Abstract
The results on systems of random equations over finite algebraic structures are reviewed. Basic definitions, concepts, and problems in this field are presented.
Similar content being viewed by others
REFERENCES
A. N. Alekseichuk, “Uniqueness of the problem of moments in the class of q-distributions,” Diskr. Matem., 10, Issue 1, 96–110 (1998).
A. N. Alekseichuk, “Limit distribution of the number of solutions of a system of stochastic linear homogeneous equations with a uniform coefficient matrix over a finite local ring,” Kibern. Sist. Analiz, No. 3, 462–466 (1998).
A. N. Alekseichuk, “Limit distributions of the length and rank of a module generated by the columns of a random equiprobable matrix over a finite chain ring,” Kibern. Sist. Analiz, No. 5, 726–731 (1998).
A. N. Alekseichuk, “The uniqueness conditions for the problem of moments in the class of q-distributions,” Diskret. Matem., 11, Issue 4, 48–57 (1999).
A. N. Alekseichuk, “Systems of linear equations with the distorted right-hand side over a ring of residues modulo 2N,” Nauk.-Tekhn. Zhurn. Zakhyst Informatsii, No. 4, 12–19 (2001).
A. N. Alekseichuk and V. V. Luk’yanov, “The method of decoding block codes in a channel with additive modulo 2N noise, using partially known input and output messages,” Collection of Scientific Papers of the Institute of Simulation Problems in the Energetics, NAS Ukr. [in Russian], No. 10 (2001), pp. 88–93.
A. N. Alekseichuk and S. M. Ignatenko, “Estimates of the efficiency of universal methods of restoring distorted linear recurrents over a ring of residues modulo 2N,” Collection of Scientific Papers of the Institute of Simulation Problems in the Energetics, NAS Ukr. [in Russian], No. 20, 40–48 (2003).
A. N. Alekseichuk and S. M. Ignatenko, “The method of optimizing algorithms for solving systems of linear equations with distorted right-hand sides over a ring of residues modulo 2N,” in: Abstracts of Papers Read at the Intern. Sci.-Pract. Conf. on Safety in Information-Telecommunication Systems, Kiev [in Russian] (2004), pp. 58–59.
G. V. Balakin, “Random matrices,” Teor. Veroyatn. i yeyo Primen., 12, Issue 2, 346–353 (1967).
G. V. Balakin, “Distribution of the rank of random matrices over a finite field,” Teor. Veroyatn. i yeyo Primen., 13, Issue 4, 631–641 (1968).
G. V. Balakin, “Distribution of the number of solutions of systems of random Boolean equations,” Teor. Veroyatn. i yeyo Primen., 13, Issue 3, 627–632 (1973).
G. V. Balakin, “Certainly compatible systems of random equations over a finite field,” in: Abstracts of Papers Read at the All-Union Conf. on Probabilistic Methods in Discrete Mathematics [in Russian], Petrozavodsk, Russia (1983), pp. 8–10.
G. V. Balakin, “A possibility of differentiating a random and a natural,” in: Abstracts of Papers Read at the All-Union Conf. on Probabilistic Methods in Discrete Mathematics [in Russian], Petrozavodsk, Russia (1988), pp. 10–11.
G. V. Balakin, “Estimates of the distribution of a rank of random matrices over a finite field,” in: Abstracts of Papers Read at the All-Union Conf. on Probabilistic Methods in Discrete Mathematics [in Russian], Petrozavodsk, Russia (1988), pp. 12–13.
G. V. Balakin, V. F. Kolchin, and V. I. Khokhlov, “Hypercycles in a random hypergraph,” Diskretn. Matem., 3, Issue 3, 102–108 (1991).
G. V. Balakin, “The number of solutions of systems of random pseudo-Boolean equations,” in: Proc. 3rd All-Union Conf. on Probabilistic Methods in Discrete Mathematics [in Russian], Petrozavodsk, Russia (1993), pp. 71–98.
G. V. Balakin, “The graphs of systems of binomial equations with Boolean unknowns,” Teor. Veroyatn. i yeyo Primen., 40, Issue 2, 241–259 (1995).
