Finding almost-satisfying assignments
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Locally Testable Codes and PCPs of Almost-Linear Length
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Robust pcps of proximity, shorter pcps and applications to coding
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Assignment Testers: Towards a Combinatorial Proof of the PCP-Theorem
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Noise stability of functions with low in.uences invariance and optimality
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
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
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
Locally Testable Codes Require Redundant Testers
SIAM Journal on Computing
Limitation on the rate of families of locally testable codes
Property testing
Limitation on the rate of families of locally testable codes
Property testing
Bounds on locally testable codes with unique tests
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
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A codeword tester is a highly query-efficient spot checking procedure for ascertaining, with good confidence, proximity of a given string to its closest codeword. We consider the problem of binary codeword testing using only two queries. It is known that three queries suffice for non-trivial codeword testing with perfect completeness (where codewords must be accepted with probability 1). It is known that two queries are not enough for testing with perfect completeness, whereas two queries suffice if one relaxes the requirement of perfect completeness (this is akin to the polynomial-time decidability of 2SAT and the APX-hardness of Max 2SAT, respectively). In this work, motivated by the parallel with 2-query PCPs and the approximability of near-satisfiable instances of Max 2SAT, we investigate 2-query testing with completeness close to 1, say 1–ε for ε→0. Our result is that, for codes of constant relative distance, such testers must also have soundness 1– O(ε) (and this is tight up to constant factors in the O(ε) term). This is to be contrasted with 2-query PCPs, where assuming the Unique Games Conjecture, one can have completeness 1–ε and soundness $1-O(\sqrt{\varepsilon})$. Hence the ratio (1–s)/(1–c) can be super-constant for 2-query PCPs while it is bounded by a constant for 2-query LTCs. Our result also shows a similar limitation of 2-query PCPs of proximity, a notion introduced in [1].