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

Let p be a prime number, Z/(p(e)) the integer residue ring, e >= 2. For a sequence a over Z/(p(e)), there is a unique decomposition a = a(0) +a(1) (.) p + (.) (.) (.) +a(e)-1 (.) p(e-1), where a(i) be the sequence over {0, 1,..., p - 1}. Let f(X) is an element of Z/(p(e))[x] be a primitive polynomial of degree n, a and b be sequences generated by f(x) over Z/(p(e)), such that a not equal 0 (mod p(e-1)). This paper shows that the distribution of zero in the sequence a(e-1) = (a(e-1) (t))(t >= 0) contains all information of the original sequence a, that is, if a(e-1) (t) = 0 if and only if b(e-1) (t) = 0 for all t >= 0, then a = b. 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/(p(e)), Finite Fields Appl. 11 (1) (2005) 30-44]. (c) 2006 Elsevier Inc. All rights reserved.