G. V. Balakin, “Efficiently solved classes of systems of Boolean equations,” Obozrenie Prikl. i Prom. Matem., 2, Issue 3, 494–501 (1995).
G. V. Balakin and Yu. B. Nikol’skii, “Sequential application of the maximum likelihood method to systems of equations with nuisance parameters,” Obozrenie Prikl. i Prom. Matem., 2, Issue 3, 468–476 (1995).
G. V. Balakin, “An introduction to the theory of random systems of equations,” in: Selected Papers in Discrete Mathematics [in Russian], 1, TVP, Moscow (1997), pp. 1–18.
G. V. Balakin, “Systems of random equations over a finite field,” in: Selected Papers in Discrete Mathematics [in Russian], 2, TVP, Moscow (1998), pp. 21–37.
G. V. Balakin, “Systems of random Boolean equations with a random sample of unknowns in each equation,” in: Selected Papers in Discrete Mathematics [in Russian], 3, TVP, Moscow (2000), pp. 21–28.
G. V. Balakin, “Systems of Boolean equations with a random sample of unknowns in each equation and distorted right-hand sides,” Obozrenie Prikl. i Prom. Matem., 7, Issue 1, 89–90 (2000).
G. V. Balakin, “Criteria separating out a certainly compatible system of equations with distorted right-hand side,” in: Selected Papers in Discrete Mathematics [in Russian], 4, TVP, Moscow (2002), pp. 7–16.
G. V. Balakin, “A sequential criterion of separating out a system of linear equations with distorted right-hand side,” in: Selected Papers in Discrete Mathematics [in Russian], 5, TVP, Moscow (2001), pp. 21–28.
G. V. Balakin, “Systems of Boolean equations with distorted right-hand sides,” Obozrenie Prikl. i Prom. Matem., 9, Issue 2, 330–331 (2002).
G. V. Balakin, “Algorithms of finding a set of least power containing a true solution with a given probability,” in: Selected Papers in Discrete Mathematics [in Russian], 7, TVP, Moscow (2003), pp. 7–21.
R. E. Blahut, Fast Algorithms for Digital Signal Processing, Addison-Wesley, Boston (1985).
M. V. Gavrilkevich and V. I. Solodovnikov, “Efficient algorithms for solving problems of linear algebra over a field of two elements,” Obozrenie Prikl. i Prom. Matem., 2, Issue 3, 399–439 (1995).
S. P. Gorshkov, “Application of the theory of NP-complete problems for estimating the complexity of the solution of systems of Boolean equations,” Obozrenie Prikl. i Prom. Matem., 2, Issue 3, 325–398 (1995).
A. M. Zubkov, “Estimates of the maximum sizes of zero submatrices of a random (0, 1)-matrix,” in: Selected Papers in Discrete Mathematics [in Russian], 4, TVP, Moscow (2001), pp. 51–56.
I. N. Kovalenko, “A limiting theorem for determinants in a class of Boolean functions,” Dokl. AN SSSR, 161, No.3, 517–519 (1965).
I. N. Kovalenko, “The limit distribution of the number of solutions of a random system of linear equations in a class of Boolean functions,” Teor. Veroyatn. i yeyo Primen., 12, Issue 1, 51–61 (1967).
I. N. Kovalenko, “Calculating the uniqueness probability of the solution of a system of random nonlinear Boolean equations,” Kibernetika, 3, 12–15 (1971).
I. N. Kovalenko, “On the distribution of the linear rank of a random matrix,” Teor. Veroyatn. i yeyo Primen., 17, Issue 2, 354–359 (1972).
I. N. Kovalenko, “On the distribution of the rank for random Boolean matrices,” Kibernetika, No. 5, 138–152 (1975).
I. N. Kovalenko and A. A. Levitskaya, “The asymptotic behavior of the number of solutions of a system of random linear equations over a finite field and a finite ring,” Dokl. AN SSSR, 221, No.4, 778–781 (1975).
I. N. Kovalenko and A. A. Levitskaya, “The asymptotic behavior of the number of solutions of a system of random linear equations over a finite field and a finite ring,” Teor. Veroyatn. Mat. Statistika, No. 13, 70–83 (1977).
