In their seminal work, Chattopadhyay and Zuckerman (STOC'16) constructed a two-source extractor with error $\varepsilon$ for $n$-bit sources having min-entropy $poly\log(n/\varepsilon)$. Unfortunately, the construction running-time is $poly(n/\varepsilon)$, which means that with polynomial-time constructions, only polynomially-large errors are possible. Our main result is a $poly(n,\log(1/\varepsilon))$-time computable two-source condenser. For any $k ... 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 >>>

We show a reduction from the existence of explicit t-non-malleable
extractors with a small seed length, to the construction of explicit
two-source extractors with small error for sources with arbitrarily
small constant rate. Previously, such a reduction was known either
when one source had entropy rate above half [Raz05] or ... more >>>

We construct explicit two-source extractors for $n$ bit sources,
requiring $n^\alpha$ min-entropy and having error $2^{-n^\beta}$,
for some constants $0 < \alpha,\beta < 1$. Previously, constructions
for exponentially small error required either min-entropy
$0.49n$ \cite{Bou05} or three sources \cite{Li15}. The construction
combines somewhere-random condensers based on the Incidence
Theorem \cite{Zuc06,Li11}, ... more >>>

We explicitly construct extractors for two independent $n$-bit sources of $(\log n)^{1+o(1)}$ min-entropy. Previous constructions required either $\mathrm{polylog}(n)$ min-entropy \cite{CZ15,Meka15} or five sources \cite{Cohen16}.

Our result extends the breakthrough result of Chattopadhyay and Zuckerman \cite{CZ15} and uses the non-malleable extractor of Cohen \cite{Cohen16}. The main new ingredient in our construction ... more >>>

A subspace-evasive set over a field ${\mathbb F}$ is a subset of ${\mathbb F}^n$ that has small intersection with any low-dimensional affine subspace of ${\mathbb F}^n$. Interest in subspace evasive sets began in the work of Pudlák and Rödl (Quaderni di Matematica 2004). More recently, Guruswami (CCC 2011) showed that ... more >>>

A $(k,\epsilon)$-biased sample space is a distribution over $\{0,1\}^n$ that $\epsilon$-fools every nonempty linear test of size at most $k$. Since they were introduced by Naor and Naor [SIAM J. Computing, 1993], these sample spaces have become a central notion in theoretical computer science with a variety of applications.

When ... more >>>

We show a generic, simple way to amplify the error-tolerance of locally decodable codes.
Specifically, we show how to transform a locally decodable code that can tolerate a constant fraction of errors
to a locally decodable code that can recover from a much higher error-rate. We also show how to ... more >>>

Recently Efremenko showed locally-decodable codes of sub-exponential
length. That result showed that these codes can handle up to
$\frac{1}{3} $ fraction of errors. In this paper we show that the
same codes can be locally unique-decoded from error rate
$\half-\alpha$ for any $\alpha>0$ and locally list-decoded from
error rate $1-\alpha$ ... more >>>