High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity∗

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 com-plexity. 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(Õ( log n)). Previously such codes were known to exist only with Ω(nβ) query complexity (for constant β> 0), and there were several, quite different, constructions known. Our codes are based on a general distance-amplification method of Alon and Luby [AL96]. 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(Õ