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Linear recurring sequences over rings and modules

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Literature Cited

  1. A. S. Ambrosimov, “On the distribution of frequencies of multigrams in linear recurring sequences over residue rings,”Uspekhi Mat. Nauk,48 (1993) (in press).

  2. M. F. Atiyah and I. G. MacDonald,Introduction to Commutative Algebra, Addison Wesley (1969).

  3. Yu. A. Bahturin,Basic Structures of Modern Algebra [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  4. E. R. Berlekamp,Algebraic Coding Theory, McGraw-Hill, New York (1968).

    Google Scholar 

  5. G. Birkhoff and T. C. Bartee,Modern Applied Algebra. McGraw-Hill, New York (1970).

    Google Scholar 

  6. Z. I. Borevich and I. R. Shafarevich,Theory of Numbers, 3rd ed. [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  7. A. Gill,Linear Sequential Circuits, McGraw-Hill, New York (1966).

    Google Scholar 

  8. N. Jacobson,The Theory of Rings, Math. Surveys, No. 2 (1943).

  9. J. Diedonne,La Geometrie des Groupes Classiques. 3rd ed., Springer (1971).

  10. J. Davenport, Y. Siret, and E. Tournier,Calcul Formel, Masson, Paris (1987).

    Google Scholar 

  11. V. P. Elizarov, “Systems of linear equations over commutative rings,”Uspekhi Mat. Nauk,48, No. 2, 181–182 (1993).

    MATH  MathSciNet  Google Scholar 

  12. V. P. Elizarov, “General solution of systems of linear homogeneous equations over a commutative ring,”Uspekhi Mat. Nauk,48 (1993) (in press).

  13. O. Zariski and P. Samuel,Commutative Algebra, I, II, Princeton (1958, 1960).

  14. F. Kasch,Moduln und Ringe, B. G. Teubner, Stuttgart (1977).

    Google Scholar 

  15. L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley, New York (1974).

    Google Scholar 

  16. Ch. W. Curtes and I. Reiner,Representation Theory of Finite Groups and Associative Algebras, Wiley, New York (1962).

    Google Scholar 

  17. A. H. Clifford and G. B. Preston,The Algebraic Theory of Semigroups, I, II, Amer. Math. Soc. (1964, 1967).

  18. D. E. Knuth,The Art of Computer Programming, Vol. 1. Fundamental Algorithms, Addison-Wesley (1968).

  19. N. M. Korobov, “Distribution of nonresidues and primitive roots in recurrent series,”Dokl. Akad. Nauk SSSR,88, No. 4, 603–606 (1953).

    MATH  MathSciNet  Google Scholar 

  20. N. M. Korobov,Trigonometric Sums and Their Applications [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  21. A. S. Kuzmin, “Polynomials of maximal period over residue rings” In:3rd International Conference in Algebra, Krasnoyarsk (1993).

  22. A. S. Kuzmin, “Distribution of elements on cycles of linear recurring sequences over residue rings,”Uspekhi Mat. Nauk,47, No. 6, 213–214 (1993).

    MathSciNet  Google Scholar 

  23. A. S. Kuzmin, “Lower bounds of ranks for coordinate sequences of linear recurring sequences over residue rings,”Uspekhi Mat. Nauk,48 (1993) (in press).

  24. A. S. Kuzmin, “On periods of binary digits of linear recurring sequences over prime finite fields,”Uspekhi Mat. Nauk,48 (1993) (in press).

  25. A. S. Kuzmin and A. A. Nechaev, “A construction of noise stable codes using linear recurrences over Galois rings,”Uspekhi Mat. Nauk,47, No. 5, 183–184 (1992).

    MathSciNet  Google Scholar 

  26. A. S. Kuzmin and A. A. Nechaev, “Linear recurring sequences over Galois rings,”Uspekhi Mat. Nauk,48, No. 1, 167–168 (1993).

    MathSciNet  Google Scholar 

  27. V. L. Kurakin, “Representations over the ringZ p n of linear recurring sequences of maximal period over the field GF(p),”Diskr. Mat.,4, No. 4, 96–116 (1992).

