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 ... more >>>

We present a Fourier-analytic approach to list-decoding Reed-Muller codes over arbitrary finite fields. We prove that the list-decoding radius for quadratic polynomials equals $1 - 2/q$ over any field $F_q$ where $q > 2$. This confirms a conjecture due to Gopalan, Klivans and Zuckerman for degree $2$. Previously, tight bounds ... more >>>

A distance estimator is a code together with a randomized algorithm. The algorithm approximates the distance of any word from the code by making a small number of queries to the word. One such example is the Reed-Muller code equipped with an appropriate algorithm. It has polynomial length, polylogarithmic alphabet ... more >>>

Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been ... more >>>

We consider the problem of testing if a given function $f : \F_q^n \rightarrow \F_q$ is close to a $n$-variate degree $d$ polynomial over the finite field $\F_q$ of $q$ elements. The natural, low-query, test for this property would be to pick the smallest dimension $t = t_{q,d}\approx d/q$ such ... more >>>

Decomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The Goldreich-Levin algorithm can be viewed as an algorithmic analogue of such a decomposition as it ... more >>>

We develop new techniques to incorporate the recently proposed ``short code" (a low-degree version of the long code) into the construction and analysis of PCPs in the classical ``Label Cover + Fourier Analysis'' framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs ... more >>>

Locally testable codes (LTCs) of constant distance that allow the tester to make a linear number of queries have become the focus of attention recently, due to their elegant connections to hardness of approximation. In particular, the binary Reed-Muller code of block length $N$ and distance $d$ is known to ... more >>>

We study the following natural question on random sets of points in $\mathbb{F}_2^m$:

Given a random set of $k$ points $Z=\{z_1, z_2, \dots, z_k\} \subseteq \mathbb{F}_2^m$, what is the dimension of the space of degree at most $r$ multilinear polynomials that vanish on all points in $Z$?

We ... more >>>

We prove that random low-degree polynomials (over $\mathbb{F}_2$) are unbiased, in an extremely general sense. That is, we show that random low-degree polynomials are good randomness extractors for a wide class of distributions. Prior to our work, such results were only known for the small families of (1) uniform sources, ... more >>>