A New Upper Bound on the Query Complexity for Testing Generalized Reed-Muller codes View Full Text


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Chapter Info

DATE

2012

AUTHORS

Noga Ron-Zewi , Madhu Sudan

ABSTRACT

Over a finite field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{F}}_q$\end{document} the (n,d,q)-Reed-Muller code is the code given by evaluations of n-variate polynomials of total degree at most d on all points (of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{F}}_q^n$\end{document}). The task of testing if a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:{\mathbb{F}}_q^n \to {\mathbb{F}}_q$\end{document} is close to a codeword of an (n,d,q)-Reed-Muller code has been of central interest in complexity theory and property testing. The query complexity of this task is the minimal number of queries that a tester can make (minimum over all testers of the maximum number of queries over all random choices) while accepting all Reed-Muller codewords and rejecting words that are δ-far from the code with probability Ω(δ). (In this work we allow the constant in the Ω to depend on d.)For codes over a prime field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb F}_q$\end{document} the optimal query complexity is well-known and known to be Θ(q⌈(d + 1)/(q − 1)⌉), and the test consists of testing if f is a degree d polynomial on a randomly chosen (⌈(d + 1)/(q − 1) ⌉)-dimensional affine subspace of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{F}}_q^n$\end{document}. If q is not a prime, then the above quantity remains a lower bound, whereas the previously known upper bound grows to O(q⌈(d + 1)/(q − q/p) ⌉) where p is the characteristic of the field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{F}}_q$\end{document}. In this work we give a new upper bound of (cq)(d + 1)/q on the query complexity, where c is a universal constant. Thus for every p and sufficiently large q this bound improves over the previously known bound by a polynomial factor.In the process we also give new upper bounds on the “spanning weight” of the dual of the Reed-Muller code (which is also a Reed-Muller code). The spanning weight of a code is the smallest integer w such that codewords of Hamming weight at most w span the code. The main technical contribution of this work is the design of tests that test a function by not querying its value on an entire subspace of the space, but rather on a carefully chosen (algebraically nice) subset of the points from low-dimensional subspaces. More... »

PAGES

639-650

Book

TITLE

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

ISBN

978-3-642-32511-3
978-3-642-32512-0

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-642-32512-0_54

DOI

http://dx.doi.org/10.1007/978-3-642-32512-0_54

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1024843922


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