2-Transitivity is Insufficient for Local Testability
- 81 Downloads
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.
KeywordsAffine invariance locally testable codes 2-transitivity
Unable to display preview. Download preview PDF.
- László Babai, Lance Fortnow, Leonid A. Levin, Mario Szegedy (1991a). Checking computations in polylogarithmic time. In The Annual ACM Symposium on Theory of Computing, 21–32.Google Scholar
- Eli Ben-Sasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, Madhu Sudan (2011a). On Sums of Locally Testable Affine Invariant Properties. Electronic Colloquium on Computational Complexity (ECCC) 18, 79. (Conference version appeared in RANDOM, 2011).Google Scholar
- Eli Ben-Sasson, Ghid Maatouk, Amir Shpilka, Madhu Sudan (2011b). Symmetric LDPC Codes are not Necessarily Locally Testable. In IEEE Conference on Computational Complexity, 55–65.Google Scholar
- Christian Borgs, Jennifer T. Chayes, László Lovász, Vera T. Sós, Balázs Szegedy, Katalin Vesztergombi (2006). Graph limits and parameter testing. In The Annual ACM Symposium on Theory of Computing, 261–270.Google Scholar
- Oded Goldreich, Shafi Goldwasser, Dana Ron (1998). Property testing and its connection to learning and approximation. Journal of the ACM 45(4), 653–750. ISSN 0004-5411.Google Scholar
- Elena Grigorescu (2010). Symmetries in Algebraic Property Testing. Ph.D. thesis, MIT.Google Scholar
- Elena Grigorescu, Tali Kaufman, Madhu Sudan (2009). Succinct representation of codes with applications to testing. In Proceedings of RANDOM-APPROX 2009, volume 5687 of Lecture Notes in Computer Science, 534–547. Springer.Google Scholar
- Tali Kaufman, Shachar Lovett (2011). New Extension of the Weil Bound for Character Sums with Applications to Coding. In The Annual IEEE Symposium on Foundations of Computer Science, 788–796.Google Scholar
- Tali Kaufman, Madhu Sudan (2008). Algebraic property testing: the role of invariance. In The Annual ACM Symposium on Theory of Computing, 403–412.Google Scholar
- F. J. MacWilliams, NeilJ. A. Sloane (1981) The Theory of Error-Correcting Codes. Elsevier/North-Holland, AmsterdamGoogle Scholar