Low Rate Is Insufficient for Local Testability

Locally testable codes are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. A linear code C ⊆ F2 is called sparse if dim(C) = O(log(n)). We say that a code C ⊆ F2 is -biased if all nonzero codewords of C have relative weight in the range ( 1 2 − , 1 2 + ), where may be a function of n. Kaufman and Sudan [10] proved that for sparse linear codes with relative distance 1 2 − n −Ω(1) are locally testable. Moreover, they showed that all sparse n−Ω(1)-biased linear codes are locally decodable. In particular sparse random codes are locally testable and are locally decodable with probability 1−o(1). Kopparty and Saraf [11] conjectured that all sparse linear codes (even with a bad distance) are locally testable. In this paper we refute this conjecture by showing that for every d(n) ranging from ω(1) to Ω(n) there exists a family of codes { C ⊂ F2 } n∈Z with linear distance and dim(C ) = Θ(d(n)) which are not locally testable (decodable). Furthermore, we show that there exists a family of codes { C ⊂ F2 } n∈Z with bias n −o(1) and dim(C) = log(n) which are not locally testable (decodable). This also shows that the results of Kaufman and Sudan [10] were surprisingly tight.