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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)


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

Author: V. V. Masol
Translated by: V. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 70 (2004).
Journal: Theor. Probability and Math. Statist. 70 (2005), 93-104
MSC (2000): Primary 60C05, 15A52, 15A03
Published electronically: August 5, 2005
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. V. V. Masol, An expansion in a small parameter of the probability that a random determinant in the field 𝐺𝐹(2) is equal to one, Teor. Ĭmovīr. Mat. Stat. 64 (2001), 102–105 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. 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.
  • 4. I. N. Kovalenko, Invariance theorems for random Boolean matrices, Kibernetika (Kiev) 5 (1975), 138–152 (Russian, with English summary). MR 0458552 (56 #16752)
  • 5. A. A. Levit\cydotskaya, Invariance theorems for a system of random linear equations over an arbitrary finite ring, Dokl. Akad. Nauk SSSR 263 (1982), no. 2, 289–291 (Russian). MR 650154 (83g:60046)
  • 6. C. Cooper, On the rank of random matrices, Random Structures Algorithms 16 (2000), no. 2, 209–232. MR 1742352 (2000k:15050),<209::AID-RSA6>3.3.CO;2-T

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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

PII: S 0094-9000(05)00633-2
Received by editor(s): April 15, 2003
Published electronically: August 5, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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