### Abstract

Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences E_{N} ∈ {-1,1}^{N} in order to measure their 'level of randomness'. Two of these parameters are the normality measure 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 N(E_{N}).

Original language | English (US) |
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Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Combinatorics Probability and Computing |

Volume | 15 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2006 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Combinatorics Probability and Computing*,

*15*(1-2), 1-29. https://doi.org/10.1017/S0963548305007170