Revision #1 Authors: Dana Moshkovitz, Ran Raz

Accepted on: 9th October 2005 00:00

Downloads: 1793

Keywords:

Given a function f:F^m rightarrow F over a finite field F, a low degree tester tests its proximity to an m-variate polynomial of total degree at most d over F. The tester is usually given access to an oracle A providing the supposed restrictions of f to affine subspaces of constant dimension (e.g., lines, planes, etc.). The tester makes very few (probabilistic) queries to f and to A (say, one query to f and one query to A), and decides whether to accept or reject based on the replies. We wish to minimize two parameters of a tester: its error and its size. The error bounds the probability that the tester accepts although the function is far from a low degree polynomial. The size is the number of bits required to write the oracle replies on all possible tester's queries. Low degree testing is a central ingredient in most constructions of probabilistically checkable proofs (PCPs) and locally testable codes (LTCs). The error of the low degree tester is related to the soundness of the PCP and its size is related to the size of the PCP (or the length of the LTC). We design and analyze new low degree testers that have both sub-constant error o(1) and almost-linear size n^{1+o(1)} (where n=|F|^m). Previous constructions of sub-constant error testers had polynomial size. These testers enabled the construction of PCPs with sub-constant soundness, but polynomial size. Previous constructions of almost-linear size testers obtained only constant error. These testers were used to construct almost-linear size LTCs and almost-linear size PCPs with constant soundness.

TR05-086 Authors: Dana Moshkovitz, Ran Raz

Publication: 14th August 2005 18:44

Downloads: 1688

Keywords:

Given a function f:F^m \rightarrow F over a finite

field F, a low degree tester tests its proximity to

an m-variate polynomial of total degree at most d

over F. The tester is usually given access to an oracle

A providing the supposed restrictions of f to

affine subspaces of constant dimension (e.g., lines, planes,

etc.). The tester makes very few (probabilistic) queries to f

and to A (say, one query to f and one query to

A), and decides whether to accept or reject based on the

replies.

We wish to minimize two parameters of a tester: its error

and its size. The error bounds the probability that

the tester accepts although the function is far from a low degree

polynomial. The size is the number of bits required to

write the oracle replies on all possible tester's queries.

Low degree testing is a central ingredient in most constructions

of probabilistically checkable proofs (PCPs) and locally

testable codes (LTCs). The error of the low degree tester is

related to the soundness of the PCP and its size is related to

the size of the PCP (or the length of the LTC).

We design and analyze new low degree testers that have both

sub-constant error o(1) and almost-linear size

n^{1+o(1)} (where n=|F|^m). Previous

constructions of sub-constant error testers had

polynomial size. These testers enabled the

construction of PCPs with sub-constant soundness, but

polynomial size. Previous

constructions of almost-linear size testers obtained only

constant error. These testers were used to

construct almost-linear size LTCs and almost-linear

size PCPs with constant soundness.