  In this paper we study pseudorandomness of a family of sequences in terms of
two measures, the family complexity ($f$-complexity) and the cross-correlation
measure of order $\ell$. We consider sequences not only on binary alphabet but
also on $k$-symbols ($k$-ary) alphabet. We first generalize some known methods
on construction of the family of binary pseudorandom sequences. We prove a
bound on the $f$-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 $f$-complexity and low cross-correlation
measure. Then we extend the results to the family of sequences on $k$-symbols
alphabet.
