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 Diskr. Mat., 1999, Volume 11, Issue 3, Pages 15–23 (Mi dm382)

Threshold property for systems of equations in finite fields

V. F. Kolchin

Abstract: We consider the system of equations in $\operatorname{GF}(q)$ with respect to unknowns $x_1,\ldots,x_N$
$$a_1^{(t)}x_{i_1(t)}+\ldots+a_r^{(t)}x_{i_r(t)}=b_t,\qquad t=1,\ldots, T,$$
where $i_1(t),\ldots, i_r(t)$, $t=1,\ldots,T$, are independent identically distributed random variables taking the values $1,…, N$ with equal probabilities, the coefficients $a_1^{(t)},\ldots,a_r^{(t)}$, $t=1,\ldots,T$, are independent identically distributed random variables independent of $i_1(t),\ldots,i_r(t)$, $t=1,\ldots,T$, and taking the non-zero values from $\operatorname{GF}(q)$ with equal probabilities, and $b_t$, $t=1,\ldots,T$, are independent random variables not depending on the left-hand side of the system and taking the values from $\operatorname{GF}(q)$ with equal probabilities.
We denote by $A_r$ the matrix of the system. A critical set of rows of $A_r$ is defined in the same way as in the case of $\operatorname{GF}(2)$ but here a critical set contains a number of rows with weights from $\operatorname{GF}(q)$. We prove that the total number $S(A_r)$ of critical sets of the matrix $A_r$ has a threshold property. Let $N,T\to \infty$ and $T/N\to\alpha$. Then for any fixed integers $r\geq 3$ and $q\geq 3$ there exists a constant $\alpha_r$ such that $\mathsf E S(A_r)\to 0$ if $\alpha<\alpha_r$, and $\mathsf E S(A_r)\to\infty$ if $\alpha>\alpha_r$.
The research was supported by the Russian Foundation for Basic Research, grants 96–01–00338 and 96–15–96092.

DOI: https://doi.org/10.4213/dm382

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English version:
Discrete Mathematics and Applications, 1999, 9:4, 355–364

Bibliographic databases:

Document Type: Article
UDC: 519.2

Citation: V. F. Kolchin, “Threshold property for systems of equations in finite fields”, Diskr. Mat., 11:3 (1999), 15–23; Discrete Math. Appl., 9:4 (1999), 355–364

Citation in format AMSBIB
\Bibitem{Kol99} \by V.~F.~Kolchin \paper Threshold property for systems of equations in finite fields \jour Diskr. Mat. \yr 1999 \vol 11 \issue 3 \pages 15--23 \mathnet{http://mi.mathnet.ru/dm382} \crossref{https://doi.org/10.4213/dm382} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1739065} \zmath{https://zbmath.org/?q=an:0980.60017} \transl \jour Discrete Math. Appl. \yr 1999 \vol 9 \issue 4 \pages 355--364 

• http://mi.mathnet.ru/eng/dm382
• https://doi.org/10.4213/dm382
• http://mi.mathnet.ru/eng/dm/v11/i3/p15

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This publication is cited in the following articles:
1. A. V. Shapovalov, “Characteristics of random systems of linear equations over a finite field”, Discrete Math. Appl., 18:6 (2008), 569–580
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