Injective maps on primitive sequences over Z/(pe)

• Sun Zhonghua
• Qi Wenfeng
Article

Abstract

Let Z/(p e) be the integer residue ring modulo p e with p an odd prime and integer e ≥ 3. For a sequence $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a}$$ over Z/(p e), there is a unique p-adic decomposition $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} = \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _0 + \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _1 \cdot p + \cdots + \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _{e - 1} \cdot p^{e - 1}$$, where each $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _i$$ can be regarded as a sequence over Z/(p), 0 ≤ ie − 1. Let f(x) be a primitive polynomial over Z/(p e) and G′(f(x), p e) the set of all primitive sequences generated by f(x) over Z/(p e). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and ged(1 + deg(μ(x)), p − 1) = 1, set
$$\varphi _{e - 1} (x_0 ,x_1 , \cdots ,x_{e - 1} ) = x_{e - 1} \cdot \left[ {\mu (x_{e - 2} ) + \eta _{e - 3} (x_0 ,x_1 , \cdots ,x_{e - 3} )} \right] + \eta _{e - 2} (x_0 ,x_1 , \cdots ,x_{e - 2} ),$$
which is a function of e variables over Z/(p). Then the compressing map
$$\varphi _{e - 1} :G'(f(x),p^e ) \to (Z/(p))^\infty ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} \mapsto \varphi _{e - 1} (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _0 ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _1 , \cdots \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _{e - 1} )$$
is injective. That is, for $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b} \in G'(f(x),p^e ),\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} = \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b}$$ and only if $$\varphi _{e - 1} (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _0 ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _1 , \cdots \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _{e - 1} ) = \varphi _{e - 1} (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b} _0 ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b} _1 , \cdots ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b} _{e - 1} )$$. As for the case of e = 2, similar result is also given. Furthermore, if functions ε e − 1 and ψ e − 1 over Z/(p) are both of the above form and satisfy
$$\varphi _{e - 1} (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _0 ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _1 , \cdots \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} _{e - 1} ) = \psi _{e - 1} (\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b} _0 ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b} _1 , \cdots ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b} _{e - 1} )$$
for $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a} ,\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b} \in G'(f(x),p^e )$$, the relations between $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{a}$$ and $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{b}$$, ε e − 1 and ψ e − 1 are discussed.

Keywords

integer residue ring linear recurring sequence primitive sequence injective map

11B50 94A55

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© Editorial Committee of Applied Mathematics 2007

Authors and Affiliations

• Sun Zhonghua
• 1
• Qi Wenfeng
• 1
1. 1.Dept. of Appl. Math.Inform. Eng. Univ.ZhengzhouChina