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Encyclopedia of Algorithms pp 446–450Cite as

Linearity Testing/Testing Hadamard Codes

1990; Blum, Luby, Rubinfeld

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Keywords and Synonyms

Linearity testing;  Testing Hadamard codes ; Homomorphism testing

Problem Definition

This problem is concerned with distinguishing functions that are homomorphisms, i. e. satisfying \( \forall x,y, f(x)+f(y)=f(x+y) \), from those functions that must be changed on at least \( \epsilon \) fraction of the domain in order to be turned into a homomorphism, given query access to the function. This problem was initially motivated by applications to testing programs which compute linear functions [8]. Since Hadamard codes are such that the codewords are exactly the evaluations of linear functions over boolean variables, a solution to this problem gives a way of distinguishing codewords of the Hadamard code from those strings that are far in relative Hamming distance from codewords. These algorithms were in turn used in the constructions of Probabilistically Checkable Proof Systems (cf. [3]). Further work has extended these techniques to testing other properties of low...

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Notes

  1. 1.

    The choice of 1/3 is arbitrary. Using standard techniques, any homomorphism tester satisfying 1/3 error probability can be turned into a homomorphism tester with \( { 0 < \beta< 1/3 } \) error probability by repeating the original tester \( { O(\log{\frac1\beta}) } \) times and taking the majority answer.

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, USA

    Ronitt Rubinfeld

Authors
  1. Ronitt Rubinfeld
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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Rubinfeld, R. (2008). Linearity Testing/Testing Hadamard Codes. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_202

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