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computational complexity

, Volume 22, Issue 1, pp 137–158 | Cite as

2-Transitivity is Insufficient for Local Testability

  • Elena Grigorescu
  • Tali Kaufman
  • Madhu Sudan
Article

Abstract

A basic goal in property testing is to identify a minimal set of features that make a property testable. For the case when the property to be tested is membership in a binary linear error-correcting code, Alon et al. (Trans Inf Theory, 51(11):4032–4039, 2005) had conjectured that the presence of a single low-weight codeword in the dual, and “2-transitivity” of the code (i.e., the code being invariant under a 2-transitive group of permutations on the coordinates of the code) suffice to get local testability. We refute this conjecture by giving a family of error-correcting codes where the coordinates of the codewords form a large field of characteristic two, and the code is invariant under affine transformations of the domain. This class of properties was introduced by Kaufman & Sudan (STOC, 2008) as a setting where many results in algebraic property testing generalize. Our result shows a complementary virtue: This family also can be useful in producing counterexamples to natural conjectures.

Keywords

Affine invariance locally testable codes 2-transitivity 

Subject classification

68Q01 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Computer SciencePurdue UniversityWest LafayetteUSA
  2. 2.Computer ScienceBar-Ilan UniversityRamat GanIsrael
  3. 3.Microsoft Research New EnglandOne Memorial DriveCambridgeUSA

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