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
Exponential lower bound for 2-query locally decodable codes via a quantum argument
Journal of Computer and System Sciences - Special issue: STOC 2003
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
General constructions for information-theoretic private information retrieval
Journal of Computer and System Sciences
An optimal lower bound for 2-query locally decodable linear codes
Information Processing Letters
Lower bounds for adaptive locally decodable codes
Random Structures & Algorithms
Towards 3-query locally decodable codes of subexponential length
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Towards 3-query locally decodable codes of subexponential length
Journal of the ACM (JACM)
The Complexity of Local List Decoding
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
An optimal lower bound for 2-query locally decodable linear codes
Information Processing Letters
High-rate codes with sublinear-time decoding
Proceedings of the forty-third annual ACM symposium on Theory of computing
Private locally decodable codes
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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
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An error-correcting code is said to be {\em locally decodable} if a randomized algorithm can recover any single bit of a message by reading only a small number of symbols of a possibly corrupted encoding of the message.Katz and Trevisan \cite{KT} showed that any such code C:\{0,1\} \rightarrow \Sigma^m with a decoding algorithm that makes at most q probes must satisfy m= \Omega((n/ \log|\Sigma|)^{q/(q-1)}). They assumed that the decoding algorithm is non-adaptive, and left open the question of proving similar bounds for adaptive decoders.We improve the results of \cite{KT} in two ways. First, we give a more direct proof of their result. Second, and this is our main result, we prove that m=\Omega({(n/\log |\Sigma|)}^{q/(q-1)}) even if the decoding algorithm is adaptive. An important ingredient of our proof is a randomized method for {\em smoothening} an adaptive decoding algorithm. The main technical tool we employ is the {\em Second Moment Method}.