The Distribution of the Rank of Random Matrices over a Finite Field

Balakin, G. V.

doi:10.1137/1113076

Cited by 16 publications

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“…Consider system (1) where the entries of the T × n matrix A = a ij are independent random variables whose distributions satisfy conditions (2) …”

Section: The Distribution Of the Rank Of A Random Matrix 105mentioning

confidence: 99%

“…Different methods are used in [1] and [2] to prove that the distribution of the rank r(A) of a sparse Boolean random matrix A approaches the Poisson distribution as T = T (n) → ∞ and n → ∞ (here n and T mean the number of columns and rows in the matrix A, respectively). If T and n are finite, then the distribution of the rank r(A) can be expressed in terms of the factorial moments of the random variable r(A).…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to the papers [1,2], we consider random matrices A whose entries have the distributions that may depend on their position in the matrix A. …”

Section: Introductionmentioning

confidence: 99%

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A theorem on the distribution of the rank of a sparse Boolean random matrix and some applications

Masol

Popereshnyak

2008

Theor. Probability and Math. Statist.

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Abstract. We consider some estimates of the rate of convergence of the distribution of a sparse Boolean random matrix to the Poisson distribution. The results obtained in the paper are applied to estimate the probability that a nonhomogeneous system of Boolean random linear equations is consistent. IntroductionDifferent methods are used in [1] and [2] to prove that the distribution of the rank r(A) of a sparse Boolean random matrix A approaches the Poisson distribution as T = T (n) → ∞ and n → ∞ (here n and T mean the number of columns and rows in the matrix A, respectively). If T and n are finite, then the distribution of the rank r(A) can be expressed in terms of the factorial moments of the random variable r(A). These expressions are found in [3]. The asymptotic behavior as n → ∞ of the factorial moments is given in [4, Theorem 4] under certain assumptions, say if n − T = const and if the distribution of entries of the matrix A depends on the number of the column only.We are interested in the bounds of the rate of convergence of the distribution of the rank of the matrix A to the Poisson distribution as n grows and in applications of these bounds for an estimation of the probability that the following nonhomogeneous system of linear equations is consistent:where the entries of the matrix A of coefficients belong to the field GF (2) consisting of two elements, the vector column B = (b 1 , . . . , b T ) is formed by the random variables b 1 , . . . , b T that do not depend on the entries of the matrix A and moreover b 1 , . . . , b T are jointly independent random variables assuming values in GF (2) and having known distributions, and where X = (x 1 , x 2 , . . . , x n ) is an n-dimensional column vector such that x i ∈ GF (2), i = 1, . . . , n.In contrast to the papers [1, 2], we consider random matrices A whose entries have the distributions that may depend on their position in the matrix A.2000 Mathematics Subject Classification. Primary 68U20; Secondary 60G10. Key words and phrases. Boolean random matrix, rank of a matrix, the probability that a system is consistent, the rate of convergence of distributions.

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“…Consider system (1) where the entries of the T × n matrix A = a ij are independent random variables whose distributions satisfy conditions (2) …”

Section: The Distribution Of the Rank Of A Random Matrix 105mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A theorem on the distribution of the rank of a sparse Boolean random matrix and some applications

Masol

Popereshnyak

2008

Theor. Probability and Math. Statist.

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“…where x is a fixed number, Bolotin [10] has presented the conditions on N and n for which the distributions of the differences between min ( , ) n N and the determinant ( ( )) rank A Nn , permanent, and linear ranks A Nn , respectively, asymptotically coincide and are either Poisson or concentrated at one point. Kovalenko has generalized the results of [119] and [10], relevant to distribution of a linear rank, in [35], where he analyzed the asymptotic behavior of the distribution of the linear rank of a (0,1)-matrix A a Nn ij = || || with independent elements distributed according to the following law: P{ } ln a n x…”

Section: Certainly Compatible Systems Of Random Equations (Ccsre)mentioning

confidence: 99%

“…Kovalenko has generalized the results of [119] and [10], relevant to distribution of a linear rank, in [35], where he analyzed the asymptotic behavior of the distribution of the linear rank of a (0,1)-matrix A a Nn ij = || || with independent elements distributed according to the following law: P{ } ln a n x…”

Section: Certainly Compatible Systems Of Random Equations (Ccsre)mentioning

confidence: 99%

Systems of Random Equations over Finite Algebraic Structures

Levitskaya

2005

Cybern Syst Anal

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519.21The results on systems of random equations over finite algebraic structures are reviewed. Basic definitions, concepts, and problems in this field are presented.Keywords: system of random equations over finite algebraic structure (finite field, finite ring, finite Abelian group), Boolean system of equations, certainly compatible system of random equations, system of random equations with independent left-and right-hand sides, system of random equations with distorted right-hand sides.This review of the results from the studies on systems of random equations over finite algebraic structures is timed to the 70th anniversary of Igor N. Kovalenko, a prominent Ukrainian mathematician, Doctor of engineering, physics, and mathematics, Academician of the National Academy of Sciences of Ukraine. He may rightfully be considered a founder of this complex and fascinating division in discrete mathematics. His fundamental and mathematically beautiful study [33] (1967) on the invariance of the limit behavior of the number of solutions of one class of systems of random equations over a GF(2) field became a basis for disciples of I. N. Kovalenko's school in extending invariance theory to random systems over arbitrary finite fields and finite rings. Over nearly 40 years since the paper [33] was published, the theory of systems of random equations over finite algebraic structures has been enriched with new problems, results, and methods. To efficiently develop this theory, we today need to sort out and systematize everything accumulated in this area. The author of the present review, a follower of I. N. Kovalenko, has tried to do this with gratitude and deep respect for the teacher.The studies by V. L. Goncharov on random substitutions and series in random (0,1)-sequences, published in the 40s of the last century, have initiated a systematic analysis of combinatory objects using methods of probability theory. Thus, they promoted the development of a new division in discrete mathematics, probabilistic combinatorial analysis, where the theory of systems of random equations over finite algebraic structures is a relatively small subdivision, despite its important applications in the theory of finite state automata, the theory of coding, the theory of random graphs, cryptography, etc. there is an explanation of this. The use of random systems as probability-theoretic models of combinatorial processes occurring in information-protection systems has vetoed publications on this subject in the available scientific literature for many years. The situation began to change for the better only in the early 90s of the last century. Indeed, a large number of papers on random systems over finite algebraic structures have appeared in the last 10 to 14 years in large mathematical journals. It is pleasant to point out that whereas this subjects was considered especially "Russian" when the review [40,41] devoted to random linear systems over finite algebraic structures was written (since the corresponding publications were mainly due to Soviet ma...

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On the distribution of rank of a random matrix over a finite field

Cooper

2000

Random Struct. Alg.

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Let M = (m ij ) be a random n × n matrix over GF(t) in which each matrix entry m ij is independently and identically distributed, with Pr(m ij = 0) = 1 − p(n) and Pr(m ij = r) = p(n)/(t − 1), r = 0. If we choose t ≥ 3, and condition on M having no zero rows or columns, then the probability that M is non-singular tends to c t ∼ ∞ j=1 (1 − t −j ) provided p ≥ (log n + d)/n, where d → −∞ slowly.

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The Distribution of the Rank of Random Matrices over a Finite Field

Cited by 16 publications

References 2 publications

A theorem on the distribution of the rank of a sparse Boolean random matrix and some applications

A theorem on the distribution of the rank of a sparse Boolean random matrix and some applications

Systems of Random Equations over Finite Algebraic Structures

On the distribution of rank of a random matrix over a finite field

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