In this paper, using p-ary bent functions defined on vector space over the finite field Fpk, we generalized the construction method of the families of p-ary bent sequences with balanced and optimal correlation properties introduced by Kumar and Moreno for an odd prime p, called generalized p-ary bent sequences. It turns out that the family of balanced p-ary sequences with optimal correlation property introduced by Moriuchi and Imamura is a special case of the newly constructed generalized p-ary bent sequences.

Binary maximal-length sequences (or m-sequences) are sequences of period 2m-1 generated by a linear recursion of degree m. Decimating an m-sequence {st} by an integer d relatively prime to 2m-1 leads to another m-sequence {sdt} of the same period. The crosscorrelation of m-sequences has many applications in communication systems and has been an important and well studied problem during more than 40 years. This paper presents an updated survey on the crosscorrelation between binary m-sequences with at most five-valued crosscorrelation and shows some of the many recent connections of this problem to several areas of mathematics such as exponential sums and Dickson polynomials.

Let m ≥ 3 be an odd positive integer, n=2m and p be an odd prime. For the decimation factor $d=rac{(p^{m}+1)(p^{m}+p-1)}{p+1}$, the cross-correlation between the p-ary m-sequence {tr1n(αt)} and its all decimated sequences {tr1n(αdt+l)} is investigated, where 0 ≤ l < gcd(d,pn-1) and α is a primitive element of Fpn. It is shown that the cross-correlation function takes values in {-1,-1±ipm|i=1,2,…p}. The result presented in this paper settles a conjecture proposed by Kim et al. in the 2012 IEEE International Symposium on Information Theory Proceedings paper (pp.1014-1018), and also improves their result.