Low Degree Local Correction Over the Boolean Cube

In this work, we show that the class of multivariate degree-d polynomials mapping t0, 1u n to any Abelian group G is locally correctable with r O d pplog nq d q queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of the authors [ABP `24] (STOC 2024) who considered the case of linear polynomials (d " 1) and gave analogous results.

Low-degree polynomials over the Boolean cube t0, 1u n arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [ABP 24] from linear polynomials to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of lowdegree polynomials over the Boolean cube. For local correction, we construct a set of points in the Boolean cube that lie between two exponentially close parallel hyperplanes and is moreover an interpolating set for degree-d polynomials. To show that the class of degree-d polynomials is list decodable up to the minimum distance, we stitch together results on anti-concentration of low-degree polynomials, the Sunflower lemma, and the Footprint bound for counting common zeroes of polynomials. Analyzing the local list corrector of [ABP 24] for higher degree polynomials involves understanding random restrictions of non-zero degree-d polynomials on a Hamming slice. In particular, we show that a simple random restriction process for reducing the dimension of the Boolean cube is a suitably good sampler for Hamming slices. Thus our