    MATH  Google Scholar 

  28. V. L. Kurakin, “Representations of linear recurring sequences and regular prime numbers,”Uspekhi Mat. Nauk,47, No. 6, 215–216 (1992).

    MATH  MathSciNet  Google Scholar 

  29. V. L. Kurakin, “Analytical structure of linear recurring sequences,”Uspekhi Mat. Nauk,48 (1993) (in press).

  30. V. L. Kurakin, “Representations over a field of linear recurrences of maximal period over a residue ring,”Uspekhi Mat. Nauk,48 (1993) (in press).

  31. V. L. Kurakin, “Convolution of linear recurring sequences,”Uspekhi Mat. Nauk,48, No. 4 235–236 (1993).

    MATH  MathSciNet  Google Scholar 

  32. V. L. Kurakin, “Structure of Hopf algebras of linear recurring sequences,”Uspekhi Mat. Nauk,48, No. 5 117–178 (1993).

    MathSciNet  Google Scholar 

  33. V. L. Kurakin, “The first coordinate sequence of a linear recurrence of maximal period over a Galois ring,”Diskr. Mat. (in press).

  34. V. L. Kurakin, “Representations of linear recurring sequences of maximal period over a finite field,”Diskr. Mat. (in press).

  35. D. Laksov, “Linear recurring sequences over finite fields,”Math. Scand. 16, 181–196 (1965).

    MATH  MathSciNet  Google Scholar 

  36. V. N. Latyshev,Combinatorial Ring Theory, Standard Bases, [In Russian], MGU, Moscow (1988).

    Google Scholar 

  37. R. Lidl and H. Niederreiter,Finite Fields, Addison-Wesley, London (1983).

    Google Scholar 

  38. E. S. Liapin,Semigroups [in Russian], Fizmatgiz, Moscow (1960).

    Google Scholar 

  39. F. J. McWilliams and N. J. A. Sloane,The Theory of Error-Correcting Codes, North-Holland (1977).

  40. Yu. I. Manin,Cubic Forms. Algebra, Geometry, Arithmetic [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  41. A. A. Markov,Finite Difference Calculates [in Russian], Odessa (1910), pp. 209–239.

  42. A. A. Nechaev, “Finite principal ideal rings,”Mat. Sb.,91, No. 3, 350–366 (1973).

    MathSciNet  Google Scholar 

  43. A. A. Nechaev, “Criteria of completeness of system of functions over finite rings and quasi-Frobenius rings,”Sib. Mat. Zh. 23, No. 3, 175–187 (1982).

    MathSciNet  Google Scholar 

  44. A. A. Nechaev, “Similarity of matrices over local commutative artinian rings,”Tr. Sem. Petrovskogo,9, 81–101 (1983).

    MATH  MathSciNet  Google Scholar 

  45. A. A. Nechaev, “Kerdoc code in a cyclic form,”Diskr. Math.,1, No. 4, 123–139 (1989).

    MATH  MathSciNet  Google Scholar 

  46. A. A. Nechaev, “Linear recurring sequences over commutative rings,”Diskr. Math.,3, No. 4, 107–121 (1991).

    Google Scholar 

  47. A. A. Nechaev, “Trace function in Galois rings and noise stable codes,” In:Fifth All-Union Symp. of Theory of Rings, Algebras and Modules, Novosibirsk (1982), p. 97.

  48. A. A. Nechaev, “The cyclic types of linear substitutions over finite commutative rings,”Mat. Sb.,184, No. 3, 21–56 (1993).

    MATH  MathSciNet  Google Scholar 

  49. A. A. Nechaev, “Linear recurring sequences over quasi-Frobenius modules,”Uspekhi Mat. Nauk,48 (1993) (in press).

  50. V. I. Nechaev, “Groups of nonsingular matrices over finite fields and recurring sequences,”Dokl. Akad. Nauk SSSR,152, No. 2, 275–277 (1963).

