ding, Parikshit Gopalan, and Richard Lipton. Error correction against computationally bounded adversaries. Manuscript, 2004. Google Scholar
  32. Zeev Dvir, Parikshit Gopalan, and Sergey Yekhanin. Matching vector codes. SIAM J. Comput., 40(4):1154-1178, 2011. Google Scholar
  33. Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Superquadratic lower bound for 3-query locally correctable codes over the reals. Theory Comput., 13(1):1-36, 2017. Google Scholar
  34. Klim Efremenko. 3-query locally decodable codes of subexponential length. SIAM J. Comput., 41(6):1694-1703, 2012. Google Scholar
  35. Anna Gál and Andrew Mills. Three-query locally decodable codes with higher correctness require exponential length. ACM Trans. Comput. Theory, 3(2):5:1-5:34, 2012. Google Scholar
  36. William I. Gasarch. A survey on private information retrieval (column: Computational complexity). Bulletin of the EATCS, 82:72-107, 2004. Google Scholar
  37. Oded Goldreich, Howard J. Karloff, Leonard J. Schulman, and Luca Trevisan. Lower bounds for linear locally decodable codes and private information retrieval. Comput. Complex., 15(3):263-296, 2006. Google Scholar
  38. Tom Gur and Oded Lachish. A lower bound for relaxed locally decodable codes. arXiv preprint, 2019. URL:
  39. Tom Gur and Oded Lachish. On the power of relaxed local decoding algorithms. SIAM J. Comput., 50(2):788-813, 2021. Google Scholar
  40. Tom Gur, Govind Ramnarayan, and Ron Rothblum. Relaxed locally correctable codes. Theory Comput., 16:1-68, 2020. Google Scholar
  41. Venkatesan Guruswami, Bernhard Haeupler, and Amirbehshad Shahrasbi. Optimally resilient codes for list-decoding from insertions and deletions. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, STOC, pages 524-537. ACM, 2020. Google Scholar
  42. Venkatesan Guruswami and Ray Li. Coding against deletions in oblivious and online models. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 625-643. SIAM, 2018. Google Scholar
  43. Venkatesan Guruswami and Ray Li. Polynomial time decodable codes for the binary deletion channel. IEEE Trans. Inf. Theory, 65(4):2171-2178, 2019. Google Scholar
  44. Venkatesan Guruswami and Adam Smith. Optimal rate code constructions for computationally simple channels. J. ACM, 63(4):35:1-35:37, September 2016. URL:
  45. Venkatesan Guruswami and Carol Wang. Deletion codes in the high-noise and high-rate regimes. IEEE Transactions on Information Theory, 63(4):1961-1970, 2017. Google Scholar
  46. Bernhard Haeupler. Optimal document exchange and new codes for insertions and deletions. In David Zuckerman, editor, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 334-347, 2019. Google Scholar
  47. Bernhard Haeupler, Aviad Rubinstein, and Amirbehshad Shahrasbi. Near-linear time insertion-deletion codes and (1+ε)-approximating edit distance via indexing. In Moses Charikar and Edith Cohen, editors, STOC, pages 697-708. ACM, 2019. Google Scholar
  48. Bernhard Haeupler and Amirbehshad Shahrasbi. Synchronization strings: codes for insertions and deletions approaching the singleton bound. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, STOC, pages 33-46. ACM, 2017. Google Scholar
  49. Bernhard Haeupler and Amirbehshad Shahrasbi. Synchronization strings: explicit constructions, local decoding, and applications. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, STOC, pages 841-854. ACM, 2018. Google Scholar
  50. Bernhard Haeupler and Amirbehshad Shahrasbi. Synchronization strings and codes for insertions and deletions - a survey, 2021. URL:
  51. Bernhard Haeupler, Amirbehshad Shahrasbi, and Madhu Sudan. Synchronization strings: List decoding for insertions and deletions. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, ICALP, volume 107 of LIPIcs, pages 76:1-76:14, 2018. Google Scholar
  52. Brett Hemenway and Rafail Ostrovsky. Public-key locally-decodable codes. In Advances in Cryptology - CRYPTO 2008, 28th Annual International Cryptology Conference, Proceedings, pages 126-143, 2008. Google Scholar
  53. Brett Hemenway, Rafail Ostrovsky, Martin J. Strauss, and Mary Wootters. Public key locally decodable codes with short keys. In 14th International Workshop, APPROX, and 15th International Workshop, RANDOM, Proceedings, pages 605-615, 2011. Google Scholar
