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 Diskr. Mat., 2000, Volume 12, Issue 1, Pages 70–81 (Mi dm318)

Limit theorems for the number of nonzero solutions of a system of random equations over the field $\mathrm{GF}(2)$

V. G. Mikhailov

Abstract: We study the properties of the number $\nu$ of non-zero solutions of system of random equations over $\mathrm{GF}(2)$ with the left-hand sides which are products of expressions of the form $a_{t1}x_1+\ldots+a_{tn}x_n+a_t$ with independent equiprobable coefficients. The right-hand sides of the system are zeros. We derive inequalities for the factorial moments of the random variable $\nu$ and necessary and sufficient conditions of the validity of the Poisson limit theorem for $\nu$.
The research was supported by the Russian Foundation for Basic Research, grants 99–01–00012 and 96–15–96092.

DOI: https://doi.org/10.4213/dm318

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English version:
Discrete Mathematics and Applications, 2000, 10:2, 115–126

Bibliographic databases:

Document Type: Article
UDC: 519.2

Citation: V. G. Mikhailov, “Limit theorems for the number of nonzero solutions of a system of random equations over the field $\mathrm{GF}(2)$”, Diskr. Mat., 12:1 (2000), 70–81; Discrete Math. Appl., 10:2 (2000), 115–126

Citation in format AMSBIB
\Bibitem{Mik00} \by V.~G.~Mikhailov \paper Limit theorems for the number of nonzero solutions of a~system of random equations over the field~$\mathrm{GF}(2)$ \jour Diskr. Mat. \yr 2000 \vol 12 \issue 1 \pages 70--81 \mathnet{http://mi.mathnet.ru/dm318} \crossref{https://doi.org/10.4213/dm318} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1778767} \zmath{https://zbmath.org/?q=an:0969.60012} \transl \jour Discrete Math. Appl. \yr 2000 \vol 10 \issue 2 \pages 115--126 

• http://mi.mathnet.ru/eng/dm318
• https://doi.org/10.4213/dm318
• http://mi.mathnet.ru/eng/dm/v12/i1/p70

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This publication is cited in the following articles:
1. V. G. Mikhailov, “The Poisson limit theorem for the number of noncollinear solutions of a system of random equations of a special form”, Discrete Math. Appl., 11:4 (2001), 391–400
2. V. G. Mikhailov, “Limit theorems for the number of points of a given set covered by a random linear subspace”, Discrete Math. Appl., 13:2 (2003), 179–188
3. V. G. Mikhailov, “Limit theorems for the number of solutions of a system of random linear equations belonging to a given set”, Discrete Math. Appl., 17:1 (2007), 13–22
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