A fast and simple randomized parallel algorithm for the maximal independent set problem
Journal of Algorithms
Checking computations in polylogarithmic time
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Pseudorandom generators without the XOR Lemma (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
On the efficiency of local decoding procedures for error-correcting codes
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Hiding Instances in Multioracle Queries
STACS '90 Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science
Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Exponential lower bound for 2-query locally decodable codes via a quantum argument
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Lower bounds for adaptive locally decodable codes
Random Structures & Algorithms
Lower bounds for linear locally decodable codes and private information retrieval
Computational Complexity
Towards 3-query locally decodable codes of subexponential length
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Corruption and Recovery-Efficient Locally Decodable Codes
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
3-query locally decodable codes of subexponential length
Proceedings of the forty-first annual ACM symposium on Theory of computing
An optimal lower bound for 2-query locally decodable linear codes
Information Processing Letters
On Matrix Rigidity and Locally Self-Correctable Codes
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
A quadratic lower bound for three-query linear locally decodable codes over any field
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Local List Decoding with a Constant Number of Queries
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
High-rate codes with sublinear-time decoding
Proceedings of the forty-third annual ACM symposium on Theory of computing
Improved lower bounds for locally decodable codes and private information retrieval
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
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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.