Paper Info
Title
Dense locally testable codes cannot have constant rate and distance
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 c3 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 c3. 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 c3. Furthermore, we show that a q-locally testable code (q 3) with a tester of density ω(1)nq-2 cannot be c3. 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 c3 question, i.e. also for nondense LTCs.
Year
DOI
Venue
2010
10.1007/978-3-642-22935-0_43
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Keywords
DocType
Citations 
nondense ltcs,dense ltcs,constant rate,constant relative distance,constant number,c3 property,q-locally testable code,linear code,c3 question,testable code
Journal
4
PageRank 
References 
Authors
0.40
8
2
Name
Order
Citations
PageRank
Irit Dinur1118785.67
Tali Kaufman249938.33