    MATH  MathSciNet  Google Scholar 

  51. V. I. Nechaev, “Linear recurring congruences with periodic coefficients,”Mat. Zametki,3, No. 6, 625–632 (1968).

    MathSciNet  Google Scholar 

  52. V. I. Nechaev, “Recurring sequences,”Uchen. Zap. Mosk. Ped. Inst.,375, 103–123 (1971).

    Google Scholar 

  53. V. I. Nechaev, “Linear congruences on powers of a prime ideal and linear recurring sequences,”Uchen. Zap. Mosk. Ped. Inst.,375, 124–132 (1971).

    Google Scholar 

  54. V. I. Nechaev, “Trigonometric sums for recurrent sequences of elements of a finite field,”Mat. Zametki,11, No. 5, 597–607 (1972).

    MathSciNet  Google Scholar 

  55. V. I. Nechaev, “Trigonometric sums for recurrent sequences,”Dokl. Akad. Nauk SSSR,206, No. 4, 811–814 (1972).

    MathSciNet  Google Scholar 

  56. V. I. Nechaev and A. M. Polosuev, “On distribution of non-residues and primitive roots in a sequence satisfying a finite difference equation with polynomial coefficients,”Vestnik MGU. Ser. 1, Mat., Mekh., No. 6, 75–84 (1964).

    Google Scholar 

  57. V. I. Nechaev and L. L. Stepanova, “Distribution of non-residues and primitive roots in recurring sequences over an algebraic number field,”Uspekhi Mat. Nauk,20, No. 3, 197–203 (1965).

    Google Scholar 

  58. V. N. Sachkov,Introduction to Combinatorical Methods of Discrete Mathematics [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  59. V. M. Sidelnikov, “On extremal polynomials used for estimating code cardinality,”Probl. Peredachi Inf.,16, No. 3, 17–30 (1980).

    MATH  MathSciNet  Google Scholar 

  60. V. M. Sidelnikov, “Bounds for number of symbols on a segment of a recurring sequence over a finite field,”Diskr. Math.,3, No. 2, 87–95 (1991).

    MATH  Google Scholar 

  61. V. M. Skornjakov and A. V. Mikhalev, “Modules,” In:Algebra. Topologiya. Geometriya. Vol. 14,Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1976), pp. 57–191.

    Google Scholar 

  62. I. M. Sobol,Numerical Monte-Carlo Methods [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  63. D. A. Suprunenko,Matrix Groups [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  64. A. I. Uzkow, “On Jordan-Hölder theorem,”Mat. Sb.,4, No. 1, 29–43 (1938).

    Google Scholar 

  65. A. I. Uzkow, “Abstract foundation of Brandt’s theory of ideals,”Mat. Sb.,6, No. 2, 253–281 (1939).

    MathSciNet  Google Scholar 

  66. A. I. Uzkov, “Zur Idealtheorie der kommutativen Ring. I,”Mat. Sb.,5, No. 3, 513–520 (1939).

    MATH  Google Scholar 

  67. A. I. Uzkov, “An algebraic lemma and normalizing E. Noether theorem,”Mat. Sb.,22, No. 2, 349–350 (1948).

    MATH  MathSciNet  Google Scholar 

  68. A. I. Uzkov, “On cyclic direct decomposability of modules over commutative rings,”Mat. Sb.,62, No. 4, 469–475 (1963).

    MATH  MathSciNet  Google Scholar 

  69. C. Faith,Algebra II. Ring Theory, Springer-Verlag (1976).

  70. N. Zierler, “Linear recurring sequences,”SIAM J. Appl. Math.,7, 31–48 (1959).

    Article  MATH  MathSciNet  Google Scholar 

  71. P. L. Chebyshev,Theory of Probabilities. Lectures 1879–1880 [in Russian], Moscow-Leningrad (1936), pp. 139–147.

  72. P. L. Chebyshev,Congruence Theory. Complete Collection of Works [in Russian], Vol. 1, Acad. Sci. USSR, Moscow-Leningrad (1944), pp. 10–172.

    Google Scholar 

  73. I. E. Sparlinsky, “Distribution of non-residues and primitive roots in recurring sequences,”Mat. Zametki,24, No. 5, 605–615 (1978).

    Google Scholar 

  74. I. E. Shparlinski, “Distribution of fractional parts of recurring sequences,”Zh. Vychisl. Mat. Mat. Fiz.,21, No. 6, 1588–1591 (1981).

