Geodesic
Limit theorems for the number of nonzero solutions of a~system of random equations over the field~$\mathrm{GF}(2)$
Diskretnaya Matematika, Tome 12 (2000) no. 1, pp. 70-81

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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. 
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     author = {V. G. Mikhailov},
     title = {Limit theorems for the number of nonzero solutions of a~system of random equations over the field~$\mathrm{GF}(2)$},
     journal = {Diskretnaya Matematika},
     pages = {70--81},
     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/item/DM_2000_12_1_a5/}
}
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V. G. Mikhailov. Limit theorems for the number of nonzero solutions of a~system of random equations over the field~$\mathrm{GF}(2)$. Diskretnaya Matematika, Tome 12 (2000) no. 1, pp. 70-81. http://geodesic.mathdoc.fr/item/DM_2000_12_1_a5/