I. N. Kovalenko, A. A. Levitskaya, and M. N. Savchuk, Selected Problems of Probabilistic Combinatorial Analysis [in Russian], Naukova Dumka, Kiev (1986).
I. N. Kovalenko and A. A. Levitskaya, “The probabilistic properties of the systems of random linear equations over finite algebraic structures,” in: Proc. 3rd All-Union Conf. on Probabilistic Methods in Discrete Mathematics [in Russian], Petrozavodsk, Russia (1993), pp. 64–70.
I. N. Kovalenko and A. A. Levitskaya, “Probabilistic properties of systems of random linear equations over finite algebraic structures,” Kibern. Sist. Analiz, No. 3, 385–390 (1993).
I. N. Kovalenko and A. A. Levitskaya, “The invariance theorems for nonlinear systems of equations over GF(2) field,” in: Abstracts of papers Read at the Symp. on Calculus Optimization Problems [in Russian], Kiev (1993), pp. 78–80.
M. V. Kozlov, “On the rank of the matrices with random Boolean elements,” Dokl. AN SSSR, 169, No.5, 1013–1016 (1966).
O. V. Kolesnikov, “Prohibitions of binary functions in solving systems of equations,” Obozrenie Prikl. i Prom. Matem., 2, Issue 3, 483–493 (1995).
V. F. Kolchin, Systems of Random Equations [in Russian], Izd. MIEM, Moscow (1988).
V. F. Kolchin, “Compatibility of the system of random equations,” Diskr. Matem., 4, Issue 3, 75–85 (1992).
V. F. Kolchin, “Random graphs and systems of linear equations in finite fields,” Random Structures and Algorithms, 5, 135–140 (1994).
V. F. Kolchin, “A classification problem in case of errors in measurements,” Diskr. Matem., 6, Issue 1, 53–66 (1994).
V. F. Kolchin and V. I. Khokhlov, “The threshold effect for special systems of random equations,” Diskr. Matem., 7, Issue 4, 29–39 (1995).
V. F. Kolchin, “Systems of random linear equations with small number of non-zero coefficients in finite fields,” in: Probabilitic Methods in Discrete Mathematics, VSP, Utrecht (1997), pp. 295–304.
V. F. Kolchin, “The threshold effect for systems of random equations,” in: Selected Papers in Discrete Mathematics [in Russian], 2, TVP, Moscow (1998), pp. 183–190.
V. F. Kolchin, “The threshold property of systems in finite fields,” Diskr. Matem., 11, Issue 3, 15–23 (1999).
V. F. Kolchin, “Compatibility probability for a special system of random equations,” in: Selected Papers in Discrete Mathematics [in Russian], 3, TVP, Moscow (2000), pp. 139–146.
V. F. Kolchin, Random Graphs [in Russian], Fizmatlit, Moscow (2000).
V. F. Kolchin, “Structures of solutions and restoration of a solution of a system of equations with distorted right-hand sides,” in: Selected Papers in Discrete Mathematics [in Russian], 5, TVP, Moscow (2002), pp. 93–102.
V. A. Kopyttsev, “Some random certainly compatible systems of equations,” Obozrenie Prikl. i Prom. Matem., 1, Issue 1, 56–84 (1994).
V. A. Kopyttsev, “The distribution of the number of solutions of random certainly compatible systems of equations,” Teor. Veroyatn. i yeyo Primen., 40, Issue 2, 430–437 (1995).
V. A. Kopyttsev, “The limit theorems for the number of solutions of a system of random equations,” Obozrenie Prikl. i Prom. Matem., 6, Issue 1, 160–161 (1999).
V. A. Kopyttsev, “The limit theorems for the number of solutions of a system of random equations,” Teor. Veroyatn. i yeyo Primen., 45, Issue 1, 52–72 (2000).
V. A. Kopyttsev, “The number of solutions of a random system of linear equations in a set of vectors of given form,” Obozrenie Prikl. i Prom. Matem., 7, Issue 1, 113–114 (2000).
