Sub-Constant Error Low Degree Test of Almost-Linear Size

38th Symposium on Theory Of Computing, Seattle, May 21-23, 2006 Sub-Constant Error Low Degree Testof Almost-Linear Size Dana Moshkovitz Weizmann Institute Ran Raz Weizmann Institute

n s(n) Probabilistically Checkable Proofs [AS92,ALMSS92] Is  satisfiable? NP: size PCP: error • Completeness:sat. )9A, Pr[acc] = 1. • Soundness:notsat. )8A, Pr[acc] ·.

Importance of PCP Theorem • Shows proofs are surprisingly powerful. • Enables hardness of approximation results. • Yields codes with local testing/decoding properties.

Decreasing Error • Prop:error¸1/||#queries • Hence, to decrease error need either: useful for applications easy! via repetition enlarge alphabet || enlarge #queries

Decreasing Error • [AS92,ALMSS92]: constant error PCP. • [ArSu97,RaSa97,DFKRS99]: sub-const error PCP (taking super-const ||). sub-const?? error o(1) 0.5

Decreasing Size size • [AS92,ALMSS92]: size=nc for large constant c. • [GS02,BSVW03,BGHSV04]: almost-linear sizen1+o(1) PCPs • [Dinur06] (based on [BS05]): size=n¢polylog n • Only constant error! nc n1+o(1) n almost linear??

Our Motivation size Want: PCP with both • sub-constant error and • almost-linear size nc almost linear?? n1+o(1) n sub-const?? error o(1)

Our Work We show: Low Degree Tester: • sub-constant error • almost-linear size Historically: Low Degree Tester(non-trivially))PCP

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|m= input size;|F|Àd,m] f : Fm!F Q(x1,…,xm) deg Q ·d

Low Degree Tester size f : Fm! F • toss coins • query f and A in O(1) places • accept/reject A low degree tester Completeness:f is deg ·d poly ! 9Aalways accept Soundness: for any A, if Pr[accept]>0(=error), 'Pr[accept] agreement of f (with poly of deg · d)

y z The Line Vs. Point Test [Rubinfeld,Sudan] Observation:FixQ:Fm!F, degQ·d. Restriction of Q to any line z+t¢y (for z,y2Fm), namely Q(z+t¢y), is a univariate polynomial of deg ·d. Moreover, this characterizesm-variate polynomials over F of deg·d. The proof A: for every line l, A(l) = univariate poly of deg·d[allegedrestriction of f to l]

Pick random z2Fm, y2Fm • Check A()(z) = f(z) y z The Line Vs. Point Test [Rubinfeld,Sudan] • makes two queries • clearly complete • quadratic size ¼|Fm|2 • [RuSu90,AS92,ALMSS92,FS93,PS94]: error <1.

Sub-Constant Error • [RaSa97]:Plane vs. Point has sub-const errorpoly(m,d)/|F|. • [ArSu97]: Line vs. Point has sub-const errorpoly(m,d)/|F|. sub-const?? error 1 0.5 0

Almost-Linear Size size • [GS02]: random (non-explicit)set of |F|m(1+o(1)) lines • [BSVW03]: -biased set of directions • |F|m¢polylog|F|m lines • constant error n2 n1+o(1) n almost linear??

y1 y2 Why Can’t Get Sub-Const. Error? Main Obstacle in taking directions =small-biased set SµFm: y1,y22S

Our Idea Fix large enough subfieldHµF. directions = Hm • DifferentHm is not -biased in Fm when HF • Short|H|·|F|o(1)!|H|m·(|F|m)o(1) • Useful Can take F=GF(2r¢k) • NaturalH=F ! standard testers

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) Ifm8¿|H|·|F|o(1), then sub-constant error and almost linear size.

Fm Sampling Lemma 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 subsetHµF • Proof via Fourier analysis

Sub-Constant Error Low Degree Test of Almost-Linear Size