Tali Kaufman (Bar-Ilan) New extension of the Weil bound for character sums, and applications to coding Tali Kaufman (Bar-Ilan) Shachar Lovett (IAS) FOCS, October 2011

Talk overview Weil bound for character sums Our new extension for the Weil bound Application: testing of affine invariant codes Proof overview Open problems

Character sums @+a++%=?

Weil bound for character sums Formal way of saying: Low-degree polynomials are either: very structured; or behave like random functions Extremely useful tool in Number theory, analysis, algebra,… Explicit constructions, derandomization, lower bounds, coding theory,…

Additive characters – finite field Additive character: map such that Example: prime field Example: non-prime field

Weil bound for character sums f(x) - degree d polynomial over additive character Then either (1) is constant for all or (2) is “uniformly distributed”

Weil bound: examples Example 1: squares are “uniform” in Example 2: f(x) lies in a subspace of co-dim 1 c≠0,

Weil bound for character sums f(x) degree d polynomial is constant; or Many generalizations: Multi-variate polynomials (Deligne), polynomials on curves (Bombieri),… Main limitation: useful only when

Going beyond root field size We can go beyond degree if we restrict ourselves to sparse polynomials Bourgain: Let f(x) be a sparse polynomial of degree . Then either is constant or

Going beyond root field size We consider polynomials over (or ) Degree of xe is e Weight degree of xe is the hamming weight of e (equiv: degree as n-variate polynomial over ) We show: Let f(x) be a sparse polynomial of low weight degree. Then either is constant or

Going beyond root field size Bourgain: sparsity ~log(n) This work: sparsity n0.1 (assumes d=O(log n)) Consider polynomials over (or ) Degree of xe is e Weight degree of xe is the hamming weight of e (equiv: degree as polynomial over ) We show: Let f(x) be a sparse polynomial of low weight degree. Then either is constant or

Good subcodes of Reed-Muller Reed-Muller code: multivariate polynomials in of degree d Equivalently: univariate polynomials in of weight degree d We show: For d=O(log n), sub-code of sparse polynomials with n0.1 monomials forms a (non-linear) code with near optimal distance

Going beyond root field size We can actually show a stronger result, combining Weil bound with sparse polynomials Let f(x)=g(x)+h(x) where g(x) has degree h(x) is sparse, low weight degree Then either is constant or

Character sums in coding theory @+a++%=!

Locally testable codes Codes where codewords can (whp) be verified to be (nearly) correct with only a few queries Combinatorial heart of many PCP constructions Big question: can locally testable codes form “good codes” (constant rate & distance)

Locally testable codes Constructions: Hadamard [BLR] Polynomials [Alon-Kaufman-Krivelevich-Litsyn-Ron, Arora-Sudan, Samorodnitsky, Raz-Safra, Moshkovitz-Raz,…] Combinatorial [Dinur-Goldenberg, Imagliazzo-Kabanets-Wigderson,…]

Locally testable codes General families of codes which guarantee local testability: Near orthogonal codes [Kaufman-Litsyn] Random sparse codes [Kaufman-Sudan, Kopparty-Saraf] Sparse affine invariant codes [Grigorescu-Kaufman-Sudan] We show: slightly less sparse affine invariant codes are still locally testable

Affine invariant codes Codeword: function Code: linear subspace Code is affine invariant if it is invariant under affine transformations of the input, i.e.

Testing of affine invariant codes Grigorescu-Kaufman-Sudan: Any affine invariant code with exp(n) codewords (i.e. sparse) can be locally testable with O(1) queries Use Bourgain character sum estimate for sparse polynomials

Testing of affine invariant codes This work: Any affine invariant code with exp(n1+) codewords (slightly less sparse) can be locally testable with poly(n) queries Use the fact that our character sum estimate works for slightly less sparse polynomials

Proof overview @+a++%=!!

Main theorem f(x)=g(x)+h(x) polynomial over Then either is constant or g(x) has degree h(x) sparse polynomial of low weight degree Then either is constant or

Main proof ideas Use derivatives + convexity to reduce degrees and wt deg f(x)=g(x)+h(x) (g – low degree, h – sparse, low weight deg) Case 1: wt-deg(g)>wt-deg(h): (less interesting case) Eliminate h by derivatives Use Weil theorem for derivatives of g Case 2: wt-deg(g)<wt-deg(h): Eliminate g by derivatives Convert h to multi-linear Use sparsity of h to “shift” it and make it have low-degree Analyze using minimal distance of low-degree polynomials Case 3: wt-deg(g)=wt-deg(h) Similar to 2, but only partially eliminate g

Summary and open problems

Summary Weil bound: estimates for character sums evaluated on low-degree polynomials Main limitation: degree bound We give a new extension, allowing a few monomials to have high degree (but low weight degree) Application to testing affine invariant codes

Open problems 1: Weil bound Extend the Weil bound to interesting families of polynomials of degree Applications Number theory Cryptography Coding Explicit constructions ?

Open problems 2: Succinctness P vs NP is really a question about succinctness Sparse polynomials: succinct algebraic circuits Bourgain, this work,…: character sum estimates for sparse polynomials which are not true for non-sparse polynomials L.-Srinivasan: correlation bounds for sparse polys Extend to less sparse polynomials, more general algebraic circuits, boolean circuits, …

Thank you