2-Transitivity is Insufficient for Local Testability

Elena Grigorescu, Tali Kaufman, Madhu Sudan

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish
Pages (from-to)137-158
Number of pages22
JournalComputational Complexity
Issue number1
StatePublished - Mar 2013


  • 2-transitivity
  • Affine invariance
  • locally testable codes


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