In 2002 Mauduit and Sárközy started to study finite sequences of $k$ symbols

$$E_N=\left(e_1,e_2,\cdots ,e_N\right)\in {\mathcal{A}}^N,$$

where

$$\mathcal{A}=\left\{a_1,a_2,\cdots ,a_k\right\},(k\in \mathbb{N},k\ge 2)$$

is a finite set of $k$ symbols. Bérczi estimated the pseudorandom measures for a truly random sequence $E_N$ of $k$ symbol. In this paper, we shall study the minimal values of correlation measures for the sequences of $k$ symbols, developing the methods similar to those introduced by Alon, Anantharam, Gyarmati, and Schmidt, among others.

We clarify a relation between the arithmetic autocorrelation and pattern distribution of binary sequences, then we apply the relation to study the upper bound of arithmetic autocorrelation for two binary sequences constructed by Fermat quotient and the generalized cyclotomic class of order $2$, respectively. Our results indicate that the sequences with large "long term" correlations may have small "short term" pattern distribution; and thus have rather small arithmetic autocorrelations.

Sequences with optimal autocorrelation properties play an important role in wireless communication, radar and cryptography. Interleaving is a very important method in constructing the optimal autocorrelation sequence. Tang and Gong gave three different constructions of interleaved sequences (generalized GMW sequences, twin prime sequences and Legendre sequences). Su et al. constructed a series of sequences with optimal autocorrelation magnitude via interleaving Ding-Helleseth-Lam sequences. In this paper we further study the correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences.

In 1997, Mauduit and Sárközy first introduced the measures of pseudorandomness for binary sequences. Since then, many pseudorandom binary sequences have been constructed and studied. In particular, Gyarmati presented a large family of pseudorandom binary sequences using the discrete logarithms. Ten years later, to satisfy the requirement from many applications in cryptography (e.g., in encrypting "bit-maps'' and watermarking), the definition of binary sequences is extended from one dimension to several dimensions by Hubert, Mauduit and Sárközy. They introduced the measure of pseudorandomness for this kind of several-dimension binary sequence which is called binary lattices. In this paper, large families of pseudorandom binary sequences and binary lattices are constructed by both discrete logarithms and multiplicative inverse modulo $p$. The upper estimates of their pseudorandom measures are based on estimates of either character sums or mixed exponential sums.