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首页> 外文期刊>IEEE Transactions on Information Theory >Injectivity of Compressing Maps on Primitive Sequences Over ${BBZ}/(p^{e})$
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Injectivity of Compressing Maps on Primitive Sequences Over ${BBZ}/(p^{e})$

机译：超过$ {BBZ} /（p ^ {e}）$的原始序列上压缩图的内射性

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Let ${bf Z}/(p^{e})$ be the integer residue ring with odd prime $p$ and integer $egeq 2$. For a sequence $underline{a}$ over ${bf Z}/(p^{e})$, one has a unique $p$-adic expansion $underline{a}=underline{a}_{0}+ underline{a}_{1}cdot p+cdots +underline{a}_{e-1}cdot p^{e-1}$, where $underline{a}_{i}$ can be regarded as a sequence over ${bf Z}/(p)$ for $0leq ileq e-1$. Let $f(x)$ be a strongly primitive polynomial over ${bf Z}/(p^{e})$ and $G^{prime}(f(x),p^{e})$ be the set of all primitive sequences generated by $f(x)$ over ${bf Z}/(p^{e})$. Recently, the authors, Xuan-Yong Zhu and Wen-Feng Qi, have proved that for a function $varphi (x_{0},ldots,x_{e-1})=g(x_{e-1})+eta (x_{0},ldots,x_{e-2})$ over ${bf Z}/(p)$ and $underline{a},$ $underline{b}in G^{^{prime}}(f(x),p^{e})$ , where $2leq deg gleq p-1,$ $varphi (underline{a}_{0}, underline{a}_{1},ldots,underline{a}_{e-1})=varphi (underline{b}_{0}, underline{b}_{1},ldots,underline{b}_{e-1})$ if and only if $underline{a} = underline{b}$. To further complete their work, we show that such injectivity also holds for $deg g=1$. That is for a function $varphi (x_{0},ldots,x_{e-1})=x_{e-1}+eta (x_{0},ldots,x_{e-2})$ over ${bf Z}/(p)$ and $underline{a},$ $underline{b}in G^{prime}(f(x),p^{e}),$ $varphi (underline{a}_{0},underline{a}_{1},ldots,underline{a}_{e-1})=varphi (underline{b}_{0},underline{b}_{1},ldots,underline{b}_{e-1})$ if and only if $underline{a}=underline{b}$ .
机译：令$ {bf Z} /（p ^ {e}）$是带有奇数素数$ p $和整数$ egeq 2 $的整数残基环。对于序列$ underline {a} $超过$ {bf Z} /（p ^ {e}）$，则具有唯一的$ p $ -adic扩展$ underline {a} = underline {a} _ {0} +下划线{a} _ {1} cdot p + cdots +下划线{a} _ {e-1} cdot p ^ {e-1} $，其中$ underline {a} _ {i} $可被视为一个序列$ 0leq ileq e-1 $超过$ {bf Z} /（p）$。假设$ f（x）$是$ {bf Z} /（p ^ {e}）$上的强本原多项式，而$ G ^ {prime}（f（x），p ^ {e}）$是集合$ f（x）$超过$ {bf Z} /（p ^ {e}）$生成的所有原始序列中的。最近，作者朱玄勇和齐文峰证明对于函数$ varphi（x_ {0}，ldots，x_ {e-1}）= g（x_ {e-1}）+ eta （x_ {0}，ldots，x_ {e-2}）$超过$ {bf Z} /（p）$和$ underline {a}，$ $ underline {b} in G ^ {^ {prime}}（ f（x），p ^ {e}）$，其中$ 2leq deg gleq p-1，$ $ varphi（下划线{a} _ {0}，下划线{a} _ {1}，ldots，下划线{a} _ {e-1}）= varphi（下划线{b} _ {0}，下划线{b} _ {1}，ldots，下划线{b} _ {e-1}）$仅当且仅当$ underline {a } =下划线{b} $。为了进一步完成他们的工作，我们证明了这样的内射性对于$ deg g = 1 $也成立。那是一个函数$ varphi（x_ {0}，ldots，x_ {e-1}）= x_ {e-1} + eta（x_ {0}，ldots，x_ {e-2}）$ over $ { bf Z} /（p）$和$下划线{a}，G $ {prime}（f（x），p ^ {e}）中的$ $下划线{b}，$ $ varphi（下划线{a} _ { 0}，下划线{a} _ {1}，ldots，下划线{a} _ {e-1}）= varphi（下划线{b} _ {0}，下划线{b} _ {1}，ldots，下划线{ b} _ {e-1}）$当且仅当$ underline {a} = underline {b} $时。

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