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

Over a finite field double-struck 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 double-struck F _q ⁿ). The task of testing if a function f : double-struck F _q ⁿ → double-struck 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 δ-far from the code with probability Ω(δ). (In this work we allow the constant in the Ω to depend on d.) For codes over a prime field double-struck F _q the optimal query complexity is well-known and known to be Θ(q ^{⌈(d+1)/(q-1)}⌉), and the test consists of testing if f is a degree d polynomial on a randomly chosen (⌉(d + 1)/(q - 1)⌈)-dimensional affine subspace of double-struck 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⌈(d+1)/(q-q/p)⌉}) where p is the characteristic of the field double-struck 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.