Uniqueness of the distribution of zeroes of primitive level sequences over Z/(pe)

Let p be a prime number, p 5, Z/(pe) the integer residue ring, e 2, Γ={0,1,…,p−1}. For a sequence over Z/(pe), there is a unique decomposition , where be the sequence over Γ. Let f(x) be a primitive polynomial with degree n over Z/(pe), and sequences generated by f(x) over Z/(pe), ; we prove that the distribution of zeroes in the sequence contains all information of the original sequence , that is, if ae−1(t)=0 if and only if be−1(t)=0 for all t 0, then . As a consequence, we have the following results: (i) two different primitive level sequences are linearly independent over Z/(p); (ii) for all positive integer k, if and only if .