High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity

Locally correctable codes (LCCs) and locally testable codes (LTCs) are error-correcting codes that admit local algorithms for correction and detection of errors. Those algorithms are local in the sense that they only query a small number of entries of the corrupted codeword. The fundamental question about LCCs and LTCs is to determine the optimal tradeoff among their rate, distance, and query complexity. In this work, we construct the first LCCs and 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(√log n)) (for LCCs) and (log n)O(log log n) (for LTCs). In addition to having small query complexity, our codes also achieve better tradeoffs 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. Over the binary alphabet, our codes meet the Zyablov bound. Such tradeoffs 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 with the same parameters.