Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over $\BBZ /(p^{e})$
Qun-Xiong Zheng, Wen‐Feng Qi · IEEE Transactions on Information Theory · 2010
Let Z/(pe) be the integer residue ring with odd prime p and integer e ¿ 2. Any sequence a over Z/(pe) has a unique p-adic expansion a = a0+ a1· p + ··· + ae-1· pe-1, where aican be regarded as a sequence over Z/(p) for 0 ¿ i ¿ e - 1. Let f(x) be a strongly primitive polynomial over Z/(pe) and a, b be two primitive sequences generated by f(x) over Z/(pe). Assume ¿(x0,..., xe-1) = xe-1+ ¿(x0,..., xe-2) is an e-variable function over Z/(p) with the monomial (p+1)/2 xe-2p-1...x1p-1not pearing in the expression of ¿(x0,x1,..., xe-2). It is shown that if there exists an s ¿ Z/(p) such that ¿(a0(t),..., ae-1(t)) = s if and only if ¿(b0(t),..., be-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/(pe), single element distribution is unique on all positions t with ¿(t) ¿ 0. In particular, when ¿(x0,x1,..., xe-2) = 0, it is a completion of the former result on the uniqueness of distribution of element 0 in highest level sequences.