    Google Scholar 

  75. I. E. Shparlinski, “On some properties of linear cyclic codes,”Probl. Peredachi Inf.,19, No. 3, 106–110 (1983).

    Google Scholar 

  76. M. Antweiler and L. Bomer, “Complex sequences over GF(q) with a two-level autocorrelation function and a large linear span,”IEEE Trans. Inf. Theory,38, No. 1, 120–130 (1992).

    MathSciNet  Google Scholar 

  77. G. Azumaya, “A duality theory for injective modules (Theory of quasi-Frobenius modules),”Am. J. Math.,81, No. 1, 249–278 (1959).

    MATH  MathSciNet  Google Scholar 

  78. B. Benzaghou and J.-P. Bezivin, “Proprietes algebriques de suites differentiellement sinies,”Bull. Soc. Mat. Fr.,120, No. 3, 327–346 (1992).

    MathSciNet  Google Scholar 

  79. D. Bollman, “Some periodicity properties of transformations on a vector space over residue class rings,”SIAM J. Appl. Math.,13, No. 3, 902–912 (1965).

    MATH  MathSciNet  Google Scholar 

  80. D. Bollman, “Some periodicity properties of modules over the ring of polynomials with coefficients in a residue class ring,”SIAM J. Appl. Math.,14, No. 2, 237–241 (1966).

    MATH  MathSciNet  Google Scholar 

  81. J. L. Brenner, “Linear recurrence relations,”Amer. Math. Monthly,61, No. 3, 171–173 (1954).

    MATH  MathSciNet  Google Scholar 

  82. A. Brousseau, “Recursion relations of products of linear recursion sequences,”Fibonacci Quart,14, No. 2, 159–166 (1976).

    MATH  MathSciNet  Google Scholar 

  83. L. Brynielsson, “On the linear complexity of combined shift register sequences,”Lect. Notes Comput. Sci.,219 (1985).

  84. S. A. Burr, “On moduli for which the Fibonacci sequence contains a complete system of residues,”Fibonacci Quart,9, 497–504 (1971).

    MATH  MathSciNet  Google Scholar 

  85. D. Calabro and J. K. Wolf, “On the synthesis of two-dimensional arrays with disarable correlation properties,”Inf. Control,11, 537–560 (1968).

    Google Scholar 

  86. R. D. Carmichael, “On sequences of integers defined by recurrence relations,”Quart. J. Pure Appl. Math.,48, 343–372 (1920).

    Google Scholar 

  87. L. Cerlienco, G. Delogu, and F. Piras, “The search for quadratic divisors of a polynomial by the method of linear recurrent sequences,”Rend. Math. Appl.,1, No. 4, 623–631 (1981).

    MathSciNet  Google Scholar 

  88. L. Cerlienco, M. Mignotte, and F. Piras, “Linear recurrent sequences: algebraic and arithmetical properties,”Enseign. Math.,33, No. 1–2, 67–108 (1987).

    MathSciNet  Google Scholar 

  89. L. Cerlienco and F. Piras, “Resultant, l.c.m. and g.c.d. of two polynomials by the method of linear recurrent sequences,”Rend. Sem. Fac. Sci. Univ. Calgari.,50, No. 3–4, 711–717 (1980).

    MathSciNet  Google Scholar 

  90. L. Cerlienco and F. Piras, “G-R-sequences and incidence coalgebras of posets of full binomial type,”J. Math. Anal. Appl.,115, No. 1, 46–56 (1986).

    MathSciNet  Google Scholar 

  91. L. Cerlienco and F. Piras, “On the continuous dual of a polynomial bialgebra,”Commun. Algebra,19, No. 10, 2707–2727 (1991).

    MathSciNet  Google Scholar 

  92. A. H. Chan and H. A. Games, “On the linear span of binary sequences obtained from finite geometries,”Lect. Notes Comput. Sci.,263, 405–417 (1987).

    MathSciNet  Google Scholar 

  93. A. H. Chan and M. Goresky, “On the linear complexity of feedback register (extended abstract),”Lect. Notes Comput. Sci.,434 (1990).

