Indeed, the point is that, for codes of constant rate and constant relative distance, local testability, let alone local correctability, was only known for a number of queries that is polynomially related to the length of the code. The current work obtains local testability and correctability for codes of constant rate and constant relative distance using query complexity that is subexponential in the logarithm of the previously known query complexity.
Note: This work grow out of a prior work by Or (see choice Nr 154), which in retrospect is defined (by the authors) as a preliminary version of it.
Update (July 2015): The local testability result (LTC), but not the local correctability result (LCC), has been improved by a recent work of the same authors [ECCC TR15-110]. Specifically, the query complexity in the new work is quasi-polylogarithmic.
In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length n, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity exp(sqrt(logn)). Previously such codes were known to exist only with n^e query complexity (for constant e>0), and there were several, quite different, constructions known.
Our codes are based on a general distance-amplification method of Alon and Luby (1996). We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of sub-constant relative distance.
Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity exp(sqrt(logn)), which additionally have the property of approaching the Singleton bound: they have almost the best-possible relationship between their rate and distance. This has the surprising consequence that asking for a large alphabet error-correcting code to further be an LCC or LTC with exp(sqrt(logn)) query complexity does not require any sacrifice in terms of rate and distance! Such a result was previously not known for any o(n) query complexity.
Our results on LCCs also immediately give locally-decodable codes (LDCs) with the same parameters.
See ECCC TR15-068.