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

Qun-Xiong Zheng,Wen-Feng Qi

doi:10.1109/tit.2009.2034782

Abstract

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 = a 0 + a 1 · 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 ¿(x 0 ,..., x e-1 ) = x e-1 + ¿(x 0 ,..., x e-2 ) is an e-variable function over Z/(p) with the monomial (p+1)/2 x e-2 p-1 ...x 1 p-1 not pearing in the expression of ¿(x 0 ,x 1 ,..., x e-2 ). It is shown that if there exists an s ¿ Z/(p) such that ¿(a 0 (t),..., a e-1 (t)) = s if and only if ¿(b 0 (t),..., b e-1 (t)) = s for all nonnegative t with ¿(i) ¿ 0, where ¿ is an m-sequence determined by f(x) and a 0 , 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 ¿(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.

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