  94. E. C. Dade, D. W. Robinson, O. Taussky, and M. Ward, “Divisors of recurrent sequences,”J. Reine Angew. Math.,214/215, 180–183 (1964).

    MathSciNet  Google Scholar 

  95. Z. D. Dai, T. Beth, and D. Gollmann, “Lower bounds for the linear complexity of sequences over residue rings,”Lect. Notes Comput. Sci.,473, 189–195 (1991).

    MathSciNet  Google Scholar 

  96. Z. D. Dai and Z. X. Wan, “A relationship between the Berlekamp-Massey and the Euclidean algorithms for linear feedback shift register synthesis,”Acta. Math. Sin. New Ser.,4, No. 1, 55–63 (1988).

    MathSciNet  Google Scholar 

  97. Z. D. Dai and K. C. Zeng, “Continued fractions and the Berlekamp-Massey algorithm,”Lect. Notes. Comput. Sci.,453, 24–31 (1990).

    MathSciNet  Google Scholar 

  98. D. J. De Carli, “A generalized Fibonacci sequence over an arbitrary ring.,”Fibonaci Quart,8, No. 2, 182–184 (1970).

    MATH  Google Scholar 

  99. D. J. De Carli, “Periodicity over the ring of matrices,”Fibonacci Quart,11, No. 5, 466–468 (1973).

    MathSciNet  Google Scholar 

  100. L. E. Dickson,History of the Theory of Numbers. Vol. 1, Carnegie Inst., Washington (1919).

    Google Scholar 

  101. L. L. Dornhoff and F. E. Hohn,Applied Modern Algebra, Macmillan, New York-London (1978).

    Google Scholar 

  102. H. J. A. Duparc, “Periodicity properties of recurring sequences. I, II,”Indagat. Math.,16, No. 3, 331–342,16, No. 4, 473–485 (1954).

    MathSciNet  Google Scholar 

  103. J. Eichenauer-Herrman, H. Grothe, and J. Lehn, “On the period length of pseudorandom vector-sequences generated by matrix generators,”Math. Comput.,2, No. 185, 145–148 (1989).

    Google Scholar 

  104. H. T. Engstrom, “Periodicity in sequences defined by linear recurrence relations,”Proc. Natl. Acad. Sci. USA,16, 663–665 (1930).

    MATH  Google Scholar 

  105. H. T. Engstrom, “On sequences defined by linear recurrence relations”Trans. Am. Math. Soc.,33, 210–218 (1931).

    MATH  MathSciNet  Google Scholar 

  106. L. Euler,Introduction to the Calculus of Infinitesimal Variables (1748),Leonardi Euleri opera omnia,8 (1922);9 (1945).

  107. H. J. Fell, “Linear complexity of transformed sequences,”Lect. Notes Comput. Sci.,514, 205–214 (1991).

    MATH  MathSciNet  Google Scholar 

  108. G. L. Feng and K. K. Tzeng, “A generalization of the Berlekamp-Massey algorithm for multisequence shift register synthesis with application to decoding cyclic codes,”IEEE Trans. Inf. Theory,37, No. 5, 1274–1287 (1991).

    MathSciNet  Google Scholar 

  109. S. D. Golic, “On the linear complexity of functions of periodic GF(q) sequences,”IEEE Trans. Inf. Theory,35, No. 1, 69–75 (1989).

    MATH  MathSciNet  Google Scholar 

  110. B. F. J. Green, J. E. K. Smith, and Klem Laura, “Empirical tests of an additive random number generator,”J. Assoc. Comput. Mach.,6, 527–537 (1959).

    MathSciNet  Google Scholar 

  111. H. Grothe, “Matrix generators for pseudo-random vector generation,”Statist. Hefte,28, No. 3, 233–238 (1987).

    MATH  MathSciNet  Google Scholar 

  112. F. G. Gustavson, “Analysis of the Berlekamp-Massey linear feedback shift register synthesis algorithm,”IBM J. Res. Dev.,20, No. 3, 204–212 (1976).

    MATH  MathSciNet  Google Scholar 

  113. M. Hall, “An isomorphism between linear recurring sequences and algebraic rings,”Trans. Am. Math. Soc.,44, No. 2, 196–218 (1938).

