Tight Bounds for Testing k-Linearity

Abstract

The function $f : \mathbb{F}_2^n \to \mathbb{F}_2$ is k-linear if it returns the sum (over $\mathbb{F}_2$) 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 $k \le \frac n2$, at least k − o(k) queries are required to test k-linearity, and we show that when $k \approx \frac n2$, this lower bound is nearly tight since $\frac43 k + o(k)$ 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.