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Measures of Pseudorandomness for Finite Sequences: Minimal Values

  • N. ALON (a1), Y. KOHAYAKAWA (a2), C. MAUDUIT (a3), C. G. MOREIRA (a4) and V. RÖDL (a5)...
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 $\cal{N}(E_N)$.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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