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Measures of Pseudorandomness: Arithmetic Autocorrelation and Correlation Measure

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Number Theory – Diophantine Problems, Uniform Distribution and Applications

Abstract

We prove a relation between two measures of pseudorandomness, the arithmetic autocorrelation, and the correlation measure of order k. Roughly speaking, we show that any binary sequence with small correlation measure of order k up to a sufficiently large k cannot have a large arithmetic correlation. We apply our result to several classes of sequences including Legendre sequences defined with polynomials.

Dedicated to Robert F. Tichy on the occasion of his 60th birthday

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Acknowledgements

The authors are partially supported by the Austrian Science Fund FWF Project 5511-N26 which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications.”

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Str. 69, 4040, Linz, Austria

    Richard Hofer, László Mérai & Arne Winterhof

Authors
  1. Richard Hofer
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  2. László Mérai
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  3. Arne Winterhof
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Corresponding author

Correspondence to Arne Winterhof .

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Editors and Affiliations

  1. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Christian Elsholtz

  2. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Peter Grabner

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Hofer, R., Mérai, L., Winterhof, A. (2017). Measures of Pseudorandomness: Arithmetic Autocorrelation and Correlation Measure. In: Elsholtz, C., Grabner, P. (eds) Number Theory – Diophantine Problems, Uniform Distribution and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-55357-3_15

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