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 $d$.)

For codes over a prime field $\F_q$ the optimal query complexity is well-known and known to be $\Theta(q^{\lceil (d+1)/(q-1)\rceil})$, and the test consists of testing if $f$ is a degree-$d$ polynomial on a randomly chosen $(\lceil (d+1)/(q-1) \rceil)$-dimensional affine subspace of $\F_q^n$. If $q$ is not a prime, then the above quantity remains a lower bound, whereas the previously known upper bound grows to $O(q^{\lceil (d+1)/(q - q/p) \rceil})$ where $p$ is the characteristic of the field $\F_q$. 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. Thus for every $p$ and sufficiently large $q$ this bound improves over the previously known bound by a polynomial factor.

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. The main technical contribution of this work is the design of tests that test a function by not querying its value on an entire subspace of the space, but rather on a carefully chosen (algebraically nice) subset of the points from low-dimensional subspaces.

An earlier version of this paper appeared in the Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM'12), pages 639-650, 2012.