Locally Testable And Locally Correctable Codes Approaching The Gilbert-Varshamov Bound

One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate R and minimum distance delta of a binary code. The best known tradeoff is the Gilbert-Varshamov bound, and says that for every delta is an element of (0,1/2), there are codes with minimum distance delta and rate R = R-GV (delta) > 0 (for a certain simple function R-GV(.)). In this paper we show that the Gilbert-Varshamov bound can be achieved by codes which support local error-detection and error correction algorithms.Specifically, we show the following results.1. Local Testing: For all delta is an element of (0, 1/2) and all R < R-GV (6), there exist codes with length n, rate R and minimum distance delta that are locally testable with quasipolylog (n) query complexity.2. Local Correction: For all epsilon > 0, for all delta < 1/2 sufficiently large, and all R < (1 - epsilon)R-GV (delta), there exist codes with length n, rate R and minimum distance delta that are locally correctable from delta/2 - o(1) fraction errors with O(n(epsilon)) query complexity.Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters.Our local testing result is obtained by combining Thommesen's random concatenation technique and the best known locally testable codes from [KMRS16]. Our local correction result, which is significantly more involved, also uses random concatenation, along with a number of further ideas: the Guruswami-Sudan-Indyk list decoding strategy for concatenated codes, Alon-Edmonds-Luby distance amplification, and the local listdecodability, local list-recoverability and local testability of Reed-Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing algorithms; this seems to be the first time local testability is used in the construction of a locally correctable code.