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.


Boolean Function Query Complexity Property Testing Full Version Tight Bound 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eric Blais
    • 1
  • Daniel Kane
    • 2
  1. 1.School of Computer ScienceCarnegie Mellon UniversityUSA
  2. 2.Department of MathematicsStanford UniversityUSA

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