Strong Locally Testable Codes with Relaxed Local Decoders

Authors Oded Goldreich, Tom Gur, Ilan Komargodski

Thumbnail PDF


  • Filesize: 0.8 MB
  • 41 pages

Document Identifiers

Author Details

Oded Goldreich
Tom Gur
Ilan Komargodski

Cite AsGet BibTex

Oded Goldreich, Tom Gur, and Ilan Komargodski. Strong Locally Testable Codes with Relaxed Local Decoders. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 1-41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


Locally testable codes (LTCs) are error-correcting codes that admit very efficient codeword tests. An LTC is said to be strong if it has a proximity-oblivious tester; that is, a tester that makes only a constant number of queries and reject non-codewords with probability that depends solely on their distance from the code. Locally decodable codes (LDCs) are complimentary to LTCs. While the latter allow for highly efficient rejection of strings that are far from being codewords, LDCs allow for highly efficient recovery of individual bits of the information that is encoded in strings that are close to being codewords. While there are known constructions of strong-LTCs with nearly-linear length, the existence of a constant-query LDC with polynomial length is a major open problem. In an attempt to bypass this barrier, Ben-Sasson et al. (SICOMP 2006) introduced a natural relaxation of local decodability, called relaxed-LDCs. This notion requires local recovery of nearly all individual information-bits, yet allows for recovery-failure (but not error) on the rest. Ben-Sasson et al. constructed a constant-query relaxed-LDC with nearly-linear length (i.e., length k^(1 + alpha) for an arbitrarily small constant alpha>0, where k is the dimension of the code). This work focuses on obtaining strong testability and relaxed decodability simultaneously. We construct a family of binary linear codes of nearly-linear length that are both strong-LTCs (with one-sided error) and constant-query relaxed-LDCs. This improves upon the previously known constructions, which obtain either weak LTCs or require polynomial length. Our construction heavily relies on tensor codes and PCPs. In particular, we provide strong canonical PCPs of proximity for membership in any linear code with constant rate and relative distance. Loosely speaking, these are PCPs of proximity wherein the verifier is proximity oblivious (similarly to strong-LTCs and every valid statement has a unique canonical proof. Furthermore, the verifier is required to reject non-canonical proofs (even for valid statements). As an application, we improve the best known separation result between the complexity of decision and verification in the setting of property testing.
  • Locally Testable Codes
  • Locally Decodable Codes
  • PCPs of Proximity


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


  1. Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil P. Vadhan. Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM Journal on Computing, 36(4):889-974, 2006. Google Scholar
  2. Eli Ben-Sasson and Madhu Sudan. Robust locally testable codes and products of codes. Random Structures & Algorithms, 28(4):387-402, 2006. Google Scholar
  3. Eli Ben-Sasson and Michael Viderman. Towards lower bounds on locally testable codes via density arguments. Computational Complexity, 21(2):267-309, 2012. Google Scholar
  4. Benny Chor, Oded Goldreich, Eyal Kushilevitz, and Madhu Sudan. Private information retrieval. Journal of the ACM, 45(6):965-981, 1998. Google Scholar
  5. Irit Dinur and Tali Kaufman. Dense locally testable codes cannot have constant rate and distance. In APPROX-RANDOM, pages 507-518, 2011. Google Scholar
  6. Irit Dinur and Omer Reingold. Assignment testers: Towards a combinatorial proof of the PCP theorem. SIAM Journal on Computing, 36(4):975-1024, 2006. Google Scholar
  7. Klim Efremenko. 3-query locally decodable codes of subexponential length. SIAM Journal on Computing, 41(6):1694-1703, 2012. Google Scholar
  8. Katalin Friedl and Madhu Sudan. Some improvements to total degree tests. In ISTCS, pages 190-198, 1995. Google Scholar
  9. William I. Gasarch. A survey on private information retrieval (column: Computational complexity). Bulletin of the EATCS, 82:72-107, 2004. Google Scholar
  10. Oded Goldreich. Short locally testable codes and proofs: A survey in two parts. In Property Testing, pages 65-104, 2010. Google Scholar
  11. Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM, 45(4):653-750, 1998. Google Scholar
  12. Oded Goldreich and Dana Ron. On proximity oblivious testing. In STOC, pages 141-150, 2009. Google Scholar
  13. Oded Goldreich and Madhu Sudan. Locally testable codes and PCPs of almost-linear length. Journal of the ACM, 53(4):558-655, 2006. Google Scholar
  14. Alan Guo, Swastik Kopparty, and Madhu Sudan. New affine-invariant codes from lifting. In ITCS, pages 529-540. ACM, 2013. Google Scholar
  15. Tom Gur and Ron D. Rothblum. Non-interactive proofs of proximity. In ITCS, pages 133-142. ACM, 2015. Google Scholar
  16. Venkatesan Guruswami and Atri Rudra. Tolerant locally testable codes. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 306-317. Springer, 2005. Google Scholar
  17. Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In STOC, pages 80-86, 2000. Google Scholar
  18. Tali Kaufman and Michael Viderman. Locally testable vs. locally decodable codes. In APPROX-RANDOM, pages 670-682, 2010. Google Scholar
  19. Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. Journal of Computer and System Sciences, 72(6):1012-1042, 2006. Google Scholar
  20. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271, 1996. Google Scholar
  21. Luca Trevisan. Some applications of coding theory in computational complexity. Electronic Colloquium on Computational Complexity (ECCC), 2004. Google Scholar
  22. Michael Viderman. A combination of testability and decodability by tensor products. In APPROX-RANDOM, pages 651-662, 2012. Google Scholar
  23. Michael Viderman. Strong LTCs with inverse poly-log rate and constant soundness. Electronic Colloquium on Computational Complexity (ECCC), 20:22, 2013. Google Scholar
  24. Sergey Yekhanin. Towards 3-query locally decodable codes of subexponential length. Journal of the ACM, 55(1):1, 2008. Google Scholar
  25. 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