The classical Direct-Product Theorem for circuits says
that if a Boolean function $f:\{0,1\}^n\to\{0,1\}$ is somewhat hard
to compute on average by small circuits, then the corresponding
$k$-wise direct product function
$f^k(x_1,\dots,x_k)=(f(x_1),\dots,f(x_k))$ (where each
$x_i\in\{0,1\}^n$) is significantly harder to compute on average by
slightly smaller circuits. We prove a \emph{fully uniform} version of the Direct-Product
Theorem with information-theoretically \emph{optimal} parameters, up
to constant factors. Namely, we show that for given $k$ and
$\epsilon$, there is an efficient randomized
algorithm $A$ with the following property. Given a circuit $C$ that
computes $f^k$ on at least $\epsilon$ fraction of inputs, the
algorithm $A$ outputs with probability at least $3/4$ a list of
$O(1/\epsilon)$ circuits such that at least one of the circuits on
the list computes $f$ on more than $1-\delta$ fraction of inputs,
for $\delta = O((\log 1/\epsilon)/k)$; moreover, each output circuit
is an $\AC^0$ circuit (of size $\poly(n,k,\log 1/\delta,1/\epsilon)$),
with oracle access to the circuit $C$.
Using the Goldreich-Levin decoding algorithm~\cite{GL89}, we also
get a \emph{fully uniform} version of Yao's XOR Lemma~\cite{Yao}
with \emph{optimal} parameters, up to constant factors. Our results
simplify and improve those in~\cite{IJK06}.
Our main result may be viewed as an efficient approximate, local,
list-decoding algorithm for direct-product codes (encoding a
function by its values on all $k$-tuples) with optimal
parameters. We generalize it to a family of ``derandomized"
direct-product codes, which we call {\em intersection codes}, where
the encoding provides values of the function only on a subfamily of
$k$-tuples. The quality of the decoding algorithm is then determined
by sampling properties of the sets in this family and their
intersections. As a direct consequence of this generalization we
obtain the first derandomized direct product result in the uniform
setting, allowing hardness amplification with only constant (as
opposed to a factor of $k$) increase in the input length. Finally,
this general setting naturally allows the decoding of concatenated
codes, which further yields nearly optimal derandomized
amplification.
