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

Let p be a prime number, Z/(pe) the integer residue ring, e⩾2. For a sequence over Z/(pe), there is a unique decomposition , where be the sequence over {0,1,…,p−1}. Let f(x)∈Z/(pe)[x] be a primitive polynomial of degree n, and be sequences generated by f(x) over Z/(pe), such that . This paper shows that the distribution of zero 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 . Here we mainly consider the case of p=3 and the techniques used in this paper are very different from those we used for the case of p⩾5 in our paper [X.Y. Zhu, W.F. Qi, Uniqueness of the distribution of zeroes of primitive level sequences over Z/(pe), Finite Fields Appl. 11 (1) (2005) 30–44].