Further Result on Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over ${\bf Z}/(p^{e})$

Let p be an odd prime number, e an integer greater than 1, and Z/(p^e) the integer residue ring modulo p^e. In this paper, we obtain an improved result of the previous paper (IEEE Trans. Inf. Theory, 56(1) (2010) 555-563) on distribution properties of compressing sequences derived from primitive sequences over Z/(p^e). It is shown that two primitive sequences α and b generated by a strongly primitive polynomial f(x) over Z/(p^e) are the same, if there exist s∈ Z/(p) and k∈ Z(p)* such that the distribution of in their compressing sequences a_e-1+η(a₀,⋯.a_e-2) and b_e-1+η(b₀,⋯,b_e-2) is coincident at the positions t with α(t)=k, where η(x₀,⋯,x_e-2) is an (e-)-variable polynomial over Z/(p) with the coefficient of x_e-2^p-1⋯x₁^p-1x₀^p-1 not equal to (-1)^e · (p+1)/2 and α is an m-sequence over Z/(p)determined by f(x) and a. Compared with the previous result, this gives a more precise characterization on the positions of a compressing sequence, i.e., of the form a_e-1+η(a₀,⋯,a_e-2), derived from a primitive sequence a over Z/(p^e) that completely determines a. In particular, the result is also true for the highest level sequence ae_-1 by taking η(x₀,⋯,x_e-2)=0.