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Diskr. Mat., 2011, Volume 23, Issue 2, Pages 3–31 (Mi dm1137)  

This article is cited in 6 scientific papers (total in 6 papers)

Reconstruction of a linear recurrence of maximal period over a Galois ring from its highest coordinate sequence

A. S. Kuzmin, A. A. Nechaev


Abstract: Let $R=GR(q^n,q^n)$ be a Galois ring of cardinality $q^n$ and characteristic $p^n$, $q=p^r$, $p$ be a prime. We call a subset $K\subset R$ a coordinate set if $0\in K$ and for any $a\in R$ there exists a unique $\varkappa(a)\in K$ such that $a\equiv\varkappa(a)\pmod{pR}$. Let $u$ be a linear recurring sequence of maximal period (MP LRS) over a ring $R$. Then any its term $u(i)$ admits a unique representation in the form
$$ u(i)=w_0(i)+pw_1(i)+…+p^{n-1}w_{n-1}(i),\qquad w_t(i)\in K,\quad t\in\{0,…,n-1\}. $$
We pose the following conjecture: the sequence $u$ can be uniquely reconstructed from the sequence $w_{n-1}$ for any choice of the coordinate set $K$. It is proved that such a reconstruction is possible under some conditions on $K$. In particular, it is possible for any $K$ if $R=\mathbf Z_{p^n}$ and for any Galois ring $R$ if $K$ is a $p$-adic (Teichmüller) coordinate set.

DOI: https://doi.org/10.4213/dm1137

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English version:
Discrete Mathematics and Applications, 2011, 21:2, 145–178

Bibliographic databases:

UDC: 519.7
Received: 16.04.2010

Citation: A. S. Kuzmin, A. A. Nechaev, “Reconstruction of a linear recurrence of maximal period over a Galois ring from its highest coordinate sequence”, Diskr. Mat., 23:2 (2011), 3–31; Discrete Math. Appl., 21:2 (2011), 145–178

Citation in format AMSBIB
\Bibitem{KuzNec11}
\by A.~S.~Kuzmin, A.~A.~Nechaev
\paper Reconstruction of a~linear recurrence of maximal period over a~Galois ring from its highest coordinate sequence
\jour Diskr. Mat.
\yr 2011
\vol 23
\issue 2
\pages 3--31
\mathnet{http://mi.mathnet.ru/dm1137}
\crossref{https://doi.org/10.4213/dm1137}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2865903}
\elib{http://elibrary.ru/item.asp?id=20730380}
\transl
\jour Discrete Math. Appl.
\yr 2011
\vol 21
\issue 2
\pages 145--178
\crossref{https://doi.org/10.1515/DMA.2011.010}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79960015715}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. D. N. Bylkov, O. V. Kamlovskii, “Parametry bulevykh funktsii, postroennykh s ispolzovaniem starshikh koordinatnykh posledovatelnostei lineinykh rekurrent”, Matem. vopr. kriptogr., 3:4 (2012), 25–53  mathnet  crossref
    2. Zheng Q.-X. Qi W.-F. Tian T., “Further Result on Distribution Properties of Compressing Sequences Derived From Primitive Sequences Over Z/(P(E))”, IEEE Trans. Inf. Theory, 59:8 (2013), 5016–5022  crossref  mathscinet  zmath  isi  elib  scopus
    3. E. M. Serebryakov, “Vosstanovlenie polinomialno uslozhnennoi lineinoi rekurrenty maksimalnogo perioda nad koltsom Galua po starshei koordinatnoi posledovatelnosti”, PDM, 2014, no. 2(24), 21–36  mathnet
    4. D. N. Bylkov, “Reconstruction of a linear recurrence of maximal period over a Galois ring of characteristic $p^3$ by its highest digital sequence”, Matem. vopr. kriptogr., 5:2 (2014), 29–35  mathnet  crossref
    5. D. N. Bylkov, “Postroenie novykh klassov filtruyuschikh generatorov, ne imeyuschikh ekvivalentnykh sostoyanii”, Matem. vopr. kriptogr., 5:4 (2014), 17–39  mathnet  crossref
    6. Tsypyschev V.N., “Lower Bounds on Linear Complexity of Digital Sequences Products of Lrs and Matrix Lrs Over Galois Ring”, Cybernetics Approaches in Intelligent Systems: Computational Methods in Systems and Software 2017, Vol. 1, Advances in Intelligent Systems and Computing, 661, ed. Silhavy R. Silhavy P. Prokopova Z., Springer International Publishing Ag, 2018, 50–61  crossref  isi  scopus
  • Дискретная математика
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