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

Abstract

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 δ of a binary code. The best known tradeoff is the Gilbert-Varshamov bound, and says that for every δ ∈ (0, 1/2), there are codes with minimum distance δ and rate R = R GV (δ) > 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 errorcorrection algorithms. Specifically, we show the following results.1. Local Testing: For all δ ∈ (0, 1/2) and all R < R GV (δ), there exist codes with length n, rate R and minimum distance δ that are locally testable with quasipolylog(n) query complexity. Local Correction:For all > 0, for all δ < 1/2 sufficiently large, and all R < (1 − )R GV (δ), there exist codes with length n, rate R and minimum distance δ that are locally correctable from 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 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, AlonEdmonds-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.

On List Recovery of High-Rate Tensor Codes

“…An instantiation of the base code to produce tensor product codes which are themselves genuinely locally list recoverable (i.e., not just approximately locally list recoverable) in the local correction version. Once more, plugging this into the machinery of [2,3,17], we get capacity-achieving locally list recoverable codes, but now in the local correction version. This now plugs in turn into the machinery of [33,19,17] to give constant-rate binary codes (with a randomized construction) approaching the Gilbert-Varshamov bound that are locally decodable with query complexity and running time N o (1) .…”

“…Once more, plugging this into the machinery of [2,3,17], we get capacity-achieving locally list recoverable codes, but now in the local correction version. This now plugs in turn into the machinery of [33,19,17] to give constant-rate binary codes (with a randomized construction) approaching the Gilbert-Varshamov bound that are locally decodable with query complexity and running time N o (1) . This improves over prior work [17] that only gave query complexity N ε with rate that is exponentially small in 1/ε.…”

“…This now plugs in turn into the machinery of [33,19,17] to give constant-rate binary codes (with a randomized construction) approaching the Gilbert-Varshamov bound that are locally decodable with query complexity and running time N o (1) . This improves over prior work [17] that only gave query complexity N ε with rate that is exponentially small in 1/ε. We obtain our result by taking the base code to be the intersection of an efficient (polytime) high-rate globally list recoverable code and a high-rate locally correctable code.…”

“…This gave the first construction of codes with rate arbitrarily close to 1 that are locally list recoverable from an Ω(1) fraction of errors (however, only in the local decoding version). Finally, using the expander-based distance amplification method of [2,3] (specialized to the setting of local list recovery [18,17]), this gave the first capacity-achieving locally list recoverable (and in particular, list decodable) codes with sublinear (and in fact NÕ (1/ log log N ) ) query complexity and running time (once more, in the local decoding version).…”

“…One could potentially hope (following [17] which implemented a local version of [33,19]) for an analogous result that would give constant-rate codes approaching the Gilbert-Varshamov bound that are locally correctable (or locally decodable) with query complexity and running time N o (1) . However, what prevented [23] from obtaining such a result was the fact that their capacity-achieving locally list recoverable codes only worked in the local decoding version (i.e., they were only able to recover message coordinates).…”