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

Abstract

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.

Locally Testable and Locally Correctable Codes Approaching the Gilbert-Varshamov Bound

“…Analogous results over large alphabets were only recently obtained [KMRS16]. In this setting, the best tradeoff between R and δ for general codes is completely known.…”

“…Our construction of locally testable codes approaching the GV bound then follows from the result of [KMRS16], which gave constructions of locally testable codes with rate and distance approaching the Singleton bound. We start with such a locally testable code from [KMRS16] as the outer code, and then concatenate it with uniformly random binary linear inner codes (independently for each coordinate of the outer code).…”

“…Our local testing result is obtained by combining Thommesen's random concatenation technique and the best known locally testable codes from [KMRS16]. Our…”

“…For binary codes, it was previously known how to do this for codes approaching the Zyablov bound [KMRS16], with the added advantage that the code also had an efficient deterministic construction. We now give formal statements to our main results.…”

“…Furthermore, Reed-Solomon codes achieve R = 1 − δ, and they can be decoded from a δ/2-fraction of errors in polynomial time. In [KMRS16], it was shown that there exist explicit locally testable codes and locally correctable codes which satisfy R = 1 − δ − (for all > 0), and which can further be locally tested and locally corrected from (δ/2 − o(1))-fraction errors in sublinear (and even subpolynomial) time.…”