Depth, ultimate period, and distribution of sequences of period pr ?? 1

In this paper, by investigating vector s ∈ F_q of length n (or equivalent sequences of period n) with infinite third depth, and the cyclic-left-shift-difference operator E-1 on s, an ultimate period sequence {(E-1)ⁱ(s)}i≥0 is constructed. Some upper bounds on the ultimate period of {(E - 1)ⁱ(s)}i≥0 and a method to determine the least ultimate period are provided. Furthermore, distributions of vectors s with period n = p^r - 1 (r > 0), are described in terms of the least ultimate periods of their respective sequences {(E - 1)ⁱ(s){i≥0.