TY - JOUR
T1 - 2-Transitivity is Insufficient for Local Testability
AU - Grigorescu, Elena
AU - Kaufman, Tali
AU - Sudan, Madhu
PY - 2013/3
Y1 - 2013/3
N2 - 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.
AB - 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.
KW - 2-transitivity
KW - Affine invariance
KW - locally testable codes
UR - http://www.scopus.com/inward/record.url?scp=84875375394&partnerID=8YFLogxK
U2 - 10.1007/s00037-012-0055-3
DO - 10.1007/s00037-012-0055-3
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:84875375394
SN - 1016-3328
VL - 22
SP - 137
EP - 158
JO - Computational Complexity
JF - Computational Complexity
IS - 1
ER -