    MATH  Google Scholar 

  114. P. Haukkanen, “On a convolution of linear recurring sequences over finite fields,”J. Algebra,149, No. 1, 179–182 (1992).

    MATH  MathSciNet  Google Scholar 

  115. K. Imamura and W. Yoshida, “A simple derivation of the Berlekamp-Massey algorithm and some applications,”IEEE Trans. Inf. Theory,33, No. 1, 146–150 (1987).

    MathSciNet  Google Scholar 

  116. T. Ikai, H. Kosako, and Y. Kojima, “Subsequences in linear recurring sequences,”Electron. Commun. Jpn.,53, No. 12, 159–166 (1970).

    MathSciNet  Google Scholar 

  117. C. J. A. Jansen and D. E. Boekee, “The shortest feedback shift register that can generate a given sequence,”Lect. Notes Comput. Sci.,435, 90–99 (1990).

    MathSciNet  Google Scholar 

  118. B. Jansson,Random Number Generators, Almqvist and Wiksell, Stockholm (1966).

    Google Scholar 

  119. S. M. Jennings, “Multiplexed sequences: some properties of the mimimum polynomial,”Lect. Notes Comput. Sci.,149 (1983).

  120. A. M. Kerdock, “A class of low-rate non-linear codes,”Inform. Control.,20, 182–187 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  121. Klove Torleiv, “Periodicity of recurring sequences in rings,”Math. Scand.,32, No. 2, 165–168 (1973).

    MathSciNet  Google Scholar 

  122. L. Kronecker,Vorlesungen uber Zahlentheorie, Vol. 1, Teubner, Leipzig (1901).

    Google Scholar 

  123. A. S. Kuzmin and A. A. Nechaev, “Linear recurrent sequences over Galois rings,” In:Proc. 2nd Internat. COnf. Algebra, Barnaul (1991).

  124. J. L. Lagrange, “Recherches sur les suites recurrentes dont les termes varient de plusieurs manieres differentes, ou sur l’integration des equations lineaires aux differences finies et partielles; et sur l’usage de ces equations dans la theorie des hasards,” In:Nouv. Memoires Acad. Roy, Berlin (1775), pp. 183–272;Oeuvres,4, Gauthier-Villars, Paris (1869), pp. 151–251.

    Google Scholar 

  125. R. R. Laxton and J. A. Anderson, “Linear recurrences and maximal length sequences,”Math. Gaz.,56, 299–309 (1972).

    MathSciNet  Google Scholar 

  126. R. Lidl and H. Niederreiter,Introduction to Finite Fields and Their Applications, Cambridge Univ. Press (1986).

  127. E. Lucas, “Theorie des fonctions numeriques simplement periodiques,”Am. J. Math.,1, 184–240, 289–321 (1878).

    MATH  MathSciNet  Google Scholar 

  128. F. J. MacWilliams and N. J. A. Sloane, “Pseudo-random sequences and arrays,”Proc. IEEE,64, No. 11, 1715–1729 (1976).

    MathSciNet  Google Scholar 

  129. M. Magidin and A. Gill, “Singular shift register over residue class ring,”Math. Syst. Theory,9, No. 4, 345–358 (1976).

    MathSciNet  Google Scholar 

  130. K. Mahler,p-Adic Numbers and Their Functions, 2nd ed., Cambridge Unv. Press, Cambridge (1981).

    Google Scholar 

  131. J. L. Massey, “Shift-register synthesis and BCH decoding,”IEEE Trans. Inf. Theory,15, No. 1, Part 1, 122–127 (1969).

    MATH  MathSciNet  Google Scholar 

  132. B. R. McDonald,Finite Rings with Identity, Marcel Dekker, New York (1974).

    Google Scholar 

  133. R. J. McEliece,Finite Fields for Computer Scientists and Engineers, Kluwer, Boston (1987).

    Google Scholar 

  134. K. Morita, “Duality for modules and its applications to the theory of rings with minimum condition,”Sci. Rep. Tokyo Kyoiku Daigaku,A6, No. 15, May, 83–142 (1958).

