Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over Z/(pe)

Let Z/(p(e)) be the integer residue ring with odd prime p and integer e >= 2. Any sequence (a) under bar over Z/)p(e)) has a unique p-adic expansion (a) under bar = (a) under bar (0) + (a) under bar (1) . p + ... + (a) under bar (e-1) . p(e-1), where (a) under bar (i) can be regarded as a sequence over Z/(p) for 0 <= i <= e - 1. Let f(x) be a strongly primitive polynomial over Z/(p(e)) and (a) under bar, (b) under bar be two primitive sequences generated by f(x) over Z/(p(e)). Assume phi(x(0), ..., x(e-1)) = x(e-1) + eta(x(0), ..., x(e-2)) is an e-variable function over Z/(p) with the monomial p+1/2x(e-2)(p-1)...x(1)(p-1)x(0)(p-1) not appearing in the expression of eta(x(0), x(1), ..., x(e-2)). It is shown that if there exists an s is an element of Z/(p) such that phi(a(0)(t), ..., a(e-1)(t)) = s if and only if phi(b(0)(t), ..., b(e-1)(t)) = s for all nonnegative t with ,alpha(t) not equal 0, where (alpha) under bar is an m-sequence determined by f(x) and (a) under bar (0), then (a) under bar = (b) under bar. This implies that for compressing sequences derived from primitive sequences generated by f(x) over Z/(p(e)), single element distribution is unique on all positions t with alpha(t) not equal 0. In particular, when eta(x(0), x(1), ..., x(e-2)) = 0, it is a completion of the former result on the uniqueness of distribution of element 0 in highest level sequences.