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SCIENCE CHINA Information Sciences, Volume 60 , Issue 5 : 052102(2017) https://doi.org/10.1007/s11432-015-5472-x

On $\boldsymbol s$-uniform property of compressing sequences derived from primitive sequences modulo odd prime powers

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  • ReceivedAug 27, 2015
  • AcceptedOct 13, 2015
  • PublishedSep 13, 2016

Abstract

Let $\mathbb{Z}/(p^e)$ be the integer residue ring modulo $p^e$ with $p$ an odd prime and $e\ge 2$.We consider the $s$-uniform property of compressing sequences derived from primitivesequences over $\mathbb{Z}/(p^e)$. We give necessary and sufficient conditions fortwo compressing sequences to be $s$-uniform with $\underline{\alpha}$ provided that the compressing map is of the form$\phi(x_0, x_1, \ldots, x_{e-1})=g(x_{e-1})+\eta(x_0, x_1,\ldots, x_{e-2})$,where $g(x_{e-1})$ is a permutation polynomial over $\mathbb{Z}/(p)$ and $\eta$ is an $(e-1)$-variablepolynomial over $\mathbb{Z}/(p)$.


Acknowledgment

This work was supported by National Basic Research Program of China (973 Program) (Grant No. 2011CB302400), Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA06010701), China Postdoctoral Science Foundation Funded Project (Grant No. 2014M560130), National Natural Science Foundation of China (Grant Nos. 61402524, 61502483), and Science and Technology on Information Assurance Laboratory (Grant No. KJ-13-006).

  • Table 1   $0$-uniform with $\underline{\alpha}|_1$
    $\underline{a}$ $ 0, 1, 8, 8, 3, 8, 4, 7, 3, 1, 2, 5, 0, 8, 1, 1, 6, 1, 5, 2, 6, 8, 7, 4$
    $\underline{b}$ $3, 8, 4, 7, 3, 1, 2, 5, 0, 8, 1, 1, 6, 1, 5, 2, 6, 8, 7, 4, 0, 1, 8, 8$
    $\underline{\alpha}$ $\mspace{-1.1mu}\boldsymbol{1}, 0, \mspace{-1.1mu}\boldsymbol{1}, 2, 2, 0, 2, \mspace{-1.1mu}\boldsymbol{1},\mspace{-1.1mu}\boldsymbol{1}, 0, \mspace{-1.1mu}\boldsymbol{1}, 2, 2, 0, 2, \mspace{-1.1mu}\boldsymbol{1}, \mspace{-1.1mu}\boldsymbol{1}, 0, \mspace{-1.1mu}\boldsymbol{1}, 2, 2, 0, 2, \mspace{-1.1mu}\boldsymbol{1}$
    2*$d_1=0$ $\phi(\underline{a})$ $1, 0, 2, 2, 2, 2, 1, 2, 2, 0, \mspace{-1.1mu}\boldsymbol{0}, 1, 1, 2, 0, \mspace{-1.1mu}\boldsymbol{0}, \mspace{-1.1mu}\boldsymbol{0}, 0, 1, 0, 0, 2, 2, 1$
    $\phi(\underline{b})$ $2, 2, 1, 2, 2, 0, 0, 1, 1, 2, \mspace{-1.1mu}\boldsymbol{0}, 0, 0, 0, 1, \mspace{-1.1mu}\boldsymbol{0}, \mspace{-1.1mu}\boldsymbol{0}, 2, 2, 1, 1, 0, 2, 2$
    2*$d_1=1$ $\phi(\underline{a})$ $1, 1, 1, 1, 2, 1, 2, \mspace{-1.1mu}\boldsymbol{0}, 2, 1, 2, 0, 1, 1, 1, 1, \mspace{-1.1mu}\boldsymbol{0}, 1, \mspace{-1.1mu}\boldsymbol{0}, 2, 0, 1, 0, 2$
    $\phi(\underline{b})$ $2, 1, 2, 0, 2, 1, 2, \mspace{-1.1mu}\boldsymbol{0}, 1, 1, 1, 1, 0, 1, 0, 2, \mspace{-1.1mu}\boldsymbol{0}, 1, \mspace{-1.1mu}\boldsymbol{0}, 2, 1, 1, 1, 1 $
    2*$d_1=2$ $\phi(\underline{a})$ $1, 2, \mspace{-1.1mu}\boldsymbol{0}, 0, 2, 0, 0, 1, 2, 2, 1, 2, 1, 0, 2, 2, \mspace{-1.1mu}\boldsymbol{0}, 2, 2, 1, 0, 0, 1, \mspace{-1.1mu}\boldsymbol{0}$
    $\phi(\underline{b})$ $ 2, 0, \mspace{-1.1mu}\boldsymbol{0}, 1, 2, 2, 1, 2, 1, 0, 2, 2, 0, 2, 2, 1, \mspace{-1.1mu}\boldsymbol{0}, 0, 1, 0, 1, 2, 0, \mspace{-1.1mu}\boldsymbol{0}$

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