Tight bounds for testing k-linearity

Abstract

The function f : double-struck F 2n → double-struck F 2 is k-linear if it returns the sum (over double-struck F 2) of exactly k coordinates of its input. We introduce strong lower bounds on the query complexity for testing whether a function is k-linear. We show that for any k ≤ n/2, at least k - o(k) queries are required to test k-linearity, and we show that when k ≈ n/2, this lower bound is nearly tight since 4/3k+o(k) queries are sufficient to test k-linearity. We also show that non-adaptive testers require 2k - O(1) queries to test k-linearity. We obtain our results by reducing the k-linearity testing problem to a purely geometric problem on the boolean hypercube. That geometric problem is then solved with Fourier analysis and the manipulation of Krawtchouk polynomials. © 2012 Springer-Verlag.