TY - GEN

T1 - Locally Testable Codes Require Redundant Testers

AU - Ben-Sasson, Eli

AU - Guruswami, Venkatesan

AU - Kaufman, Tali

AU - Sudan, Madhu

AU - Viderman, Michael

PY - 2009

Y1 - 2009

N2 - Locally testable codes (LTCs) are errorcorrecting codes for which membership, in the code, of a given word can be tested by examining it in very few locations. Most known constructions of locally testable codes are linear codes, and give error-correcting codes whose duals have (superlinearly) many small weight codewords. Examining this feature appears to be one of the promising approaches to proving limitation results for (i.e., upper bounds on the rate of) LTCs. Unfortunately till now it was not even known if LTCs need to be non-trivially redundant, i.e., need to have one linear dependency among the low-weight codewords in its dual. in this paper we give the first lower bound of this form, by showing that every positive rate constant query strong LTC must have linearly many redundant low-weight codewords in its dual. We actually prove the stronger claim that the actual test itself must use a linear number of redundant dual codewords (beyond the minimum number of basis elements required to characterize the code); in other words, non-redundant (in fact, low redundancy) local testing is impossible.

AB - Locally testable codes (LTCs) are errorcorrecting codes for which membership, in the code, of a given word can be tested by examining it in very few locations. Most known constructions of locally testable codes are linear codes, and give error-correcting codes whose duals have (superlinearly) many small weight codewords. Examining this feature appears to be one of the promising approaches to proving limitation results for (i.e., upper bounds on the rate of) LTCs. Unfortunately till now it was not even known if LTCs need to be non-trivially redundant, i.e., need to have one linear dependency among the low-weight codewords in its dual. in this paper we give the first lower bound of this form, by showing that every positive rate constant query strong LTC must have linearly many redundant low-weight codewords in its dual. We actually prove the stronger claim that the actual test itself must use a linear number of redundant dual codewords (beyond the minimum number of basis elements required to characterize the code); in other words, non-redundant (in fact, low redundancy) local testing is impossible.

KW - Dual codes

KW - Ldpc codes

KW - Linear codes

KW - Lower bounds

KW - Property testing

UR - http://www.scopus.com/inward/record.url?scp=70350624763&partnerID=8YFLogxK

U2 - 10.1109/CCC.2009.6

DO - 10.1109/CCC.2009.6

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:70350624763

SN - 9780769537177

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 52

EP - 61

BT - Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009

T2 - 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009

Y2 - 15 July 2009 through 18 July 2009

ER -