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

摘要

Over a finite field Fq the (ndq) -Reed-Muller code is the code given by evaluations of n-variate polynomials of total degree at most d on all points (of Fnq). The task of testing if a function f:FnqFq is close to a codeword of an (ndq) -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 -far from the code with probability (). (In this work we allow the constant in the to depend on d.) For codes over a prime field Fq the optimal query complexity is well-known and known to be (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 Fnq. 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?qp)) where p is the characteristic of the field Fq. In this work we give a new upper bound of (cq)(d+1)q on the query complexity, where c is a universal constant. Thus for every p and sufficiently large constant q this bound improves over the previously known bound by a polynomial factor, as we let d. 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 {em 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.