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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 LCCs and LTCs with block length n, constant rate (which can even be taken arbitrarily close to 1) and constant relative distance, whose query complexity is exp(Õ(√logn)) (for LCCs) and (logn)O(loglogn) (for LTCs). Previously such codes were known to exist only with Ω(nβ) query complexity (for constant β>0). In addition to having small query complexity, our codes also achieve better trade-offs between the rate and the relative distance than were previously known to be achievable by LCCs or LTCs. Specifically, over large (but constant size) alphabet, our codes approach the Singleton bound, that is, 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 sub-polynomial query complexity does not require any sacrifice in terms of rate and distance! Over the binary alphabet, our codes meet the Zyablov bound. Such trade-offs between the rate and the relative distance were previously not known for any o(n) query complexity. Our results on LCCs also immediately give locally-decodable codes (LDCs) with the same parameters. Our codes are based on a technique of Alon, Edmonds and Luby. We observe that this technique can be used as a general distance-amplification method, and show that it interacts well with local correctors and testers. We obtain our main results by applying this method to suitably constructed LCCs and LTCs in the non-standard regime of sub-constant relative distance.

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