# Optimal Testing of Reed-Muller Codes

Ontology type: schema:Chapter      Open Access: True

### Chapter Info

DATE

2010

AUTHORS ABSTRACT

We consider the problem of testing if a given function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f : \mathbb{F}_2^n \rightarrow \mathbb{F}_2$\end{document} is close to any degree d polynomial in n variables, also known as the problem of testing Reed-Muller codes. We are interested in determining the query-complexity of distinguishing with constant probablity between the case where f is a degree d polynomial and the case where f is Ω(1)-far from all degree d polynomials. Alon et al. [AKK+05] proposed and analyzed a natural 2d + 1-query test T0, and showed that it accepts every degree d polynomial with probability 1, while rejecting functions that are Ω(1)-far with probability Ω(1/(d 2d)). This leads to a O(d 4d)-query test for degree d Reed-Muller codes.We give an asymptotically optimal analysis of T0, showing that it rejects functions that are Ω(1)-far with Ω(1)-probability (so the rejection probability is a universal constant independent of d and n). In particular, this implies that the query complexity of testing degree d Reed-Muller codes is O(2d).Our proof works by induction on n, and yields a new analysis of even the classical Blum-Luby-Rubinfeld [BLR93] linearity test, for the setting of functions mapping \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_2^n$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_2$\end{document}. Our results also imply a “query hierarchy” result for property testing of affine-invariant properties: For every function q(n), it gives an affine-invariant property that is testable with O(q(n))-queries, but not with o(q(n))-queries, complementing an analogous result of [GKNR08] for graph properties.This is a brief overview of the results in the paper [BKS+09]. More... »

PAGES

269-275

### Book

TITLE

Property Testing

ISBN

978-3-642-16366-1
978-3-642-16367-8

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-642-16367-8_19

DOI

http://dx.doi.org/10.1007/978-3-642-16367-8_19

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1052623414

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