  Curve samplers are sampling algorithms that proceed by viewing the domain as
a vector space over a finite field, and randomly picking a low-degree curve in
it as the sample. Curve samplers exhibit a nice property besides the sampling
property: the restriction of low-degree polynomials over the domain to the
sampled curve is still low-degree. This property is often used in combination
with the sampling property and has found many applications, including PCP
constructions, local decoding of codes, and algebraic PRG constructions.
  The randomness complexity of curve samplers is a crucial parameter for its
applications. It is known that (non-explicit) curve samplers using $O(\log
N+\log(1/\delta))$ random bits exist, where $N$ is the domain size and $\delta$
is the confidence error. The question of explicitly constructing
randomness-efficient curve samplers was first raised in \cite{TU06} where they
obtained curve samplers with near-optimal randomness complexity.
  We present an explicit construction of low-degree curve samplers with {\em
optimal} randomness complexity (up to a constant factor), sampling curves of
degree $\left(m\log_q(1/\delta)\right)^{O(1)}$ in $\mathbb{F}_q^m$. Our
construction is a delicate combination of several components, including
extractor machinery, limited independence, iterated sampling, and
list-recoverable codes.
