2-Transitivity Is Insufficient for Local Testability

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. [N. Alon et al., 2003] had conjectured that the presence of a single low weight code in the dual, and "2-transitivity" of the code (i.e., the code is 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 and Sudan [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.