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

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Over a finite field 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 ⁿ). The task of testing if a function f:ⁿ → 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.) 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.