On the efficiency of local decoding procedures for error-correcting codes
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Some 3CNF Properties Are Hard to Test
SIAM Journal on Computing
Lower bounds for adaptive locally decodable codes
Random Structures & Algorithms
The PCP theorem by gap amplification
Journal of the ACM (JACM)
Sparse Random Linear Codes are Locally Decodable and Testable
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Combinatorial construction of locally testable codes
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Short PCPs with Polylog Query Complexity
SIAM Journal on Computing
Locally Testable Codes Require Redundant Testers
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
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
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
Local list-decoding and testing of random linear codes from high error
Proceedings of the forty-second ACM symposium on Theory of 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
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Three results are shown regarding locally testable and locally decodable linear codes. All three results rely on the observation that repetition codes have the same local testability and local decodability parameters as the unrepeated base code used to create them. The first two results deal with families of sparse linear codes, i.e., codes with dimension logarithmic in the code blocklength n. Such codes have been shown by Kaufman and Sudan [8] to be locally testable and decodable as long as all nonzero codewords have Hamming weight n ċ (1/2 ± n-Ω(1)). Our first result shows that certain sparse codes are neither locally testable, nor locally decodable. This refutes a conjecture of Kopparty and Saraf [9] which postulated that all sparse codes are locally testable. Our second result shows that the result of Kaufman and Sudan is surprisingly tight, and for any function h(n) = o(1) there exist families of sparse codes all of whose codewords have weight n ċ (1/2 ± n-h(n)) and these codes are neither locally testable, nor locally decodable. Our third and final result is about the redundancy of locally testable codes. Informally, the redundancy of a locally testable code is the minimal number of redundant tests sampled by a tester, where a test is said to be redundant if is a linear combination of other tests. Ben-Sasson et al. [1] introduced the notion of redundancy and showed that for every linear locally testable code the redundancy is at least linear in the dimension of the code. Our last result shows that redundancy is indeed a function of the code dimension, not blocklength, and that the bound given in [1] is nearly tight.