Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over $BBZ /(p^{e})$

Let Z/(p^e) be the integer residue ring with odd prime p and integer e ¿ 2. Any sequence a over Z/(p^e) has a unique p-adic expansion a = a₀ + a₁ · p + ··· + a_e-1 · p^e-1, where a_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, b be two primitive sequences generated by f(x) over Z/(p^e). Assume ¿(x₀,..., x_e-1) = x_e-1 + ¿(x₀,..., x_e-2) is an e-variable function over Z/(p) with the monomial (p+1)/2 x_e-2 ^p-1 ...x₁ ^p-1 not pearing in the expression of ¿(x₀,x₁,..., x_e-2). It is shown that if there exists an s ¿ Z/(p) such that ¿(a₀(t),..., a_e-1 (t)) = s if and only if ¿(b₀ (t),..., b_e-1 (t)) = s for all nonnegative t with ¿(i) ¿ 0, where ¿ is an m-sequence determined by f(x) and a₀, then a = b. 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 ¿(t) ¿ 0. In particular, when ¿(x₀,x₁,..., x_e-2) = 0, it is a completion of the former result on the uniqueness of distribution of element 0 in highest level sequences.