Injectivity w.r.t. Distribution of Elements in the Compressed Sequences Derived from Primitive Sequences over $Z/p^eZ$

Let p ≥ 3 be a prime and e ≥ 2 an integer. Denote σ(x) as a primitive polynomial of degree n over Z/pZ, and G as the set of primitive linear recurring sequences generated by σ(x). A map ψ on Z/pZ naturally induces a map ψ̂ on G, mapping a sequence (. . . , st−1, st, st+1, . . . ) to (. . . , ψ(st−1), ψ(st), ψ(st+1), . . . ). Previous results constructed special maps inducing injective maps on G. Comparatively, for most primitive polynomials, injectivity of any induced map ψ̂ on G is determined in this article. Furthermore, provided with ( x n −1 − 1 )2 /p 6≡ a mod (p, σ(x)) for any a ∈ Z/pZ, a lower bound is given for the number of maps from Z/pZ to a finite set which induce injective maps on G. Additionally, three families of maps on Z/pZ are shown to induce injective maps on G, improving previous results.