$p$ be an odd prime number, $e$ an integer greater than 1, and'/> Further Result on Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over ${bf Z}/(p^{e})$
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首页> 外文期刊>IEEE Transactions on Information Theory >Further Result on Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over ${bf Z}/(p^{e})$
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Further Result on Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over ${bf Z}/(p^{e})$

机译：从$ {bf Z} /（p ^ {e}）$上的原始序列得出的压缩序列分布特性的进一步结果

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Let $p$ be an odd prime number, $e$ an integer greater than 1, and ${bf Z}/(p^{e})$ 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 ${bf Z}/(p^{e})$. It is shown that two primitive sequences $underline{a}$ and $underline{b}$ generated by a strongly primitive polynomial $fleft(xright)$ over ${bf Z}/(p^{e})$ are the same, if there exist $sin{bf Z}/(p)$ and $kin{bf Z}/(p)^{ast}$ such that the distribution of $s$ in their compressing sequences $underline{a}_{e-1}+eta (underline{a}_{0},ldots,underline{a}_{e-2})$ and $underline{b}_{e-1}+eta (underline{b}_{0},ldots,underline{b}_{e-2})$ is coincident at the positions $t$ with $alpha (t)=k$, where $etale- t(x_{0},ldots,x_{e-2}right)$ is an $(e-1)$-variable polynomial over ${bf Z}/(p)$ with the coefficient of $x_{e-2}^{p-1}cdots x_{1}^{p-1}x_{0}^{p-1}$ not equal to $left(-1right)^{e}cdotleft(p+1right)/2$ and $underline{alpha}$ is an $m$ -sequence over ${bf Z}/(p)$ determined by $f(x)$ and $underline{a}$. Compared with the previous result, this gives a more precise characterization on the positions of a compressing sequence, i.e., of the form $underline{a}_{e-1}+eta (underline{a}_{0},ldots,underline{a}_{e-2})$ , derived from a primitive sequence $underline{a}$ over ${bf Z}/(p^{e})$ that completely determines $underline{a}$ . In particular, the result is also true for the highest level sequence $underline{a}_{e-1}$ by taking $eta (x_{0},ldots,x_{e-2})=0$.
机译：令 $ p $ 为奇数质数， $ e $ 大于1的整数，并且 $ {bf Z} /（p ^ {e}）$ 整数残基环模 $ p ^ {e} $ 。在本文中，我们获得了前一篇论文（IEEE Trans。Inf。Theory，56（1）（2010）555–563）的改进结果，该结果涉及在 $ {bf Z} /（p ^ {e}）$ 。显示了两个基本序列 $ underline {a} $ 和 $ underline {b} $ 由强本原多项式 $ fleft（xright）$ 生成在 $ {bf Z} /（p ^ {e}）$ 相同，如果存在 $ sin {bf Z} /（p）$ 和 $ kin {bf Z} /（p）^ {ast} $ ，这样 $ s $ < / tex> 的压缩顺序 $ underline {a} _ {e-1} + eta（下划线{a} _ {0}， ldots，下划线{a} _ {e-2}）$ 和 $ underline {b} _ {e-1} + eta（下划线{b} _ {0}，点号，下划线{b} _ {e-2}）$ 在位置 $ t $ 和 $ alpha（t）= k $ ，其中 $ etale- t（x_ {0}，ldots，x_ {e-2} right）$ 是 $（e-1）$ - $ {bf Z} /（p）$ < / formula>的系数为 $ x_ {e-2} ^ {p-1} cdots x_ {1} ^ {p-1} x_ {0 } ^ {p-1} $ 不等于 $ left（-1right）^ {e} cdotleft（p + 1right ）/ 2 $ 和 $ underline {alpha} $ 是 $ m $ -序列在 $ {bf Z} /（p）$ 由 $ f（x）$ 确定和 $ underline {a} $ 。与先前的结果相比，这可以更精确地描述压缩序列的位置，即形式为<公式> typetype =“ inline”> $ underline {a} _ {e- 1} + eta（下划线{a} _ {0}，ldots，下划线{a} _ {e-2}）$ ，源自原始序列 $下划线{a} $ 在 $ {bf Z} /（p ^ {e }）$ 完全确定 $ underline {a} $ 。特别地，对于最高级别的序列 $ underline {a} _ {e-1} $ ，结果也是正确的取 $ eta（x_ {0}，ldots，x_ {e-2}）= 0 $ 。

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