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.