Abstract
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 rejects non-codewords with a probability that depends solely on their distance from the code.
Locally decodable codes (LDCs) are complementary 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.
Constructions of strong-LTCs with nearly-linear length are known, but 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 k1+α for an arbitrarily small constant α > 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 either obtain only 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.
- Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil P. Vadhan. 2006. Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM J. Comput. 36, 4 (2006), 889--974. Google ScholarDigital Library
- Eli Ben-Sasson and Madhu Sudan. 2006. Robust locally testable codes and products of codes. Random Struct. Algorithms 28, 4 (2006), 387--402. Google ScholarDigital Library
- Eli Ben-Sasson and Michael Viderman. 2012. Towards lower bounds on locally testable codes via density arguments. Comput. Complexity 21, 2 (2012), 267--309. Google ScholarDigital Library
- Clément L. Canonne and Tom Gur. 2017. An adaptivity hierarchy theorem for property testing. In Proceedings of the 32nd Computational Complexity Conference (CCC’17). 27:1--27:25. Google ScholarDigital Library
- Benny Chor, Oded Goldreich, Eyal Kushilevitz, and Madhu Sudan. 1998. Private information retrieval. J. ACM 45, 6 (1998), 965--981. Google ScholarDigital Library
- Irit Dinur and Tali Kaufman. 2011. Dense locally testable codes cannot have constant rate and distance. In APPROX-RANDOM. 507--518. Google ScholarDigital Library
- Irit Dinur and Omer Reingold. 2006. Assignment testers: Towards a combinatorial proof of the PCP theorem. SIAM J. Comput. 36, 4 (2006), 975--1024. Google ScholarDigital Library
- Klim Efremenko. 2012. 3-query locally decodable codes of subexponential length. SIAM J. Comput. 41, 6 (2012), 1694--1703.Google ScholarDigital Library
- Katalin Friedl and Madhu Sudan. 1995. Some improvements to total degree tests. In ISTCS. 190--198. Google ScholarDigital Library
- William I. Gasarch. 2004. A survey on private information retrieval (Column: Computational Complexity). Bulletin of the EATCS 82 (2004), 72--107.Google Scholar
- Oded Goldreich. 2010. Short locally testable codes and proofs: A survey in two parts. In Property Testing. 65--104.Google Scholar
- Oded Goldreich, Shafi Goldwasser, and Dana Ron. 1998. Property testing and its connection to learning and approximation. J. ACM 45, 4 (1998), 653--750. Google ScholarDigital Library
- Oded Goldreich and Tom Gur. 2016. Universal locally verifiable codes and 3-round interactive proofs of proximity for CSP. Electronic Colloquium on Computational Complexity (ECCC) 23 (2016), 192.Google Scholar
- Oded Goldreich and Tom Gur. 2018. Universal locally testable codes. Chicago J. Theor. Comput. Sci. (2018).Google Scholar
- Oded Goldreich and Dana Ron. 2009. On proximity oblivious testing. In STOC. 141--150. Google ScholarDigital Library
- Oded Goldreich and Madhu Sudan. 2006. Locally testable codes and PCPs of almost-linear length. J. ACM 53, 4 (2006), 558--655. Google ScholarDigital Library
- Alan Guo, Swastik Kopparty, and Madhu Sudan. 2013. New affine-invariant codes from lifting. In Proceedings of the 2013 Conference on Innovations in Theoretical Computer Science (ITCS’13). ACM, 529--540. Google ScholarDigital Library
- Tom Gur, Govind Ramnarayan, and Ron D. Rothblum. 2018. Relaxed locally correctable codes. In 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA. 27:1--27:11.Google Scholar
- Tom Gur and Ron D. Rothblum. 2015. Non-interactive proofs of proximity. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science (ITCS’15). ACM, 133--142. Google ScholarDigital Library
- Venkatesan Guruswami and Atri Rudra. 2005. Tolerant locally testable codes. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. Springer, 306--317. Google ScholarDigital Library
- Jonathan Katz and Luca Trevisan. 2000. On the efficiency of local decoding procedures for error-correcting codes. In STOC. 80--86. Google ScholarDigital Library
- Tali Kaufman and Michael Viderman. 2010. Locally testable vs. locally decodable codes. In APPROX-RANDOM. 670--682. Google ScholarDigital Library
- Iordanis Kerenidis and Ronald de Wolf. 2004. Exponential lower bound for 2-query locally decodable codes via a quantum argument. J. Comput. System Sci. 69, 3 (2004), 395--420. Google ScholarDigital Library
- Michal Parnas, Dana Ron, and Ronitt Rubinfeld. 2006. Tolerant property testing and distance approximation. J. Comput. System Sci. 72, 6 (2006), 1012--1042. Google ScholarDigital Library
- Alexander Polishchuk and Daniel A. Spielman. 1994. Nearly-linear size holographic proofs. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing. 194--203. Google ScholarDigital Library
- Ronitt Rubinfeld and Madhu Sudan. 1996. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput. 25, 2 (1996), 252--271. Google ScholarDigital Library
- Luca Trevisan. 2004. Some applications of coding theory in computational complexity. Electronic Colloquium on Computational Complexity (ECCC) (2004).Google Scholar
- Michael Viderman. 2012. A combination of testability and decodability by tensor products. In APPROX-RANDOM. 651--662.Google Scholar
- Michael Viderman. 2013. Strong LTCs with inverse poly-log rate and constant soundness. Electronic Colloquium on Computational Complexity (ECCC) 20 (2013), 22.Google Scholar
- David P. Woodruff. 2012. A quadratic lower bound for three-query linear locally decodable codes over any field. J. Comput. Sci. Technol. 27, 4 (2012), 678--686.Google ScholarCross Ref
- Sergey Yekhanin. 2008. Towards 3-query locally decodable codes of subexponential length. J. ACM 55, 1 (2008), 1. Google ScholarDigital Library
- Sergey Yekhanin. 2012. Locally decodable codes. Found. Trends Theor. Comput. Sci. 6, 3 (2012), 139--255. Google ScholarDigital Library
Index Terms
- Strong Locally Testable Codes with Relaxed Local Decoders
Recommendations
Locally testable codes with constant rate, distance, and locality
STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of ComputingA locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads q bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter q is called the ...
Strong locally testable codes with relaxed local decoders
CCC '15: Proceedings of the 30th Conference on Computational ComplexityLocally 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-...
Locally testable vs. locally decodable codes
APPROX/RANDOM'10: Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniquesWe study the relation between locally testable and locally decodable codes. Locally testable codes (LTCs) are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. ...
Comments