# Linearity and Group Homomorphism Testing/Testing Hadamard Codes

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## 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 :â€‰{ 0,â€‰1}nâ€‰â†’â€‰{ 0,â€‰1} is linear if for some a 1,â€‰a 2,â€‰â€¦,â€‰a n â€‰âˆˆâ€‰{ 0,â€‰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 $$\mathbb{F}_{2}$$. That is,...

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## Acknowledgements

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

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### Cite this entry

Raskhodnikova, S., Rubinfeld, R. (2015). Linearity and Group Homomorphism Testing/Testing Hadamard Codes. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27848-8_202-2

• DOI: https://doi.org/10.1007/978-3-642-27848-8_202-2

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• Publisher Name: Springer, Berlin, Heidelberg

• Online ISBN: 978-3-642-27848-8

• eBook Packages: Springer Reference Computer SciencesReference Module Computer Science and Engineering