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Encyclopedia of Algorithms pp 1107–1110Cite as

Linearity Testing/Testing Hadamard Codes

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  • First Online:

Years and Authors of Summarized Original Work

  • 1993; Blum, Luby, Rubinfeld

Problem Definition

In this article, we discuss the problem of testing linearity of functions and, more generally, testing whether a given function is a group homomorphism. An algorithm for this problem, given by [9], is one of the most celebrated property testing algorithms. It is part of or is a special case of many important property testers for algebraic properties. Originally designed for program checkers and self-correctors, it has found uses in probabilistically checkable proofs (PCPs), which are an essential tool in proving hardness of approximation.

We start by formulating an important special case of the problem, testing the linearity of Boolean functions. A function f : { 0, 1}n → { 0, 1} is linear if for some \(a_{1},a_{2},\ldots,a_{n} \in \{ 0,1\}\),

$$\displaystyle{f(x_{1},x_{2},\ldots,x_{n}) = a_{1}x_{1} + a_{2}x_{2} + \cdots a_{n}x_{n}.}$$

The operations in this definition are over \(\mathbb{F}_{2}\)...

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Acknowledgements

The first author was supported in part by NSF award CCF-1422975 and by NSF CAREER award CCF-0845701.

Author information

Authors and Affiliations

  1. Computer Science and Engineering Department, Pennsylvania State University, University Park, State College, PA, USA

    Sofya Raskhodnikova

  2. Massachusetts Institute of Technology (MIT), Cambridge, MA, USA

    Ronitt Rubinfeld

  3. Tel Aviv University, Tel Aviv-Yafo, Israel

    Ronitt Rubinfeld

Authors
  1. Sofya Raskhodnikova
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  2. Ronitt Rubinfeld
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Raskhodnikova, S., Rubinfeld, R. (2016). Linearity Testing/Testing Hadamard Codes. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_202

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