# Encyclopedia of Algorithms

Living Edition
| Editors: Ming-Yang Kao

# Linearity and Group Homomorphism Testing/Testing Hadamard Codes

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27848-8_202-2

## 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,  ,  a n ∈ { 0, 1},
$$\displaystyle{f(x_{1},x_{2},\ldots ,x_{n}) = a_{1}x_{1} + a_{2}x_{2} + \cdots a_{n}x_{n}.}$$

## Keywords

Property testing Sublinear-time algorithms Linearity of functions Group homomorphism Error-correcting codes
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## Notes

### Acknowledgements

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

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