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.

翻译：本文研究序列族在两种测度下的伪随机性，即族复杂度（$f$-复杂度）与$\ell$阶互相关测度。我们不仅考虑二元字母表上的序列，还考虑$k$-符号（$k$元）字母表上的序列。首先，我们推广了一些关于二元伪随机序列族构造的已知方法。针对特定不可约多项式的勒让德符号构成的二元序列大族，我们证明了其$f$-复杂度的上界。结果表明，该序列族及其对偶族同时具有较大的族复杂度和较小的互相关测度（直至相当高的阶数）。其次，我们提出另一个具有高$f$-复杂度和低互相关测度的二元序列族。最后，我们将上述结果推广至$k$-符号字母表上的序列族。