An estimate for the rate of convergence of the distribution of the number of false solutions of a system of nonlinear random equations in the field $GF(2)$

Authors:
V. I. Masol and M. V. Slobodyan

Translated by:
S. Kvasko

Journal:
Theor. Probability and Math. Statist. **77** (2008), 121-134

MSC (2000):
Primary 60C05, 15A52, 15A03

DOI:
https://doi.org/10.1090/S0094-9000-09-00751-0

Published electronically:
January 16, 2009

MathSciNet review:
2432776

Full-text PDF Free Access

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Abstract: We prove a result on the rate of convergence as $n\to \infty$ of the distribution of the number of false solutions of a system of nonlinear random equations in the field $GF(2)$ to the Poisson distribution with parameter $2^{m}$. We assume, in particular, that the difference between the number $n$ of unknowns and the number $N$ of equations of the system is a constant $m$.

References
- V. I. Masol,
*A theorem on the limit distribution of the number of false solutions of a system of nonlinear random Boolean equations*, Teor. Veroyatnost. i Primenen. **43** (1998), no. 1, 41–56 (Russian, with Russian summary); English transl., Theory Probab. Appl. **43** (1999), no. 1, 75–88. MR **1669972**, DOI https://doi.org/10.1137/S0040585X97976672
- William Feller,
*An introduction to probability theory and its applications. Vol. I*, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR **0228020**
- V. N. Sachkov,
*Vvedenie v kombinatornye metody diskretnoĭ matematiki*, “Nauka”, Moscow, 1982 (Russian). MR **700691**

References
- V. I. Masol,
*A theorem on the limit distribution of the number of false solutions of a system of nonlinear random Boolean equations*, Teor. Veroyatnost. i Primenen. **43** (1998), no. 1, 41–56; English transl. in Theory Probab. Appl. **43** (1999), no. 1, 75–88. MR **1669972 (2000f:60040)**
- W. Feller,
*An Introduction to Probability Theory and its Applications*, 3rd ed., vol. I, John Wiley & Sons, New York–London–Sydney, 1968. MR **0228020 (37:3604)**
- V. N. Sachkov,
*Introduction to Combinatorial Methods of Discrete Mathematics*, “Nauka”, Moscow, 1982. (Russian) MR **700691 (85g:05001)**

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Additional Information

**V. I. Masol**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
vimasol@ukr.net

**M. V. Slobodyan**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
mslob@ukr.net

Keywords:
System of nonlinear random equations,
the field $GF(2)$,
rate of convergence

Received by editor(s):
February 10, 2006

Published electronically:
January 16, 2009

Article copyright:
© Copyright 2009
American Mathematical Society