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

AbstractLet p be a prime number, Z/(pe) the integer residue ring, e⩾2. For a sequence a̲ over Z/(pe), there is a unique decomposition a̲=a̲0+a̲1⋅p+⋯+a̲e−1⋅pe−1, where a̲i be the sequence over {0,1,…,p−1}. Let f(x)∈Z/(pe)[x] be a primitive polynomial of degree n, a̲ and b̲ be sequences generated by f(x) over Z/(pe), such that a̲≠0̲(modpe−1). This paper shows that the distribution of zero in the sequence a̲e−1=(ae−1(t))t⩾0 contains all information of the original sequence a̲, that is, if ae−1(t)=0 if and only if be−1(t)=0 for all t⩾0, then a̲=b̲. 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]