TY - GEN

T1 - Locally testable vs. locally decodable codes

AU - Kaufman, Tali

AU - Viderman, Michael

PY - 2010

Y1 - 2010

N2 - 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 non-sparse 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.

AB - 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 non-sparse 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.

UR - http://www.scopus.com/inward/record.url?scp=78149326061&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-15369-3_50

DO - 10.1007/978-3-642-15369-3_50

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AN - SCOPUS:78149326061

SN - 3642153682

SN - 9783642153686

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 670

EP - 682

BT - Approximation, Randomization, and Combinatorial Optimization

T2 - 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010

Y2 - 1 September 2010 through 3 September 2010

ER -