V. A. Kopyttsev, “The number of solutions of a system of linear Boolean equations in a set of vectors with a given number of units,” Diskr. Matem., 14, Issue 4, 87–110 (2002).
V. A. Kopyttsev, “The number of solutions of a system of linear equations in a set of vectors with a given number of nonzero components,” Obozrenie Prikl. i Prom. Matem., 10, Issue 3, 677–678 (2003).
A. V. Lapshin, “The sufficient condition of the asymptotic equivalence of the average number of solutions of a random comparison system,” Obozrenie Prikl. i Prom. Matem., 4, Issue 3, 366–368 (1997).
A. V. Lapshin, “The sufficient conditions of the asymptotic equivalence of the average number of solutions of a random comparison system for one type of the distribution of indices of unknowns,” Obozrenie Prikl. i Prom. Matem., 5, Issue 2, 244–245 (1998).
A. V. Lapshin, “The asymptotic value of the average number of solutions of a random system of binomial equations,” Obozrenie Prikl. i Prom. Matem., 7, Issue 1, 118 (2000).
A. V. Lapshin, “The moments of the number of solutions of a random comparison system,” Obozrenie Prikl. i Prom. Matem., 8, Issue 2, 784–785 (2001).
A. A. Levitskaya, “The invariance theorems for the asymptotic behavior of the number of solutions of a system of random linear equations over a finite ring,” Kibernetika, No. 2, 140–141 (1978).
A. A. Levitskaya, “The invariance theorems for a system of random linear equations over an arbitrary finite ring,” Dokl. AN SSSR, 263, No.2, 279–281 (1982).
A. A. Levitskaya, “The invariance theorems for a system of random linear equations over an arbitrary finite ring not containing the left unity,” Dokl. AN USSR, Ser. A, No. 9, 71–72 (1984).
A. A. Levitskaya, “Compatibility probability for a system of random linear equations over a finite ring,” Teor. Veroyatn. i yeyo Primen., 30, Issue 2, 339–350 (1985).
A. A. Levitskaya, “The invariance theorems for systems of random linear equations in finite Abelian groups,” in: Abstracts of papers read at the All-Union Conf. on Probabilistic Methods in Discrete Mathematics [in Russian], Petrozavodsk, Russia (1988), pp. 56–57.
A. A. Levitskaya, “The limit distribution of the rank of a random Boolean matrix with a fixed number of units in rows,” Dokl. AN USSR, Ser. A, No. 8, 45–48 (1991).
A. A. Levitskaya, “The invariance theorems for the rank of a random Boolean matrix with a fixed number of units in rows,” Dokl. AN USSR, Ser. A, No. 9, 51–54 (1991).
A. A. Levitskaya, “Invariance theorems for a class of nonlinear systems of equations over an arbitrary finite field,” Kibern. Sist. Analiz, No. 2, 239–247 (1996).
A. A. Levitskaya, “The invariance theorems for systems of random nonlinear Boolean equations,” in: Proc. Ukrainian Mathem. Congr. on Algebraic Structures and Their Application [in Ukrainian], Kiev (2002), pp. 76–79.
A. A. Levitskaya, “Theorems of invariance for systems of random Boolean equations,” Abstracts of Papers Read at the Intern. Gnedenko Conf. (June 3–7), Kiev (2002), pp. 210–211.
L. A. Lyapkov and B. A. Sevast’yanov, “The probability distribution of a permanent of a random Boolean matrix,” Diskr. Matem., 2, Issue 2, 138–144 (1990).
L. A. Lyapkov and B. A. Sevast’yanov, “The limiting probability distribution of a permanent of a random matrix in a GF(p) field,” Diskr. Matem., 8, Issue 2, 3–13 (1996).
V. V. Masol, “Expansion in powers of a small parameter of the distribution of a random determinant in a GF(2) field,” Obozrenie Prikl. i Prom. Matem., 7, Issue 2, 511–512 (2000).
V. V. Masol, “Explicit representation of some coefficients in the expansion of the random matrix rank distribution in the field GF(2),” Theory Stoch. Proc., 6(22), Issue 3–4, 122–126 (2000).
