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

Over a finite field ${\text{F}}_q$ 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 ${\text{F}}_q^n$). The task of testing if a function $f:{\text{F}}_q^n\to {\text{F}}_q$ 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 $\mathrm{\Omega }(\delta )$. (In this work we allow the constant in the $\mathrm{\Omega }$ to depend on $d$.) In this work we give a new upper bound of $(cq{)}^{(d+1)\mathrm{/}q}$ 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.