Abstract
Locally decodable codes are error-correcting codes with the extra property that, in order to retrieve the value of a single input position, it is sufficient to read a small number of positions of the codeword. We refer to the probability of getting the correct value as the correctness of the decoding algorithm.
A breakthrough result by Yekhanin [2007] showed that 3-query linear locally decodable codes may have subexponential length. The construction of Yekhanin, and the three query constructions that followed, achieve correctness only up to a certain limit which is 1 − 3δ for nonbinary codes, where an adversary is allowed to corrupt up to δ fraction of the codeword. The largest correctness for a subexponential length 3-query binary code is achieved in a construction by Woodruff [2008], and it is below 1 − 3δ.
We show that achieving slightly larger correctness (as a function of δ) requires exponential codeword length for 3-query codes. Previously, there were no larger than quadratic lower bounds known for locally decodable codes with more than 2 queries, even in the case of 3-query linear codes. Our lower bounds hold for linear codes over arbitrary finite fields and for binary nonlinear codes. Considering larger number of queries, we obtain lower bounds for q-query codes for q > 3, under certain assumptions on the decoding algorithm that have been commonly used in previous constructions. We also prove bounds on the largest correctness achievable by these decoding algorithms, regardless of the length of the code. Our results explain the limitations on correctness in previous constructions using such decoding algorithms. In addition, our results imply trade-offs on the parameters of error-correcting data structures.
- Alon, N., Babai, L., and Itai, A. 1986. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algor. 7, 567--583. Google Scholar
Digital Library
- Babai, L., Fortnow, L., Levin, L. A., and Szegedy, M. 1991. Checking computations in polylogarithmic time. In Proceedings of the ACM Symposium on Theory of Computing (STOC’91). 21--32. Google Scholar
Digital Library
- Beaver, D. and Feigenbaum, J. 1990. Hiding instances in multioracle queries. In Proceedings of the Symposium on Theoritical Aspects of Computer Science (STACS’90). 37--48. Google Scholar
Digital Library
- Ben-Aroya, A., Regev, O., and Wolf, R. d. 2008. A hypercontractive inequality for matrix-valued functions with applications to quantum computing and ldcs. In Proceedings of the IEEE Annual Symposium on Foundations of Computer Science (FOCS’08). 477--486. Google Scholar
Digital Library
- Ben-Aroya, A., Efremenko, K., and Ta-Shma, A. 2010. Local list decoding with a constant number of queries. In Proceedings of the IEEE Annual Symposium on Foundations of Computer Science (FOCS’10). 715--722. Google Scholar
Digital Library
- de Wolf, R. 2009. Error-correcting data structures. In Proceedings of the Symposium on Theoritical Aspects of Computer Science (STACS’09). 313--324.Google Scholar
- Deshpande, A., Jain, R., Kavitha, T., Radhakrishnan, J., and Lokam, S. V. 2005. Lower bounds for adaptive locally decodable codes. Random Struct. Algor. 27, 3, 358--378. Google Scholar
Digital Library
- Dvir, Z. 2010. On matrix rigidity and locally self-correctable codes. In Proceedings of the IEEE Conference on Computational Complexity. 291--298. Google Scholar
Digital Library
- Dvir, Z. and Shpilka, A. 2005. Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits. In Proceedings of the ACM Symposium on Theory of Computing (STOC’05). 592--601. Google Scholar
Digital Library
- Dvir, Z., Gopalan, P., and Yekhanin, S. 2010. Matching vector codes. In Proceedings of the IEEE Annual Symposium on Foundations of Computer Science (FOCS’10). 705--714. Google Scholar
Digital Library
- Efremenko, K. 2009. 3-query locally decodable codes of subexponential length. In Proceedings of the ACM Symposium on Theory of Computing (STOC’09). 39--44. Google Scholar
Digital Library
- Gal, A. and Mills, A. 2011. Three query linear locally decodable codes with higher correctness require exponential length. In Proceedings of the Symposium on Theoritical Aspects of Computer Science (STACS’11). 673--684.Google Scholar
- Goldreich, O., Karloff, H., Schulman, L. J., and Trevisan, L. 2006. Lower bounds for linear locally decodable codes and private information retrieval. Comput. Complex. 15, 3, 263--296. Google Scholar
Digital Library
- Katz, J. and Trevisan, L. 2000. On the efficiency of local decoding procedures for error-correcting codes. In Proceedings of the ACM Symposium on Theory of Computing (STOC’00). 80--86. Google Scholar
Digital Library
- Kerenidis, I. and de Wolf, R. 2003. Exponential lower bound for 2-query locally decodable codes via a quantum argument. In Proceedings of the ACM Symposium on Theory of Computing (STOC’03). 106--115. Google Scholar
Digital Library
- Kopparty, S., Saraf, S., and Yekhanin, S. 2011. High-rate codes with sublinear-time decoding. In Proceedings of the ACM Symposium on Theory of Computing (STOC’11). 167--176. Google Scholar
Digital Library
- Miltersen, P. B. 1999. Cell probe complexity - a survey. In Advances in Data Structures Workshop.Google Scholar
- Obata, K. 2002. Optimal lower bounds for 2-query locally decodable linear codes. In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX-RANDOM’02). 39--50. Google Scholar
Digital Library
- Raghavendra, P. 2007. A note on Yekhanin’s locally decodable codes. In Proceedings of the Electronic Colloquium on Computational Complexity (ECCC TR07-016).Google Scholar
- Shiowattana, D. and Lokam, S. V. 2006. An optimal lower bound for 2-query locally decodable linear codes. Inf. Process. Lett. 97, 6, 244--250. Google Scholar
Digital Library
- Sudan, M., Trevisan, L., and Vadhan, S. 1999. Pseudorandom generators without the xor lemma. In Proceedings of the ACM Symposium on Theory of Computing (STOC’99). 537--546. Google Scholar
Digital Library
- Trevisan, L. 2004. Some applications of coding theory in computational complexity. In Proceedings of the Electronic Colloquium on Computational Complexity (ECCC TR04-043).Google Scholar
- Wehner, S. and de Wolf, R. 2005. Improved lower bounds for locally decodable codes and private information retrieval. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP’05). 1424--1436. Google Scholar
Digital Library
- Woodruff, D. 2006. Some new lower bounds for general locally decodable codes. In Proceedings of the Electronic Colloquium on Computational Complexity (ECCC TR07-006).Google Scholar
- Woodruff, D. 2008. Corruption and recovery-efficient locally decodable codes. In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX-RANDOM’08). 584--595. Google Scholar
Digital Library
- Woodruff, D. 2010. A quadratic lower bound for three-query linear locally decodable codes over any field. In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX-RANDOM’10). 766--779. Google Scholar
Digital Library
- Yekhanin, S. 2007. Towards 3-query locally decodable codes of subexponential length. In Proceedings of the ACM Symposium on Theory of Computing (STOC’07). 266--274. Google Scholar
Digital Library
Index Terms
(auto-classified)Three-Query Locally Decodable Codes with Higher Correctness Require Exponential Length
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