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

Qun-Xiong Zheng Qun-Xiong Zheng,Tian Tian Tian Tian,Wen-Feng Qi Wen-Feng Qi

doi:10.1109/tit.2013.2258712

Abstract

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(a 0 ,⋯.a e-2 ) and b e-1 +η(b 0 ,⋯,b e-2 ) is coincident at the positions t with α(t)=k, where η(x 0 ,⋯,x e-2 ) is an (e-)-variable polynomial over Z/(p) with the coefficient of x e-2 p-1 ⋯x 1 p-1 x 0 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(a 0 ,⋯,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(x 0 ,⋯,x e-2 )=0.

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