Further results on the compression maps of primitive sequences over Z=(<italic>p</italic><sup><italic>e</italic></sup>)

Abstract

Let Z/( p e ) be the integer residue ring with p prime and e ≥ 2. Let f ( x ) be a primitive polynomial over Z/( p e ) with degree n and G ( f ( x ), p e ) the set of all primitive sequences over Z/( p e ) generated by f ( x ). For any sequence a∈ G ( f ( x ), p e ), it has a unique p -adic expansion a = a0 + a1 p +…+ a e -1 p e -1. Let o( x 0, x 1,…, x e -1) = g ( x e -1)+ μ ( x 0, x 1,…, x e -2) be a function from F p e to F p . Then o can induce a compression mapping from G ( f ( x ), p e ) to F p ∞. In recent years, Zhu, Tian and Qi have proved that the compression mapping is injective under the condition that p is an odd prime and f ( x ) is a strongly primitive polynomial. In this paper, we improve their results. More exactly, we prove that the compression mapping is also injective only under the condition that p is an odd prime and f ( x ) is a primitive polynomial. Of course we should ask that deg( g ( x )) is an odd number or g ( x ) = x k +Σi=0 k -2 c i x i.