Sub-Constant Error
Low Degree Test
of Almost-Linear Size
Dana Moshkovitz
Weizmann Institute
Ran Raz
Weizmann Institute
Probabilistically Checkable Proofs
[AS92,ALMSS92]
Is  satisfiable?
n
NP:
s(n)
PCP:
siz
e

• Completeness:  sat. ) 9A, Pr[acc] = 1.
error
• Soundness:  not sat. ) 8A, Pr[acc] · .
2
Importance of PCP Theorem



Shows proofs are surprisingly powerful.
Enables hardness of approximation results.
Yields codes with local testing/decoding
properties.
3
Decreasing Error

Prop: error ¸ 1/||#queries

Hence, to decrease error need either:
useful for
application
s
enlarge
alphabet
||
easy! via
repetition
enlarge
#queries
4
Decreasing Error


[AS92,ALMSS92]: constant error PCP.
[ArSu97,RaSa97,DFKRS99]: sub-const error
PCP (taking super-const ||).
sub-const??
o(1)
0.5
error
5
Decreasing Size
size
nc


n1+o(1)
n

[AS92,ALMSS92]: size=nc for
large constant c.
[GS02,BSVW03,BGHSV04]:
almost-linear size n1+o(1) PCPs
[Dinur06] (based on [BS05]):
size=n¢polylog n
almost  Only constant error!
linear??
6
Our Motivation
size
n
c
Want: PCP with both
 sub-constant error and
almost  almost-linear size
linear??
n1+o(1)
n
sub-const??
o(1)
error
7
Our Work
We show:
Low Degree Tester:
 sub-constant error
 almost-linear size
Historically:
Low Degree Tester
(non-trivially))
PCP
8
Low Degree Testing


F = finite field; m = dimension; d = degree.
Goal: verify f : Fm!F is close to being
polynomial of total degree · d.
f : Fm!F
Q(x1,…,xm)
deg Q ·d
[|F|m = input size; |F|Àd,m]
9
Low Degree Tester
f : Fm! F
A
size
low degree tester
1.
2.
3.
toss coins
query f and A in O(1) places
accept/reject
Completeness: f is deg ·d poly ! 9A always accept
Soundness: for any A, if Pr[accept]>0 (=error),
agreement of f
(with poly of deg · d)
' Pr[accept]
10
The Line Vs. Point Test
[Rubinfeld,Sudan]
Observation: Fix Q:Fm!F, degQ·d.
Restriction of Q to any line z+t¢y (for
y
m
z,y2F ), namely Q(z+t¢y), is a univariate
polynomial of deg ·d.
Moreover, this characterizes m-variate
polynomials over F of deg·d.
The proof A: for every line l,
A(l) = univariate poly of deg·d [alleged
restriction of f to l]
z
11
The Line Vs. Point Test
[Rubinfeld,Sudan]
1. Pick random z2Fm, y2Fm
2. Check A(
y
z
)(z) = f(z)
• makes two queries
• clearly complete
• quadratic size ¼|Fm|2
• [RuSu90,AS92,ALMSS92,FS93,PS94]: error <1.
12
Sub-Constant Error


[RaSa97]: Plane vs. Point has sub-const error
poly(m,d)/|F|.
[ArSu97]: Line vs. Point has sub-const error
poly(m,d)/|F|.
sub-const??
0
0.5
1
error
13
Almost-Linear Size
size
n
2


[GS02]: random (non-explicit)
set of |F|m(1+o(1)) lines
[BSVW03]: -biased set of
directions

n1+o(1)
n

|F|m¢polylog|F|m lines
constant error
almost
linear??
14
Why Can’t Get Sub-Const. Error?
Main Obstacle in taking
directions = small -biased set SµFm:
y1,y22S
y1
y2
15
Our Idea
Fix large enough subfield HµF.
directions =
m
H
1. Different Hm is not -biased in Fm when HF
2. Short
|H|·|F|o(1) ! |H|m·(|F|m)o(1)
3. Useful Can take F=GF(2r¢k)
4. Natural H=F ! standard testers
16
Our Results
Construction: Plane vs. Point low degree tester
with:
 size
|F|m¢|H|2m
 error
c¢m((1/|H|)1/8 + (md/|F|)1/4)
If m8¿|H|·|F|o(1), then sub-constant error and
almost linear size.
17
Sampling Lemma
Fm
Lemma (Sampling):
Let AµFm.
Pick random z2Fm, y2Hm, l = { z+t¢y | t2F }.




For H=F, follows from pairwise independence
[BSVW03]: when y 2 -biased set : same lemma
with (1/|F| + ) instead of 1/|H|
Holds for any subset HµF
Proof via Fourier analysis
18