A New Upper Bound on the Query Complexity of Testing Generalized Reed-Muller Codes.

Noga Ron-Zewi,Madhu Sudan

doi:10.4086/toc.2013.v009a025

Abstract

Over a finite field \({\mathbb{F}}_q\) 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 \({\mathbb{F}}_q^n\)). The task of testing if a function \(f:{\mathbb{F}}_q^n \to {\mathbb{F}}_q\) 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.)

Highlights

In this work we present new upper bounds on the query complexity of testing Reed-Muller codes, the codes obtained by evaluations of multivariate low-degree polynomials, over general fields
It is well-known that the query complexity of testing a linear code C is lower bounded by the “minimum distance” of its dual, where the minimum distance of a code is the minimum weight of a non-zero codeword. (The weight of a word is the number of non-zero coordinates.) Applied to the Reed-Muller code RM[n, d, q] this suggests a lower bound on the query complexity via the minimum distance of its dual, which turns out to be a Reed-Muller code
The minimum distance of the latter is well-known and is q(d+1)/(q−1) and this leads to the tight analysis of the query complexity of Reed-Muller codes over prime fields

Summary

Introduction

In this work we present new upper bounds on the query complexity of testing Reed-Muller codes, the codes obtained by evaluations of multivariate low-degree polynomials, over general fields. In the process we give new upper bounds on the spanning weight of Reed-Muller codes. We explain these terms and our results below. We start with the definition of Reed-Muller codes. Let Fq denote the finite field on q elements. The Reed-Muller codes have two parameters in addition to the order of the field, namely the degree d and number n of variables. The (n, d, q)-ReedMuller code RM[n, d, q] is the set of functions from Fnq to Fq that are evaluations of n-variate polynomials of total degree at most d

Testing Reed-Muller codes

Spanning weight

Qualitative description and techniques

Background and restatement of problem

Single-orbit characterizations

Spanning weight of the dual code is a lower bound on query complexity

Canonical monomials and a new constraint

Tightness of our analysis

Rejecting all canonical monomials in the border simultaneously

Rejecting all monomials in the border