A new upper bound on the query complexity for testing generalized Reed-Muller codes

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.