Geodesic
On the number of solutions of a system of random linear equations in a set of vectors of a special form
Diskretnaya Matematika, Tome 18 (2006) no. 1, pp. 40-62.

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We analyse the distribution of the number of solutions of a system of random linear equations over $\mathit{GF}(q)$ in the set of vectors which have a given number of nonzero coordinates and in some subsets of this set. We deduce sufficient conditions for convergence of the distribution to the Poisson law, as well as to some other limit distributions related to this law, and to the standard normal law. Here we extend the results which the author have proved earlier for the case $q=2$. 
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V. A. Kopyttsev. On the number of solutions of a system of random linear equations in a set of vectors of a special form. Diskretnaya Matematika, Tome 18 (2006) no. 1, pp. 40-62. http://geodesic.mathdoc.fr/item/DM_2006_18_1_a3/

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