A hard-core predicate for all one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Self-testing/correcting with applications to numerical problems
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Improved low-degree testing and its applications
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Pseudorandom generators without the XOR Lemma (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Learning Polynomials with Queries: The Highly Noisy Case
SIAM Journal on Discrete Mathematics
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
Noise-tolerant learning, the parity problem, and the statistical query model
Journal of the ACM (JACM)
Almost Orthogonal Linear Codes are Locally Testable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Sub-constant error low degree test of almost-linear size
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
New Results for Learning Noisy Parities and Halfspaces
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Low-degree tests at large distances
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Sparse Random Linear Codes are Locally Decodable and Testable
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
List-decoding reed-muller codes over small fields
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Uniform direct product theorems: simplified, optimized, and derandomized
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Locally Testing Direct Product in the Low Error Range
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
New direct-product testers and 2-query PCPs
Proceedings of the forty-first annual ACM symposium on Theory of computing
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Linearity testing in characteristic two
IEEE Transactions on Information Theory - Part 1
IEEE Transactions on Information Theory
Low rate is insufficient for local testability
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Guest column: testing linear properties: some general theme
ACM SIGACT News
Limitation on the rate of families of locally testable codes
Property testing
Limitation on the rate of families of locally testable codes
Property testing
SIAM Journal on Discrete Mathematics
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In this paper, we give surprisingly efficient algorithms for list-decoding and testing random linear codes. Our main result is that random sparse linear codes are locally list-decodable and locally testable in the high-error regime with only a constant number of queries. More precisely, we show that for all constants c 0 and γ 0, and for every linear code C ⊆ 0,1N which is: sparse: |C| ≤ Nc, and unbiased: each nonzero codeword in C has weight in (1/2 - N-γ, 1/2 + N-γ), C is locally testable and locally list-decodable from (1/2 - ε)-fraction worst-case errors using only poly(1/ε) queries to a received word. We also give subexponential time algorithms for list-decoding arbitrary unbiased (but not necessarily sparse) linear codes in the high-error regime. In particular, this yields the first subexponential time algorithm even for the problem of (unique) decoding random linear codes of inverse-polynomial rate from a fixed positive fraction of errors. Earlier, Kaufman and Sudan had shown that sparse, unbiased codes can be locally (unique) decoded and locally tested from a constant fraction of errors, where this constant fraction tends to 0 as the number of codewords grows. Our results significantly strengthen their results, while also having significantly simpler proofs. At the heart of our algorithms is a natural "self-correcting" operation defined on codes and received words. This self-correcting operation transforms a code C with a received word w into a simpler code C' and a related received word w', such that w is close to C if and only if w' is close to C'. Starting with a sparse, unbiased code C and an arbitrary received word w, a constant number of applications of the self-correcting operation reduces us to the case of local list-decoding and testing for the Hadamard code, for which the well known algorithms of Goldreich-Levin and Blum-Luby-Rubinfeld are available. This yields the constant-query local algorithms for the original code C. Our algorithm for decoding unbiased linear codes in subexponential time proceeds similarly. Applying the self-correcting operation to an unbiased code C and an arbitrary received word a super-constant number of times, we get reduced to the problem of learning noisy parities, for which non-trivial subexponential time algorithms were recently given by Blum-Kalai-Wasserman and Feldman-Gopalan-Khot-Ponnuswami. Our result generalizes a result of Lyubashevsky, which gave a subexponential time algorithm for decoding random linear codes of inverse-polynomial rate from random errors.