The Unique Distribution of Zeros in 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 >= 3. 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 (w) be a strongly primitive polynomial over Z/(p(e)) and let (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)), where the degree of x(e-2) in eta(x(0), ... , x(e-2)) is less than p-1. It is shown that if phi(a(0)(t), ... ,a(e-1) (t)) = 0 if and only if phi(b(0)(t), ... , b(e-1)(t)) = 0 for all nonnegative integer 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. In particular, when eta(x(0), ... , x(e-2)) = 0, it is just the former result on the unique distribution of zeros in the highest level sequences.