1. 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

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

3. Character sums
@+a++%=?

4. 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,…

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

6. 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”

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

8. 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

9. 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

10. 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

11. Going beyond root field size
Bourgain: sparsity ~log(n)
This work: sparsity n (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

12. 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

13. 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

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

15. 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)

16. 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,…]

17. 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

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

19. 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

20. 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

21. Proof overview
@+a++%=!!

22. 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

23. 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

24. Summary and open problems

25. 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

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

27. 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, …

28. Thank you