V. I. Masol, “Expansion of the domain of invariance for random Boolean matrices,” Kibernetika, No. 3, 445–449 (1980).
V. I. Masol, “On the probability of the unique solution of a system of linear random Boolean equations,” Visnyk Kyiv. Universitetu, Issue 30, 58–62 (1988).
V. I. Masol, “Theorems of invariance for systems of random Boolean equations,” in: Abstracts of Papers Read at 6th Intern. Conf. of Probability Theory and Math. Statist., Vilnius (1993), pp. 19–20.
V. I. Masol, “Moments of the number of solutions of system of random Boolean equations,” Random Oper. and Stoch. Equations, 1, No.2, 171–179 (1993).
V. I. Masol, “The Poisson theorems for the limit distribution of the number of solutions of the system of nonlinear random Boolean equations. I,” in: Abstracts of Papers Read at the 2nd All-Russian Colloquium School on Stochastic Methods [in Russian], 95–96 (1995).
V. I. Masol, “The Poisson theorems for the limit distribution of the number of solutions of a system of nonlinear random Boolean equations. II,” in: Abstracts of Papers Read at the 3rd All-Russian Colloquium School on Stochastic Methods [in Russian], 115–116 (1996).
V. I. Masol, “The order of the rate of convergence to the limit distribution of the number of false solutions of the system of nonlinear random Boolean equations,” Obozrenie Prikl. i Prom. Matem., 4, Issue 3, 379–380 (1997).
V. I. Masol, “The limit distribution of the number of solutions of a system of random Boolean equations with the linear part,” Ukr. Mat. Zhurn., 50, No.9, 1214–1226 (1998).
V. I. Masol, “Theorem on the limit distribution of the number of false solutions of a system of nonlinear random Boolean equations,” Teor. Veroyatn. i yeyo Primen., 43, Issue 1, 41–56 (1998).
V. I. Masol, “Estimates of the probability of the existence of false solutions of a system of nonlinear random Boolean equations,” Obozrenie Prikl. i Prom. Matem., 5, Issue 2, 252–253 (1998).
V. I. Masol, “Some probabilistic properties of a system of nonlinear random Boolean equations,” Obozrenie Prikl. i Prom. Matem., 7, Issue 2, 512–513 (2000).
V. G. Mikhailov, “The limiting theorems for a random covering of a finite set,” Teor. Veroyatn. i yeyo Primen., 41, Issue 2, 272–283 (1996).
V. G. Mikhailov, “The limiting theorems for the number of nontrivial solutions of a system of random equations over a GF(2) field,” Teor. Veroyatn. i yeyo Primen., 43, Issue 3, 598–606 (1998).
V. G. Mikhailov, “The limiting theorems for the number of nontrivial solutions of a system of random equations over a GF(2) field,” Diskr. Matem., 12, Issue 1, 70–81 (2000).
V. G. Mikhailov, “The limiting Poisson theorem for the number of noncollinear solutions of a system of random equations of a special form,” Diskr. Matem., 13, Issue 3, 81–90 (2001).
M. N. Pokhlin, “On the applicability conditions for a method of the solution of systems of binomial equations,” Obozrenie Prikl. i Prom. Matem., 5, Issue 2, 274–275 (1998).
V. N. Sachkov, Probabilistic Methods in Combinatorial Theory [in Russian], Nauka, Moscow (1978).
V. N. Sachkov, An Introduction to the Combinatorial Methods of Discrete Mathematics [in Russian], Nauka, Moscow (1982).
V. N. Sachkov and V. E. Tarakanov, Combinatorial Analysis of Non-Negative Matrices [in Russian], TVP, Moscow (2000).
B. A. Sevast’yanov, “The probability distribution of the permanents of random matrices with independent elements in a GF(p) field,” in: Selected Papers in Discrete Mathematics [in Russian], 3, TVP, Moscow (2000), pp. 235–248.
D. Slepyan, “The class of binary signal alphabets,” in: The Theory of Message Transmission [in Russian], Izd. Inostr. Lit., Moscow (1957), pp. 82–113.
