High rate locally correctable codes via lifting

TL;DR

This paper introduces a flexible lifting framework for constructing high-rate, locally correctable error-correcting codes that surpass previous rate barriers and require fewer queries, with applications to algebraic geometry codes.

Contribution

The paper generalizes lifting techniques for codes, overcoming rate barriers and enabling the construction of explicit high-rate locally correctable codes from algebraic geometry codes.

Findings

Codes achieve $N^ ext{epsilon}$ query complexity with high rate.

Lifting framework overcomes rate barriers of previous methods.

Constructs explicit codes with small alphabet size.

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, suggested by Ben-Sasson et al, 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 is extremely flexible, requiring no special properties (e.g. linearity, invariance) of the base code, and requiring 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, 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.