Dense locally testable codes cannot have constant rate and distance

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-fix...

• Type: Journal
• Year: 2011
• Where: ECCC
• Authors: Irit Dinur, Tali Kaufman
• Keywords: ECCC 2011, ECommerce, Error Correcting Code, Ltcs, Randomized Algorithm