2 2 A pr 2 01 5 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 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(Õ( √ logn)). Previously such codes were known to exist only with Ω(n) 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( √ 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(Õ( √ 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. ∗A preliminary version of this work appeared as [Mei14]. †Department of Mathematics & Department of Computer Science, Rutgers University, Piscataway NJ 08854, USA. Supported in part by a Sloan Fellowship and NSF grant CCF-1253886. [email protected] ‡Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. This research was carried out when Meir was supported in part by the Israel Science Foundation (grant No. 460/05). [email protected] §School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA. Supported in part by the Rothschild fellowship and NSF grant CCF-1412958. [email protected] ¶Department of Mathematics & Department of Computer Science, Rutgers University, Piscataway NJ 08854, USA. Supported in part by NSF grant CCF-1350572. [email protected]