A Generalization of the Family of $p$-ary Decimated Sequences With Low Correlation

Let
$p$
be a prime and
$n$
a positive integer. Let
$e\vert p^{n}-1$
and
$N={{p^{n}-1}\over{e}}$
. In this paper, we construct a family
$S$
of
$e^{2}N~p$
-ary sequences, each member of
$S$
has period
$N$
and the magnitudes of correlations of members of
$S$
are upper bounded by
$2\sqrt{p^{n}}=2\sqrt{eN+1}$
.