Some 3CNF properties are hard to test
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Robust pcps of proximity, shorter pcps and applications to coding
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Simple PCPs with poly-log rate and query complexity
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Almost Orthogonal Linear Codes are Locally Testable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
A Randomness-Efficient Sampler for Matrix-valued Functions and Applications
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
The Complexity of Online Memory Checking
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
The PCP theorem by gap amplification
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Robust locally testable codes and products of codes
Random Structures & Algorithms
The PCP theorem by gap amplification
Journal of the ACM (JACM)
Low-degree tests at large distances
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Non-Abelian homomorphism testing, and distributions close to their self-convolutions
Random Structures & Algorithms
Tensor Products of Weakly Smooth Codes Are Robust
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
Tolerant Linearity Testing and Locally Testable Codes
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Incrementally verifiable computation or proofs of knowledge imply time/space efficiency
TCC'08 Proceedings of the 5th conference on Theory of cryptography
Short locally testable codes and proofs: a survey in two parts
Property testing
Short locally testable codes and proofs: a survey in two parts
Property testing
Short locally testable codes and proofs
Studies in complexity and cryptography
Tolerant locally testable codes
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
The tensor product of two codes is not necessarily robustly testable
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
Robust local testability of tensor products of LDPC codes
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
On 2-query codeword testing with near-perfect completeness
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
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Locally testable codes are error-correcting codes that admit very efficient codeword tests. Specifically, using a constant number of (random) queries, non-codewords are rejected with probability proportional to their distance from the code.Locally testable codes are believed to be the combinatorial core of PCPs. However, the relation is less immediate than commonly believed. Nevertheless, we show that certain PCP systems can be modified to yield locally testable codes. On the other hand, we adapt techniques we develop for the construction of the latter to yield new PCPs. Our main results are locally testable codes and PCPs of almost-linear length. Specifically, we present:Locally testable (linear) codes in which \kappainformation bits are encoded by a codeword of length approximately \kappa\cdot \exp (\sqrt {\log \kappa } ). This improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large non-binary alphabet.PCP systems of almost-linear length for SAT. The length of the proof is approximately n \cdot \exp (\sqrt {\log n} ) and verification in performed by a constant number (i.e., 19) of queries, as opposed to previous results that used proof length n^{1 + ({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 q}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$q$}})} for verification by q queries.The novel techniques in use include a random projection of certain codewords and PCP-oracles, an adaptation of PCP constructions to obtain "linear PCP-oracles" for provingconjunctions of linear conditions, and a direct construction of locally testable (linear) codes of sub-exponential length.