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

DOI:
https://doi.org/10.1090/S0094-9000-05-00633-2

Published electronically:
August 5, 2005

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a random matrix in the field 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 and , .

**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****2.**V. V. Masol,*Explicit representation of some coefficients in the expansion of the random matrix rank distribution in the field*, 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****5.**A. A. Levit⋅skaya,*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****6.**C. Cooper,*On the rank of random matrices*, Random Structures Algorithms**16**(2000), no. 2, 209–232. MR**1742352**, https://doi.org/10.1002/(SICI)1098-2418(200003)16:2<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

Email:
vicamasol@pochtamt.ru

DOI:
https://doi.org/10.1090/S0094-9000-05-00633-2

Received by editor(s):
April 15, 2003

Published electronically:
August 5, 2005

Article copyright:
© Copyright 2005
American Mathematical Society