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Diskr. Mat., 2010, Volume 22, Issue 2, Pages 3–21 (Mi dm1091)  

This article is cited in 13 scientific papers (total in 13 papers)

Poisson-type theorems for the number of special solutions of a random linear inclusion

V. A. Kopyttsev, V. G. Mikhailov


Abstract: For given sets $D$ and $B$ of vectors of linear spaces over a finite field of dimensions $n$ and $T$, respectively, and a random $T\times n$ matrix $A$ over this field, we consider the distribution of the number of vectors satisfying the system of relations $x\in D$, $Ax\in B$ (that is, the number of solutions of the random linear inclusion $Ax\in B$ belonging to the set $D$). The conditions of convergence of this distribution, as $n,T\to\infty$, to the simple and compound Poisson distributions are given. These conditions require that the distribution of the matrix $A$ converge to the uniform distribution and at least one of the sets $D$ and $B$ satisfy the condition which is called here the condition of asymptotic freedom from linear combinations. These results generalise the known limit theorems on the number of special solutions of a system of random linear equations. In particular, they give a possibility to describe the asymptotic behaviour of the number of approximate solutions of a priori solvable systems.

DOI: https://doi.org/10.4213/dm1091

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English version:
Discrete Mathematics and Applications, 2010, 20:2, 191–211

Bibliographic databases:

Document Type: Article
UDC: 519.2
Received: 11.03.2010

Citation: V. A. Kopyttsev, V. G. Mikhailov, “Poisson-type theorems for the number of special solutions of a random linear inclusion”, Diskr. Mat., 22:2 (2010), 3–21; Discrete Math. Appl., 20:2 (2010), 191–211

Citation in format AMSBIB
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\by V.~A.~Kopyttsev, V.~G.~Mikhailov
\paper Poisson-type theorems for the number of special solutions of a~random linear inclusion
\jour Diskr. Mat.
\yr 2010
\vol 22
\issue 2
\pages 3--21
\mathnet{http://mi.mathnet.ru/dm1091}
\crossref{https://doi.org/10.4213/dm1091}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2730124}
\elib{http://elibrary.ru/item.asp?id=20730331}
\transl
\jour Discrete Math. Appl.
\yr 2010
\vol 20
\issue 2
\pages 191--211
\crossref{https://doi.org/10.1515/DMA.2010.011}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77953018262}


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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. V. A. Kopyttsev, V. G. Mikhailov, “Teoremy puassonovskogo tipa dlya chisla reshenii sluchainykh vklyuchenii”, Matem. vopr. kriptogr., 1:4 (2010), 63–84  mathnet  crossref
    2. V. A. Kopyttsev, V. G. Mikhailov, “O raspredelenii chisel reshenii sluchainykh vklyuchenii”, Matem. vopr. kriptogr., 2:2 (2011), 55–80  mathnet  crossref
    3. V. A. Kopyttsev, V. G. Mikhailov, “Poisson-type limit theorems for the generalised linear inclusion”, Discrete Math. Appl., 22:4 (2012), 477–491  mathnet  crossref  crossref  mathscinet  elib  elib
    4. V. A. Kopyttsev, V. G. Mikhailov, “Usloviya skhodimosti k raspredeleniyu Puassona dlya chisel reshenii sluchainykh vklyuchenii”, Matem. vopr. kriptogr., 3:3 (2012), 35–55  mathnet  crossref
    5. A. M. Zubkov, V. I. Kruglov, “Momentnye kharakteristiki vesov vektorov v sluchainykh dvoichnykh lineinykh kodakh”, Matem. vopr. kriptogr., 3:4 (2012), 55–70  mathnet  crossref
    6. A. M. Zubkov, V. I. Kruglov, “Statisticheskie kharakteristiki vesovykh spektrov sluchainykh lineinykh kodov nad $\mathrm{GF}(p)$”, Matem. vopr. kriptogr., 5:1 (2014), 27–38  mathnet  crossref
    7. V. A. Kopyttsev, V. G. Mikhailov, “Ob odnom asimptoticheskom svoistve sfer v diskretnykh prostranstvakh bolshoi razmernosti”, Matem. vopr. kriptogr., 5:1 (2014), 73–83  mathnet  crossref
    8. V. A. Kopyttsev, V. G. Mikhailov, “An estimate of the approximation accuracy in B. A. Sevastyanov's limit theorem and its application in the problem of random inclusions”, Discrete Math. Appl., 25:3 (2015), 149–156  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    9. A. M. Zubkov, V. I. Kruglov, “Veroyatnostnye kharakteristiki vesovykh spektrov sluchainykh lineinykh podkodov nad $\mathrm{GF}(p)$”, PDM. Prilozhenie, 2014, no. 7, 118–121  mathnet  elib
    10. V. A. Kopyttsev, V. G. Mikhailov, “Yavnye otsenki tochnosti puassonovskoi approksimatsii dlya raspredeleniya chisla reshenii sluchainykh vklyuchenii”, Matem. vopr. kriptogr., 6:1 (2015), 57–79  mathnet  crossref  mathscinet  elib
    11. V. A. Kopyttsev, V. G. Mikhailov, “Estimates for distribution of the minimal distance of a random linear code”, Discrete Math. Appl., 26:4 (2016), 203–211  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    12. V. A. Kopyttsev, “Mnogomernaya teorema Puassona dlya chisel reshenii sluchainykh vklyuchenii, blizkikh k zadannym vektoram”, Matem. vopr. kriptogr., 7:4 (2016), 67–80  mathnet  crossref  mathscinet  elib
    13. V. G. Mikhailov, “Formuly dlya odnoi kharakteristiki sfer i sharov v dvoichnykh prostranstvakh bolshoi razmernosti”, Diskret. matem., 30:2 (2018), 62–72  mathnet  crossref  elib
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