Locally Testable Codes Require Redundant Testers
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Sudan_Locally Testable.PDF
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Author(s)
Viderman, Michael
Guruswami, Venkatesan
Ben-Sasson, Eli
Kaufman-Halman, Tali
Sudan, Madhu
Date Issued
September 2009
Journal
2009 Computational Complexity
Publisher
Institute of Electrical and Electronics Engineers
Citation
Ben-Sasson, E. et al. “Locally Testable Codes Require Redundant Testers.” Computational Complexity, 2009. CCC '09. 24th Annual IEEE Conference on. 2009. 52-61. © 2009 IEEE
Version
Author's final manuscript
Abstract
Locally testable codes (LTCs) are error- correcting 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.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
http://dx.doi.org/10.1109/CCC.2009.6
