Distribution of elements on cycles of linear recurrences over Galois field

Let nu be a linear recurring sequence (LRS) over the field P of q = p(s) elements, which has irreducible characteristic polynomial of degree m and period Delta = q(m)-1/d, where d epsilon Z divides q(m) -1 and p(t) + 1 for some t epsilon N. For this sequence nu we described possible frequences of appearance of element a epsilon P in the cycle (nu(0), nu(1),..., nu(Delta - 1)). The proofs are based on properties of Gauss sums and generalize the results of works [2, 3].