  54. Brett Hemenway, Rafail Ostrovsky, and Mary Wootters. Local correctability of expander codes. Inf. Comput., 243:178-190, 2015. Google Scholar
  55. Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In STOC, pages 80-86, 2000. Google Scholar
  56. Iordanis Kerenidis and Ronald de Wolf. Exponential lower bound for 2-query locally decodable codes via a quantum argument. J. Comput. Syst. Sci., 69(3):395-420, 2004. Google Scholar
  57. Swastik Kopparty, Or Meir, Noga Ron-Zewi, and Shubhangi Saraf. High-rate locally correctable and locally testable codes with sub-polynomial query complexity. J. ACM, 64(2):11:1-11:42, 2017. Google Scholar
  58. Swastik Kopparty and Shubhangi Saraf. Guest column: Local testing and decoding of high-rate error-correcting codes. SIGACT News, 47(3):46-66, 2016. Google Scholar
  59. Vladimir Iosifovich Levenshtein. Binary codes capable of correcting deletions, insertions and reversals. Soviet Physics Doklady, 10(8):707-710, 1966. Doklady Akademii Nauk SSSR, V163 No4 845-848 1965. Google Scholar
  60. Richard J. Lipton. A new approach to information theory. In STACS, pages 699-708, 1994. Google Scholar
  61. Shu Liu, Ivan Tjuawinata, and Chaoping Xing. On list decoding of insertion and deletion errors. CoRR, abs/1906.09705, 2019. URL:
  62. Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. J. ACM, 39(4):859-868, 1992. Google Scholar
  63. Jiri Matousek Marcos Kiwi, Martin Loebl. Expected length of the longest common subsequence for large alphabets. Advances in Mathematics, 197(2):480-498, 2005. Google Scholar
  64. Hugues Mercier, Vijay K. Bhargava, and Vahid Tarokh. A survey of error-correcting codes for channels with symbol synchronization errors. IEEE Communications Surveys and Tutorials, 12, 2010. Google Scholar
  65. Silvio Micali, Chris Peikert, Madhu Sudan, and David A. Wilson. Optimal error correction against computationally bounded noise. In Theory of Cryptography, Second Theory of Cryptography Conference, TCC 2005, Cambridge, MA, USA, February 10-12, 2005, Proceedings, pages 1-16, 2005. Google Scholar
  66. Michael Mitzenmacher. A survey of results for deletion channels and related synchronization channels. Probability Surveys, 6:1-3, July 2008. Google Scholar
  67. Rafail Ostrovsky, Omkant Pandey, and Amit Sahai. Private locally decodable codes. In ICALP, pages 387-398, 2007. Google Scholar
  68. Rafail Ostrovsky and Anat Paskin-Cherniavsky. Locally decodable codes for edit distance. In Anja Lehmann and Stefan Wolf, editors, Information Theoretic Security, pages 236-249, Cham, 2015. Springer International Publishing. Google Scholar
  69. L. J. Schulman and D. Zuckerman. Asymptotically good codes correcting insertions, deletions, and transpositions. IEEE Transactions on Information Theory, 45(7):2552-2557, 1999. Google Scholar
  70. Ronen Shaltiel and Jad Silbak. Explicit list-decodable codes with optimal rate for computationally bounded channels. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM, pages 45:1-45:38, 2016. Google Scholar
  71. N.J.A. Sloane. On single-deletion-correcting codes. arXiv, 2002. URL:
  72. Madhu Sudan, Luca Trevisan, and Salil P. Vadhan. Pseudorandom generators without the XOR lemma (abstract). In CCC, page 4, 1999. Google Scholar
  73. Luca Trevisan. Some applications of coding theory in computational complexity. CoRR, cs.CC/0409044, 2004. URL:
  74. Stephanie Wehner and Ronald de Wolf. Improved lower bounds for locally decodable codes and private information retrieval. In ICALP, volume 3580 of Lecture Notes in Computer Science, pages 1424-1436. Springer, 2005. Google Scholar
  75. David P. Woodruff. New lower bounds for general locally decodable codes. Technical report, Weizmann Institute of Science, Israel, 2007. Google Scholar
  76. 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
  77. S. M. Hossein Tabatabaei Yazdi, Ryan Gabrys, and Olgica Milenkovic. Portable and error-free dna-based data storage. Scientific Reports, 7:2045-2322, 2017. URL:
  78. Sergey Yekhanin. Towards 3-query locally decodable codes of subexponential length. J. ACM, 55(1):1:1-1:16, 2008. Google Scholar
  79. Sergey Yekhanin. Locally decodable codes. Foundations and Trends in Theoretical Computer Science, 6(3):139-255, 2012. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail