The main contribution of this work is a new type of graph product, which we call the zig-zag
product. Taking a product of a large graph with a small graph, the resulting graph inherits
(roughly) its size from the large one, its degree from the small one, and ... more >>>

Optimal dispersers have better dependence on the error than
optimal extractors. In this paper we give explicit disperser
constructions that beat the best possible extractors in some
parameters. Our constructions are not strong, but we show that
having such explicit strong constructions implies a solution
to the Ramsey graph construction ... more >>>

We present new explicit constructions of *deterministic* randomness extractors, dispersers and related objects. We say that a
distribution $X$ on binary strings of length $n$ is a
$\delta$-source if $X$ assigns probability at most $2^{-\delta n}$
to any string of length $n$. For every $\delta>0$ we construct the
following poly($n$)-time ... more >>>

Let $F$ be the field of $q$ elements, where $q=p^{\ell}$ for prime $p$. Informally speaking, a polynomial source is a distribution over $F^n$ sampled by low degree multivariate polynomials. In this paper, we construct extractors for polynomial sources over fields of constant size $q$ assuming $p \ll q$.

More generally, ... more >>>

In this paper we motivate the study of Boolean dispersers for quadratic varieties by showing that an explicit construction of such objects gives improved circuit lower bounds. An $(n,k,s)$-quadratic disperser is a function on $n$ variables that is not constant on any subset of $\mathbb{F}_2^n$ of size at least $s$ ... more >>>

A code $\mathcal{C}$ is $(1-\tau,L)$ erasure list-decodable if for every codeword $w$, after erasing any $1-\tau$ fraction of the symbols of $w$,
the remaining $\tau$-fraction of its symbols have at most $L$ possible completions into codewords of $\mathcal{C}$.
Non-explicitly, there exist binary $(1-\tau,L)$ erasure list-decodable codes having rate $O(\tau)$ and ... more >>>

The best known circuit lower bounds against unrestricted circuits remained around $3n$ for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than $5n$. In this work, we suggest a first non-gate-elimination approach for ... more >>>

We identify a new notion of pseudorandomness for randomness sources, which we call the average bias. Given a distribution $Z$ over $\{0,1\}^n$, its average bias is: $b_{\text{av}}(Z) =2^{-n} \sum_{c \in \{0,1\}^n} |\mathbb{E}_{z \sim Z}(-1)^{\langle c, z\rangle}|$. A source with average bias at most $2^{-k}$ has min-entropy at least $k$, and ... more >>>

For $S\subseteq \mathbb{F}^n$, consider the linear space of restrictions of degree-$d$ polynomials to $S$. The Hilbert function of $S$, denoted $\mathrm{h}_S(d,\mathbb{F})$, is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets $S$ of arbitrary finite grids in $\mathbb{F}^n$ ... more >>>