431 research outputs found
Families of sequences with good family complexity and cross-correlation measure
In this paper we study pseudorandomness of a family of sequences in terms of
two measures, the family complexity (-complexity) and the cross-correlation
measure of order . We consider sequences not only on binary alphabet but
also on -symbols (-ary) alphabet. We first generalize some known methods
on construction of the family of binary pseudorandom sequences. We prove a
bound on the -complexity of a large family of binary sequences of
Legendre-symbols of certain irreducible polynomials. We show that this family
as well as its dual family have both a large family complexity and a small
cross-correlation measure up to a rather large order. Next, we present another
family of binary sequences having high -complexity and low cross-correlation
measure. Then we extend the results to the family of sequences on -symbols
alphabet.Comment: 13 pages. Comments are welcome
Maximum-order Complexity and Correlation Measures
We estimate the maximum-order complexity of a binary sequence in terms of its
correlation measures. Roughly speaking, we show that any sequence with small
correlation measure up to a sufficiently large order cannot have very small
maximum-order complexity
The correlation measures of finite sequences: limiting distributions and minimum values
Three measures of pseudorandomness of finite binary sequences were introduced
by Mauduit and S\'ark\"ozy in 1997 and have been studied extensively since
then: the normality measure, the well-distribution measure, and the correlation
measure of order r. Our main result is that the correlation measure of order r
for random binary sequences converges strongly, and so has a limiting
distribution. This solves a problem due to Alon, Kohayakawa, Mauduit, Moreira,
and R\"odl. We also show that the best known lower bounds for the minimum
values of the correlation measures are simple consequences of a celebrated
result due to Welch, concerning the maximum nontrivial scalar products over a
set of vectors.Comment: 19 pages, this version contains small changes taking into account
referee comment
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