On periodic sequences from GF(q)

AbstractIf s(t) is a periodic sequence from GF(q) = F, and if N is the number of times a non-zero element from F appears in a period of s, we present a theorem which says N ≡ 0(mod pϵ), where ϵ is an integer which depends upon the support of the Fourier transform of s. An easy corollary deals with G(f), the set of all sequences from F which satisfy the linear recurrence with characteristic polynomial f ϵ F[x]. It says that, for all s ϵ G(f), N ≡ 0(mod pϵ), where now ϵ depends upon the smallest integer ω for which it is possible to write 1 as a product of ω conjugates of roots of f