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Low Degree Local Correction Over the Boolean Cube

Prashanth Amireddy, Amik Raj Behera, Manaswi Paraashar, Srikanth Srinivasan, Madhu Sudan

Abstract

In this work, we show that the class of multivariate degree-$d$ polynomials mapping $\{0,1\}^{n}$ to any Abelian group $G$ is locally correctable with $\widetilde{O}_{d}((\log n)^{d})$ 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 [ABPSS24] (STOC 2024) who considered the case of linear polynomials and gave analogous results. Low-degree polynomials over the Boolean cube $\{0,1\}^{n}$ arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [ABPSS24] from linear to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of low-degree 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 [ABPSS24] 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.

Low Degree Local Correction Over the Boolean Cube

Abstract

In this work, we show that the class of multivariate degree- polynomials mapping to any Abelian group is locally correctable with 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 [ABPSS24] (STOC 2024) who considered the case of linear polynomials and gave analogous results. Low-degree polynomials over the Boolean cube arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [ABPSS24] from linear to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of low-degree 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- polynomials. To show that the class of degree- 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 [ABPSS24] for higher degree polynomials involves understanding random restrictions of non-zero degree- 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.

Paper Structure

This paper contains 63 sections, 38 theorems, 99 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Low degree polynomials over groups.
  3. Local correction of polynomials.
  4. List correction of codes.
  5. Local list correction of codes.
  6. Motivation and previous work
  7. Technical challenges in extending beyond the linear case
  8. Our main results
  9. Technical tools
  10. Preliminaries
  11. Notation
  12. Basic definitions and tools
  13. Probabilistic notions.
  14. Coding theory notions.
  15. Subcubes of $\{0,1\}^n$.
  16. ...and 48 more sections

Key Result

Theorem 1.3.1

For every Abelian group $G$ and for every constant $\varepsilon > 0$, the space $\mathcal{P}_{d}$ has a $(\delta,q)$-local correction algorithm where $\delta = \frac{1}{2^{d+1}}-\varepsilon$ and $q = \widetilde{\mathcal{O}}_{\varepsilon}(\log n)^{d}.$

Theorems & Definitions (82)

  • Theorem 1.3.1: Local correction algorithms for $\mc{P}_{d}$ up to the unique decoding radius
  • Theorem 1.3.2
  • Theorem 1.3.3: Combinatorial list decoding bound for $\mc{P}_{d}$
  • Theorem 1.3.4: Local list correction for $\mathcal{P}_{d}$
  • Remark 1.3.5
  • Theorem 2.2.1: Basic facts about multilinear polynomials
  • Definition 2.2.2: Local Correction Algorithm
  • Definition 2.2.3: Local List-Correction Algorithm
  • Remark 2.2.4
  • Definition 2.2.5: Combinatorial List Decodability
  • ...and 72 more