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The Ramanujan Journal

, Volume 39, Issue 3, pp 639–645 | Cite as

Family complexity and cross-correlation measure for families of binary sequences

  • Arne Winterhof
  • Oğuz Yayla
Article

Abstract

We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al. in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family \((e_{i,1},\ldots ,e_{i,N})\in \{-1,+1\}^N\), \(i=1,\ldots ,F\), of binary sequences of length \(N\) in terms of the cross-correlation measure of its dual family \((e_{1,n},\ldots ,e_{F,n})\in \{-1,+1\}^F\), \(n=1,\ldots ,N\). We apply this result to the family of sequences of Legendre symbols with irreducible quadratic polynomials modulo \(p\) with middle coefficient \(0\), that is, \(e_{i,n}=\big (\frac{n^2-bi^2}{p}\big )_{n=1}^{(p-1)/2}\) for \(i=1,\ldots ,(p-1)/2\), where \(b\) is a quadratic nonresidue modulo \(p\), showing that this family as well as its dual family has both a large family complexity and a small cross-correlation measure up to a rather large order.

Keywords

Pseudorandomness Binary sequences Family complexity  Cross-correlation measure Legendre sequence  Polynomials over finite fields 

Mathematics Subject Classification

11K45 11T24 

Notes

Acknowledgments

The authors thank the anonymous referee for some useful comments.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria

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