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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.  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 bydoi:10.1007/s00037-012-0055-3 fatcat:rawo37tevje7thtiogoaagrjey