Revision #1 Authors: Ido Ben-Eliezer, Rani Hod, Shachar Lovett

Accepted on: 3rd December 2008 00:00

Downloads: 4772

Keywords:

We study the problem of how well a typical multivariate polynomial

can be approximated by lower degree polynomials over~$\F$. We prove

that, with very high probability, a random degree~$d+1$ polynomial

has only an exponentially small correlation with all polynomials of

degree~$d$, for all degrees~$d$ up to $\Theta(n)$. That is, a random

degree~$d+1$ polynomial does not admit a good approximation of lower

degree. In order to prove this, we provide far tail estimates on the

distribution of the bias of a random low degree polynomial.

Recently, several results regarding the weight distribution of

Reed--Muller codes were obtained. Our results can be interpreted as

a new large deviation bound on the weight distribution of

Reed--Muller codes.

TR08-080 Authors: Ido Ben-Eliezer, Rani Hod, Shachar Lovett

Publication: 10th September 2008 00:55

Downloads: 3000

Keywords:

We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over $\F$.

We prove that, with very high probability, a random degree $d$ polynomial has only an exponentially small correlation with all polynomials of degree $d-1$, for all degrees $d$ up to $\Theta(n)$.

That is, a random degree $d$ polynomial does not admit good approximations of lesser degree.

In order to prove this, we prove far tail estimates on the distribution of the bias of a random low degree polynomial.

As part of the proof, we also prove tight lower bounds on the dimension of truncated Reed--Muller codes.