  Over a finite field $\F_q$ the $(n,d,q)$-Reed-Muller code is the code given
by evaluations of $n$-variate polynomials of total degree at most $d$ on all
points (of $\F_q^n$). The task of testing if a function $f:\F_q^n \to \F_q$ is
close to a codeword of an $(n,d,q)$-Reed-Muller code has been of central
interest in complexity theory and property testing. The query complexity of
this task is the minimal number of queries that a tester can make (minimum over
all testers of the maximum number of queries over all random choices) while
accepting all Reed-Muller codewords and rejecting words that are $\delta$-far
from the code with probability $\Omega(\delta)$. (In this work we allow the
constant in the $\Omega$ to depend on $d$.) In this work we give a new upper
bound of $(c q)^{(d+1)/q}$ on the query complexity, where $c$ is a universal
constant. In the process we also give new upper bounds on the "spanning weight"
of the dual of the Reed-Muller code (which is also a Reed-Muller code). The
spanning weight of a code is the smallest integer $w$ such that codewords of
Hamming weight at most $w$ span the code.
