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MEASURES OF PSEUDORANDOMNESS FOR FINITE SEQUENCES: MINIMAL VALUES
 

Summary: MEASURES OF PSEUDORANDOMNESS FOR FINITE
SEQUENCES: MINIMAL VALUES
N. ALON, Y. KOHAYAKAWA, C. MAUDUIT, C. G. MOREIRA, AND V. R ¨ODL
Dedicated to Professor B´ela Bollob´as on the occasion of his 60th birthday
Abstract. Mauduit and S´ark¨ozy introduced and studied certain nu-
merical parameters associated to finite binary sequences EN {-1, 1}N
in order to measure their `level of randomness'. Two of these parameters
are the normality measure N(EN ) and the correlation measure Ck(EN )
of order k, which focus on different combinatorial aspects of EN . In their
work, amongst others, Mauduit and S´ark¨ozy 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 Ck(EN ) (k even) for arbitrary
sequences EN , establishing one of their conjectures. We also give an
algebraic construction for a sequence EN with small normality mea-
sure N(EN ).
Contents
1. Introduction and statement of results 2
1.1. Typical and minimal values of correlation 3
1.2. Typical and minimal values of normality 5

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics