  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
ccc-property: constant relative rate, constant relative distance, and that can
be tested with a constant number of queries. Such codes are sometimes referred
to as "asymptotically good".
  We show that dense LTCs cannot be ccc. 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 >> 1.
  More precisely, we show that a 3-query locally testable code with a tester of
density >> 1 cannot be ccc. Moreover, we show that a q-query locally testable
code (q>3) with a tester of density >> n^{q-2} cannot be ccc. Our results hold
when the tester has the following two properties: 1) "no weights": Every
q-tuple of queries occurs with the same probability. 2) "last-one-fixed": In
every `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 ccc-question, i.e. also for non-dense LTCs.
