Abstract

We study the local decodability and (tolerant) local testability of low‐degree n‐variate polynomials over arbitrary fields, evaluated over the domain {0,1}n. We show that for every field there is a tolerant local test whose query complexity depends only on the degree. In contrast we show that decodability is possible over fields of positive characteristic, but not over the reals.

Highlights

  • [28, 8, 5, 21, 24, 19, 29, 3, 2] for some of the early applications.) One of the key properties of low-degree n-variate polynomials underlying many of the applications is the “DeMillo-Lipton

  • Schwartz-Zippel” distance lemma [11, 27, 30] which upper bounds the number of zeroes that a non-zero low-degree polynomial may have over “grids”, i.e., over domains of the form A1 × · · · × An. This turns the space of polynomials into an error-correcting code and many applications are built around this class of codes. These applications have motivated a rich collection of tools including polynomial time decoding algorithms for these codes, and “local decoding” [4, 18, 9] and “local testing” [26, 1, 15] procedures for these codes

  • The specific question we address below is: when is the family of degree d n-variate polynomials locally decodable and testable when the domain is {0, 1}n?

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Summary

Introduction

Low-degree polynomials have played a central role in computational complexity. (See for instance [28, 8, 5, 21, 24, 19, 29, 3, 2] for some of the early applications.) One of the key properties of low-degree n-variate polynomials underlying many of the applications is the “DeMillo-Lipton-. When A1 = · · · = An = F (and so F is finite) it was shown by Kaufman and Ron [15] (with similar results in Jutla et al [14]) that the family of n-variate degree d polynomials over F is (δ, q)locally decodable and (δ, q)-locally testable for some δ = exp(−d) and q = exp(d) In particular both q and 1/δ are bounded for fixed d, independent of n and F. The specific question we address below is: when is the family of degree d n-variate polynomials locally decodable and (tolerantly) testable when the domain is {0, 1}n? Working with domains of other and possibly varying sizes of Ai’s would lead to quantitative changes and we do not consider that setting in this paper.)

Main Results
Overview of proofs
Basic notation
Local Testers and Decoders
Facts about binomial coefficients
Results
Local linear spans of balanced vectors
Tolerant Local Testing
Full Text
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