Checking computations in polylogarithmic time
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Efficient checking of polynomials and proofs and the hardness of approximation problems
Efficient checking of polynomials and proofs and the hardness of approximation problems
Nearly-linear size holographic proofs
STOC '94 Proceedings of the twenty-sixth 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
Breaking the O(n1/(2k-1)) Barrier for Information-Theoretic Private Information Retrieval
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Upper Bound on Communication Complexity of Private Information Retrieval
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Better Lower Bounds for Locally Decodable Codes
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Exponential lower bound for 2-query locally decodable codes via a quantum argument
Journal of Computer and System Sciences - Special issue: STOC 2003
General constructions for information-theoretic private information retrieval
Journal of Computer and System Sciences
A tight lower bound for restricted PIR protocols
Computational Complexity
An \Omega(n^1/3 ) Lower Bound for Bilinear Group Based Private Information Retrieval
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Lower bounds for linear locally decodable codes and private information retrieval
Computational Complexity
A Geometric Approach to Information-Theoretic Private Information Retrieval
SIAM Journal on 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
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
3-query locally decodable codes of subexponential length
Proceedings of the forty-first annual ACM symposium on Theory of computing
Decoding frequency permutation arrays under infinite norm
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4
Communications of the ACM
Two-query PCP with subconstant error
Journal of the ACM (JACM)
Quantum private queries: security analysis
IEEE Transactions on Information Theory
Testing non-uniform k-wise independent distributions over product spaces
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
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 techniques
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
Decoding frequency permutation arrays under Chebyshev distance
IEEE Transactions on Information Theory
High-rate codes with sublinear-time decoding
Proceedings of the forty-third annual ACM symposium on Theory of computing
Locally decodable codes: a brief survey
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
Limits on the rate of locally testable affine-invariant codes
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
SIAM Journal on Computing
A novel elementary construction of matching vectors
Information Processing Letters
From irreducible representations to locally decodable codes
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Optimally robust private information retrieval
Security'12 Proceedings of the 21st USENIX conference on Security symposium
New bounds for matching vector families
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
A new family of locally correctable codes based on degree-lifted algebraic geometry codes
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Local correctability of expander codes
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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A q-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new constructions of three query LDCs of vastly shorter length than that of previous constructions. Specifically, given any Mersenne prime p = 2t − 1, we design three query LDCs of length N = exp(O(n1/t)), for every n. Based on the largest known Mersenne prime, this translates to a length of less than exp(O(n10 − 7)) compared to exp(O(n1/2)) in the previous constructions. It has often been conjectured that there are infinitely many Mersenne primes. Under this conjecture, our constructions yield three query locally decodable codes of length N = exp(nO(1/log log n)) for infinitely many n. We also obtain analogous improvements for Private Information Retrieval (PIR) schemes. We give 3-server PIR schemes with communication complexity of O(n10 − 7) to access an n-bit database, compared to the previous best scheme with complexity O(n1/5.25). Assuming again that there are infinitely many Mersenne primes, we get 3-server PIR schemes of communication complexity nO(1/log logn)) for infinitely many n. Previous families of LDCs and PIR schemes were based on the properties of low-degree multivariate polynomials over finite fields. Our constructions are completely different and are obtained by constructing a large number of vectors in a small dimensional vector space whose inner products are restricted to lie in an algebraically nice set.