V. G. Smirnov, “The systems of Boolean equations of recurrent type,” Obozrenie Prikl. i Prom. Matem., 2, Issue 3, 477–482 (1995).
V. G. Smirnov, “Some classes of efficiently solved systems of Boolean equations,” in: Selected Papers in Discrete Mathematics [in Russian], 3, TVP, Moscow (2000), pp. 269–282.
A. V. Tarasov, “On the average number of solutions of certainly compatible random twice bijunctive systems of equations,” Obozrenie Prikl. i Prom. Matem., 9, Issue 3, 657 (2002).
A. N. Timashov, “The law of large numbers for the permanents of random twice stochastic matrices,” Obozrenie Prikl. i Prom. Matem., 5, Issue 2, 284–285 (1998).
A. N. Timashov, “The law of large numbers for the permanents of random twice stochastic matrices,” Diskr. Matem., 11, Issue 3, 91–98 (1999).
A. N. Timashov, “Permanents of random twice stochastic matrices and asymptotic estimates of the number of Latin rectangles and Latin squares,” Diskr. Matem., 14, Issue 4, 62–87 (2002).
A. V. Shapovalov, “Connectivity and threshold functions for subgraphs of random homogeneous hypergraphs,” Diskr. Matem., 5, Issue 4, 131–142 (1993).
A. V. Shapovalov, “The probability of compatibility of random systems of Boolean equations,” Diskr. Matem., 7, Issue 2, 146–159 (1995).
V. S. Shevelev, “On “projection” of the Erdos-Renyi theorem about permanent of stochastic (0, 1)-matrices into the subset of stochastic (0, 1)-matrices with equal row sums,” in: Abstracts of Papers Read at the 3rd All-Russian Colloquium School on Stochastic Methods (1996), pp. 177–178.
W. Ahn and B. Choi, “Asymptotics about permanent of random (0, 1)-matrices,” Math. Jap., 31, 167–174 (1986).
J. Blomer, R. Karp, and E. Wezl, “The rank of sparse random matrices over finite fields,” Random Structures and Algorithms, 10, No.4, 407–419 (1997).
N. Calkin, “Dependent sets of constant weight vectors in GF(q),” Random Structures and Algorithms, 9, No.1, 49–53 (1996).
N. Calkin, “Dependent sets of constant weight binary vectors,” in: Combinatorics Probability and Computing (1996).
L. S. Charlap, H. D. Rees, and D. P. Robbins, “The asymptotic probability that a random biased matrix is invertible,” Discrete Math., 82, 153–163 (1990).
C. Cooper, “Asymptotics for dependent sums of random vectors,” Random Structures and Algorithms, 14, No.3, 267–292 (1999).
C. Cooper, “On the rank of random matrices,” Random Structures and Algorithms, 16, 209–232 (2000).
C. Cooper, “On the asymptotic distribution of rank on random matrices over a finite field,” Random Structures and Algorithms, 17, 1–18 (2001).
P. Erdos and A. Renyi, “On random matrices, Series A,” Magyar Tud. Akad. Mat. Kutato Int. Kozl., 8, No.3, 455–461 (1963).
M. Grinfeld and P. A. Knight, “Weak lumpability in the k-SAT problem,” Appl. Math. Lett., 13, No.6, 49–53 (2000).
J. Kahn and K. Komlos, “Singularity probabilities for random matrices over finite fields,” Combinatorics Probability and Computing, 10, 137–157 (2001).
G. Landsberg, “Uber eine Anzahlbestimmung und eine damit Zusammenhangende Reihe,” J. Reine Angew. Math., III, 87–88 (1895).
I. Palasti, “On a system of random Boolean equations,” Magyar Tud. Akad., Nat. es. fiz. tud. oszt. Kozl., 19, No.1, 2, 33–72 (1970).
J. M. Van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge (1992).
Author information
Authors and Affiliations
Additional information
__________
Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 82–116, January–February 2005.
Rights and permissions
About this article
Cite this article
Levitskaya, A.A. Systems of Random Equations over Finite Algebraic Structures. Cybern Syst Anal 41, 67–93 (2005). https://doi.org/10.1007/s10559-005-0042-7
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10559-005-0042-7
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