Abstract
A q-query locally testable code (LTC) is an error correcting code that can be tested by a randomized algorithm that reads at most q symbols from the given word. An important question is whether there exist LTCs that have the c 3 property: constant rate, constant relative distance, and that can be tested with a constant number of queries. Such LTCs are sometimes referred to as “asymptotically good”.
We show that dense LTCs cannot be c 3. The density of a tester is roughly the average number of distinct local views in which a coordinate participates. An LTC is dense if it has a tester with density ω(1).
More precisely, we show that a 3-query locally testable code with a tester of density ω(1) cannot be c 3. Furthermore, we show that a q-locally testable code (q > 3) with a tester of density ω(1)n q − 2 cannot be c 3. Our results hold when the tester has the following two properties:
-
(no weights:) Every q-tuple of queries occurs with the same probability.
-
(‘last-one-fixed’:) In every q-query ‘test’ of the tester, the value to any q − 1 of the symbols determines the value of the last symbol. (Linear codes have constraints of this type).
We also show that several natural ways to quantitatively improve our results would already resolve the general c 3 question, i.e. also for non-dense LTCs.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ben-Sasson, E., Sudan, M.: Simple PCPs with poly-log rate and query complexity. In: Proc. 37th ACM Symp. on Theory of Computing, pp. 266–275 (2005)
Ben-Sasson, E., Guruswami, V., Kaufman, T., Sudan, M., Viderman, M.: Locally testable codes require redundant testers. SIAM J. Comput. 39(7), 3230–3247 (2010)
Ben-Sasson, E., Goldreich, O., Sudan, M.: Bounds on 2-query codeword testing. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 216–227. Springer, Heidelberg (2003)
Dinur, I.: The PCP theorem by gap amplification. Journal of the ACM 54(3) (2007)
Goldreich, O., Sudan, M.: Locally testable codes and PCPs of almost-linear length. J. ACM 53(4), 558–655 (2006)
Meir, O.: Combinatorial construction of locally testable codes. SIAM J. Comput. 39(2), 491–544 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dinur, I., Kaufman, T. (2011). Dense Locally Testable Codes Cannot Have Constant Rate and Distance. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2011 2011. Lecture Notes in Computer Science, vol 6845. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22935-0_43
Download citation
DOI: https://doi.org/10.1007/978-3-642-22935-0_43
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22934-3
Online ISBN: 978-3-642-22935-0
eBook Packages: Computer ScienceComputer Science (R0)