    Google Scholar 

  135. M. Nagata,Local Rings, Int. Publ., New York (1962).

    Google Scholar 

  136. M. B. Nathanson, “Difference operators and periodic sequences over finite modules,”Acta Math. Acad. Sci. Hungary,28, No. 3–4, 219–224 (1976).

    MATH  MathSciNet  Google Scholar 

  137. A. A. Nechaev, “Linear recurring sequences and quasi-Frobenius modules,” In:International School in Algebra and Analysis, Baikal (1992).

  138. H. Niederreiter, “Some new exponential sums with applications to pseudorandom numbers,” In:Topics in Number Theory (Debrecen, 1974), Colloquia math. Soc. Janos Bolyai, North-Holland, Amsterdam (1976), pp. 209–232.

    Google Scholar 

  139. H. Niederreiter, “On the cycle structure of linear recurring sequences,”Math. Scand.,38, No. 1, 53–77 (1976).

    MATH  MathSciNet  Google Scholar 

  140. H. Niederreiter and S. S. Shiue, “Equidistribution of linear recurring sequences in finite fields,”Acta Arithm.,38, No. 2, 197–207 (1980).

    MathSciNet  Google Scholar 

  141. H. Niederreiter, “Distribution properties of feedback shift register sequences,”Probl. Contr. Inform. Theory (Hungary),15, No. 1, 19–34 (1986).

    MATH  MathSciNet  Google Scholar 

  142. H. Niederreiter, “A simple and general approach to the decimation of feedback shift register sequences,”Probl. Contr. Inform. Theory (Hungary),17, No. 5, 327–331 (1988).

    MATH  MathSciNet  Google Scholar 

  143. T. Nomura and A. Fukuda, “Linear recurring planes and two-dimensional cyclic codes,”Electron. Commun. Jpn.,54, No. 3, 23–30 (1971).

    MathSciNet  Google Scholar 

  144. T. Nomura, H. Miyakawa, H. Imai, and A. Fukuda, “A method of construction and some properties of planes having maximum area matrix,”Electron. Commun. Jpn.,54, No. 5, 18–25 (1971).

    MathSciNet  Google Scholar 

  145. T. Nomura, H. Miyakawa, H. Imai, and A. Fukuda, “Some properties of thep-plane and its extension to three-dimensional space,”Electron. Commun. Jpn.,54, No. 8, 27–34 (1971).

    MathSciNet  Google Scholar 

  146. T. Nomura, H. Miyakawa, H. Imai, and A. Fukuda, “A theory of two-dimensional linear recurring arrays,”IEEE Trans. Inf. Theory,18, No. 6, 775–785 (1972).

    MathSciNet  Google Scholar 

  147. B. Peterson and E. Y. Taft, “The Hopf algebra of linear recursive sequences,”Aequat. Math.,20, 1–17 (1980).

    MathSciNet  Google Scholar 

  148. R. Raghavendran, “A class of finite rings,”Compos. Math.,22, No. 1, 49–57 (1970).

    MATH  MathSciNet  Google Scholar 

  149. F. Rhodes, “Regular mappings of sequence space over finite fields,”Q. J. Math. Oxford, Ser. 2,37, No. 146, 231–238 (1976).

    MathSciNet  Google Scholar 

  150. D. W. Robinson, “A note on linear recurrent sequences modulom,”Am. Math. Monthly,73, No. 6, 619–621 (1966).

    MATH  Google Scholar 

  151. D. W. Robinson, “The rank and period of a linear recurrent sequence over a ring,”Fibonacci Quart.,14, No. 3, 210–214 (1976).

    MATH  MathSciNet  Google Scholar 

  152. R. A. Rueppel and O. J. Staffelbach, “Products of linear recurring sequences with maximum complexity,”IEEE Trans. Inf. Theory,33, No. 1, 126–131 (1987).

    Google Scholar 

  153. S. Sakata, “Doubly linear recurring arrays andM-arrays,”Trans. Inst. Electron. Comm. Eng.,A60, No. 10, 918–925 (1977).

    Google Scholar 

  154. S. Sakata, “General theory of doubly periodic arrays over an arbitrary finite field and its applications,”IEEE Trans. Inf. Theory,24, 719–730 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  155. S. Sakata, “On determining the independent point set for doubly periodic arrays and encoding two-dimensional cyclic codes and their duals,”IEEE Trans. Inf. Theory,27, No. 5, 556–565 (1981).

    MATH  MathSciNet  Google Scholar 

  156. S. Sakata, “Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array,”J. Symbolic Comput.,34, No. 3, 321–337 (1988).

    MathSciNet  Google Scholar 

  157. S. Sakata, “Cycle representatives of quasi-irreducible two-dimensional cyclic codes,”IEEE Trans. Inf. Theory,34, No. 4, 871–875 (1988).

    MathSciNet  Google Scholar 

  158. S. Sakata, “Synthesis of two-dimensional linear feedback shift registers and Groebner bases,”Lect. Notes Comput. Sci.,356, 394–407 (1989).

    MATH  MathSciNet  Google Scholar 

  159. S. Sakata, “Extension of the Berlekamp-Massey algorithm to N dimensions,”Inform. Comput.,84, No. 2, 207–239 (1990).

    MATH  MathSciNet  Google Scholar 

  160. E. S. Selmer,Linear Recurrence Relations over Finite Fields, Univ. of Bergen (1966).

  161. J. S. Shiue and T. L. Sheu, “On the periodicity of linear recurrence of second order in commutative rings,”Tamkang J. Math.,4, 105–107 (1973).

    MathSciNet  Google Scholar 

  162. N. J. A. Sloane and J. A. Reeds, “Shift register synthesis (modulom),”SIAM J.,14, No. 3, 505–513 (1985).

    MathSciNet  Google Scholar 

  163. M. Sweedler,Hopf algebras, Benjamin, New York (1969).

    Google Scholar 

  164. I. Vajda and T. Nemetz, “Substitution of characters inq-arym-sequences,”Lect. Notes Comput. Sci.,508, 96–105 (1991).

    MathSciNet  Google Scholar 

  165. A. Vince, “Period of a linear recurrence,”Acta Arithm.,39, No. 4, 303–311 (1981).

    MATH  MathSciNet  Google Scholar 

  166. M. Ward, “The arithmetical theory of linear recurring series,”Trans. Am. Math. Soc.,35, No. 3, 600–628 (1933).

    MATH  Google Scholar 

  167. M. Ward, “Arithmetical properties of sequences in rings,”Ann. Math.,39, 210–219 (1938).

    MATH  Google Scholar 

  168. W. A. Webb and C. T. Long, “Distribution modulop of the general linear second-order recurrence,”Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.,58, No. 2, 92–100 (1975).

    MathSciNet  Google Scholar 

  169. R. Wisbauer,Grundlagen der Modul und Ringtheorie, Verl. Reinhard Fischer, Munich (1988).

    Google Scholar 

  170. R. B. Yale, “Error correcting codes and linear recurring sequences,”Report 34-37, M. I. T. Lincoln Laboratory, Lexington, Massachusetts (1958).

    Google Scholar 

  171. L. A. Zadeh and E. Polak,System Theory, McGraw-Hill, New York (1969).

    Google Scholar 

  172. K. C. Zeng and M. Q. Huang, “Solving equations in sequences,”Lect. Notes Comput. Sci.,453, 327–332 (1990).

    MathSciNet  Google Scholar 

  173. N. Zierler, “Linear recurring sequences and error-correcting codes,” In:Error Correcting Codes, Wiley, New York (1968), pp. 47–59.

    Google Scholar 

  174. N. Zierler and W. H. Mills, “Products of linear recurring sequences,”J. Algebra,27, No. 1, 147–157 (1973).

    MathSciNet  Google Scholar 

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Authors
  1. V. L. Kurakin
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  2. A. S. Kuzmin
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  3. A. V. Mikhalev
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  4. A. A. Nechaev
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Additional information

To the 80th anniversary of the birth of Alexander Illarionovich Uzkow (1913–1990)

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 10, Algebra-2, 1994.

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Kurakin, V.L., Kuzmin, A.S., Mikhalev, A.V. et al. Linear recurring sequences over rings and modules. J Math Sci 76, 2793–2915 (1995). https://doi.org/10.1007/BF02362772

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