Measures of Pseudorandomness for Finite Sequences: Minimal Values

  • Authors:

  • N. Alon;Y. Kohayakawa;C. Mauduit;C. G. Moreira;V. Rödl

  • Affiliations:

  • Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: noga@math.tau.ac.il);Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo, Brazil (e-mail: yoshi@ime.usp.br);Institut de Mathématiques de Luminy, CNRS-UPR9016, 163 av. de Luminy, case 907, F-13288, Marseille Cedex 9, France (e-mail: mauduit@iml.univ-mrs.fr);IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil (e-mail: gugu@impa.br);Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (e-mail: rodl@mathcs.emory.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2006

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Abstract

Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences $E_N\in\{-1,1\}^N$ in order to measure their ‘level of randomness’. Two of these parameters are the normality measure $\cal{N}(E_N)$ and the correlation measure $C_k(E_N)$ of order k, which focus on different combinatorial aspects of $E_N$. In their work, amongst others, Mauduit and Sárközy investigated the minimal possible value of these parameters.In this paper, we continue the work in this direction and prove a lower bound for the correlation measure $C_k(E_N)$ (k even) for arbitrary sequences $E_N$, establishing one of their conjectures. We also give an algebraic construction for a sequence $E_N$ with small normality measure $\cal{N}(E_N)$.