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Local Correction of Linear Functions over the Boolean Cube

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

TL;DR

This work studies local correction and local list correction for linear functions on the Boolean cube mapped to arbitrary Abelian groups, establishing near-optimal sublinear-query algorithms. The authors generalize Goldreich–Levin to group-valued polynomials, expanding from the binary case to $ ext{Abelian}$ groups and even real-valued codomains, while achieving a $( frac14-\

Abstract

We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain $\{0,1\}^n$ over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance $1/2$ and we give local-correction algorithms correcting up to nearly $1/4$-fraction errors making $\widetilde{\mathcal{O}}(\log n)$ queries. This query complexity is optimal up to $\mathrm{poly}(\log\log n)$ factors. We also give local list-correcting algorithms correcting $(1/2 - \varepsilon)$-fraction errors with $\widetilde{\mathcal{O}}_{\varepsilon}(\log n)$ queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is $\mathbb{Z}_2$. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). The central challenge in constructing the local corrector is constructing "nearly balanced vectors" over $\{-1,1\}^n$ that span $1^n$ -- we show how to construct $\mathcal{O}(\log n)$ vectors that do so, with entries in each vector summing to $\pm1$. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius $(1/2-\varepsilon)$ is $\mathcal{O}_{\varepsilon}(1)$. Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls.

Local Correction of Linear Functions over the Boolean Cube

TL;DR

This work studies local correction and local list correction for linear functions on the Boolean cube mapped to arbitrary Abelian groups, establishing near-optimal sublinear-query algorithms. The authors generalize Goldreich–Levin to group-valued polynomials, expanding from the binary case to groups and even real-valued codomains, while achieving a $( frac14-\

Abstract

We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance and we give local-correction algorithms correcting up to nearly -fraction errors making queries. This query complexity is optimal up to factors. We also give local list-correcting algorithms correcting -fraction errors with queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is . By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). The central challenge in constructing the local corrector is constructing "nearly balanced vectors" over that span -- we show how to construct vectors that do so, with entries in each vector summing to . The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius is . Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls.
Paper Structure (67 sections, 36 theorems, 111 equations, 3 algorithms)

This paper contains 67 sections, 36 theorems, 111 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Group valued polynomials.
  3. Local Correction.
  4. Local List Correction.
  5. Motivation for Our Work
  6. Our Main Results
  7. Proof Overview
  8. Local Correction - \ref{['thm:uniquedeg1']}
  9. First step
  10. Second and third steps
  11. Combinatorial List Decoding Bound - \ref{['thm:comblistdeg1']}
  12. Case 1: $p=2$
  13. Case 2: Large $p$
  14. Putting it together
  15. Local List Correction - \ref{['thm:listdecoding']}
  16. ...and 52 more sections

Key Result

Theorem 1.1

The space $\mathcal{P}_1$ has a $(\delta,q)$-local correction algorithm where $\delta = \frac{1}{4}-\varepsilon$ for any constant $\varepsilon > 0$ and $q = \tilde{O}(\log n).$

Theorems & Definitions (90)

  • Theorem 1.1: Local correction algorithms for $\mathcal{P}_1$ up to the unique decoding radius
  • Theorem 1.2: Combinatorial list decoding bound for $\mathcal{P}_1$
  • Theorem 1.3: Local List Correction for degree-$1$
  • Theorem 2.1
  • Definition 1: Local Correction Algorithm
  • Definition 2: Local List-Correction Algorithm
  • Remark 2.2
  • Definition 3: Combinatorial List Decodability
  • Definition 4: Noise distribution
  • Theorem 2.3: odonnellbook
  • ...and 80 more