Journal of the ACM (JACM)
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
Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Simple PCPs with poly-log rate and query complexity
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Some 3CNF Properties Are Hard to Test
SIAM Journal on Computing
Locally testable codes and PCPs of almost-linear length
Journal of the ACM (JACM)
Lower bounds for linear locally decodable codes and private information retrieval
Computational Complexity
The PCP theorem by gap amplification
Journal of the ACM (JACM)
Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits
SIAM Journal on Computing
Towards 3-query locally decodable codes of subexponential length
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
Algebraic property testing: the role of invariance
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
2-Transitivity Is Insufficient for Local Testability
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
3-query locally decodable codes of subexponential length
Proceedings of the forty-first annual ACM symposium on Theory of computing
Locally Testable Codes Require Redundant Testers
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
An optimal lower bound for 2-query locally decodable linear codes
Information Processing Letters
On 2-query codeword testing with near-perfect completeness
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
Limitation on the rate of families of locally testable codes
Property testing
Limitation on the rate of families of locally testable codes
Property testing
A new family of locally correctable codes based on degree-lifted algebraic geometry codes
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We study the relation between locally testable and locally decodable codes. Locally testable codes (LTCs) are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. Locally decodable codes (LDCs) allow to recover each message entry with high probability by reading only a few entries of a slightly corrupted codeword. A linear code C ⊆ F2n is called sparse if n ≥ 2Ω(dim(C)). It is well-known that LTCs do not imply LDCs and that there is an intersection between these two families. E.g. the Hadamard code is both LDC and LTC. However, it was not known whether LDC implies LTC. We show the following results. - Two-transitive codes with a local constraint imply LDCs, while they do not imply LTCs. - Every non-sparse LDC contains a large subcode which is not LTC, while every subcode of an LDC remains LDC. Hence, every nonsparse LDC contains a subcode that is LDC but is not LTC. The above results demonstrate inherent differences between LDCs and LTCs, in particular, they imply that LDCs do not imply LTCs.