Binary and ternary sequences with a few cross correlations

Let m be a positive integer, r ≡ 1 (mod 3) be a prime number, and the order of p modulo r m be ϕ ( r m ) 3 , where ϕ is the Euler function. Let q = p ϕ ( r m ) 3 and d = q − 1 r m . We investigate the cross correlation distribution between a p -ary m -sequence and its d -decimated sequences. In this paper, we deal with the cases of p = 2 and p = 3. Our results show that the binary sequences have two-valued cross correlations and the ternary sequences have at most three-valued cross correlations, see Theorems 3.2 and 4.2. As a byproduct, we also explicitly compute the Gauss periods η 0 ( q − 1 r m , q ) .