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Series expansion for the probability that a random Boolean matrix is of maximal rank

Author(s): V. V. Masol
Translated by: V. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 70 (2004).
Journal: Theor. Probability and Math. Statist. No. 70 (2005), 93-104.
MSC (2000): Primary 60C05, 15A52, 15A03
Posted: August 5, 2005
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Abstract: We consider a random $(N\times n)$ matrix in the field $GF(2)$ and establish relations that allow one to find the coefficients of the expansion of the probability that a given matrix is of maximal rank into a series in powers of a small parameter. We give explicit formulas for the cases of $n=1$ and $n=2$, $N\geq n$.

References:

1.
V. V. Masol, An expansion in a small parameter of the probability that a random determinant in the field $GF(2)$ is 1, Teor. Imovirnost. Mat. Stat. 64 (2001), 102-105; English transl. in Theor. Probability Math. Statist. 64 (2002), 117-121. MR 1922957 (2003g:60015)

2.
V. V. Masol, Explicit representation of some coefficients in the expansion of the random matrix rank distribution in the field $GF(2)$, Theory Stoch. Process. 6(22) (2000), no. 3-4, 122-126.

3.
V. V. Masol, Expansion in terms of powers of small parameter of the maximum rank distribution of a random Boolean matrix, Kibernetika i Sistemnyi Analiz 38 (2002), no. 6, 176-180; English transl. in Cybernetics and Systems Analysis 38 (2003), no. 6, 938-942.

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I. N. Kovalenko, Invariance theorems for random Boolean matrices Kibernetika 11 (1975), no. 5, 138-152; English transl. in Cybernetics 11 (1976), no. 5, 818-834. MR 0458552 (56:16752)

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A. A. Levitskaya, Invariance theorems for a system of random linear equations over an arbitrary finite ring, Dokl. AN SSSR 263 (1982), no. 2, 289-291; English transl. in Soviet Math. Dokl. 25 (1982), 340-342. MR 0650154 (83g:60046)

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C. Cooper, On the rank of random matrices, Random Structures and Algorithms 16 (2000), no. 2, 209-232. MR 1742352 (2000k:15050)

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Additional Information:

V. V. Masol
Affiliation: Department of Probability Theory and Mathematical Statistics, Mechanics and Mathematics Faculty, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: vicamasol@pochtamt.ru

DOI: 10.1090/S0094-9000-05-00633-2
PII: S 0094-9000(05)00633-2
Received by editor(s): 15/APR/2003
Posted: August 5, 2005
Copyright of article: Copyright 2005, American Mathematical Society

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