Nearly-linear size holographic proofs
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Robust locally testable codes and products of codes
Random Structures & Algorithms
Locally testable codes and PCPs of almost-linear length
Journal of the ACM (JACM)
The PCP theorem by gap amplification
Journal of the ACM (JACM)
Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding
SIAM Journal on Computing
Assignment Testers: Towards a Combinatorial Proof of the PCP Theorem
SIAM Journal on Computing
Short PCPs with Polylog Query Complexity
SIAM Journal on Computing
Composition of Semi-LTCs by Two-Wise Tensor Products
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Combinatorial Construction of Locally Testable Codes
SIAM Journal on Computing
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
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Given two codes R and C, their tensor product R@?C consists of all matrices whose rows are codewords of R and whose columns are codewords of C. The product R@?C is said to be robust if for every matrix M that is far from R@?C it holds that the rows and columns of M are far on average from R and C respectively. Ben-Sasson and Sudan (RSA 28 (4) (2006)) have asked under which conditions the product R@?C is robust. So far, a few important families of tensor products were shown to be robust, and a counter-example of a product that is not robust was also given. However, a precise characterization of codes whose tensor product is robust is yet unknown. In this work, we highlight a common theme in the previous works on the subject, which we call ''the rectangle method''. In short, we observe that all proofs of robustness in the previous works are done by constructing a certain ''rectangle'', while in the counter-example no such rectangle can be constructed. We then show that a rectangle can be constructed if and only if the tensor product is robust, and therefore the proof strategy of constructing a rectangle is complete.