High-Rate Locally Correctable Codes via Lifting

Abstract

We present a general framework for constructing high-rate error correcting codes that are locally correctable (and hence locally decodable if linear) with a sublinear number of queries, based on lifting codes with respect to functions on the coordinates. Our approach generalizes the lifting of affine-invariant codes (of Guo, Kopparty, and Sudan) and its generalization automorphic lifting (alluded to in the work of Ben-Sasson et al., but distinct from their degree lifting), which lifts algebraic geometry codes with respect to a group of automorphisms of the code. Our notion of lifting is a natural alternative to the degree lifting of Ben-Sasson et al. and it carries two advantages. First, it overcomes the rate barrier inherent in degree lifting. Second, it requires no special properties (e.g. linearity and invariance) of the base code, and requires a very little structure on the set of functions on the coordinates of the code. As an application, we construct new explicit families of locally correctable codes by lifting algebraic geometry codes. Like the multiplicity codes of Kopparty, Saraf, Yekhanin, and the affine-lifted codes of Guo, Kopparty, and Sudan, our codes of block length $N$ can achieve $N^\epsilon $ query complexity and $1-\alpha $ rate for any given $\epsilon , \alpha > 0$ , while correcting a constant fraction of errors, in contrast to the Reed–Muller codes and the degree-lifted AG codes of Ben-Sasson et al., which face a rate barrier of $\epsilon ^{O(1/\epsilon )}$ . However, like the degree-lifted AG codes, our codes are over an alphabet significantly smaller than that obtained by Reed–Muller codes, affine-lifted codes, and multiplicity codes.