Living Reference Work Entry

Encyclopedia of Algorithms

pp 1-6

# Linearity and Group Homomorphism Testing/Testing Hadamard Codes

• Sofya RaskhodnikovaAffiliated withComputer Science and Engineering Department, Pennsylvania State University Email author
• , Ronitt Rubinfeld

### Keywords

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

## 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 a1, a2, , an ∈ { 0, 1}, ...

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