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Fundam. Prikl. Mat., 1995, Volume 1, Issue 1, Pages 229–254 (Mi fpm53)  

This article is cited in 19 scientific papers (total in 19 papers)

Finite quasi-Frobenius modules, applications to codes and linear recurrences

A. A. Nechaev

Abstract: A simple exposition of the main properties of the quasi-Frobenius modules over finite commutative rings with identity elements. The presented results show the special role of such modules in the theory of linear recurrences and in the theory of linear codes over rings and modules. In particular it is proved that the general weight functions of the linear code over a ring and the dual code over the corresponding $QF$-module are connected by the Mac-Williams identity.

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Received: 01.01.1995

Citation: A. A. Nechaev, “Finite quasi-Frobenius modules, applications to codes and linear recurrences”, Fundam. Prikl. Mat., 1:1 (1995), 229–254

Citation in format AMSBIB
\by A.~A.~Nechaev
\paper Finite quasi-Frobenius modules, applications to codes and linear recurrences
\jour Fundam. Prikl. Mat.
\yr 1995
\vol 1
\issue 1
\pages 229--254

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    This publication is cited in the following articles:
    1. V. P. Elizarov, “Sistemy lineinykh uravnenii nad kvazifrobeniusovymi koltsami”, Fundament. i prikl. matem., 1:2 (1995), 535–539  mathnet  mathscinet  zmath
    2. O. A. Logachev, A. A. Sal'nikov, V. V. Yashchenko, “Pairing of Finite Modules, Duality of Linear Codes, and the MacWilliams Identities”, Problems Inform. Transmission, 34:3 (1998), 250–258  mathnet  mathscinet  zmath
    3. A. A. Nechaev, T. Khonol'd, “Weighted Modules and Representations of Codes”, Problems Inform. Transmission, 35:3 (1999), 205–223  mathnet  mathscinet  zmath
    4. Kurakin V.L., Kuzmin A.S., Markov V.T., Mikhalev A.V., Nechaev A.A., “Linear codes and polylinear recurrences over finite rings and modules (a survey)”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Proceedings, Lecture Notes in Computer Science, 1719, 1999, 365–391  crossref  mathscinet  zmath  isi
    5. Nechaev A.A., “Recurring sequences”, Formal Power Series and Algebraic Combinatorics, 2000, 54–66  crossref  mathscinet  zmath  isi
    6. Kurakin V.L., Mikhalev A.V., Nechaev A.A., “Polylinear recurring sequences over a bimodule”, Formal Power Series and Algebraic Combinatorics, 2000, 484–495  crossref  mathscinet  zmath  isi
    7. V. L. Kurakin, “Linear complexity of polylinear sequences”, Discrete Math. Appl., 11:1 (2001), 1–51  mathnet  crossref  mathscinet  zmath
    8. A. A. Nechaev, D. A. Mikhailov, “A canonical system of generators of a unitary polynomial ideal over a commutative Artinian chain ring”, Discrete Math. Appl., 11:6 (2001), 545–586  mathnet  crossref  mathscinet  zmath
    9. V. L. Kurakin, A. A. Nechaev, “Quasi-Frobenius bimodules of functions on a semigroup”, Russian Math. Surveys, 57:6 (2002), 1230–1231  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    10. V. P. Elizarov, “Sistemy lineinykh uravnenii nad modulyami”, Fundament. i prikl. matem., 8:4 (2002), 979–991  mathnet  mathscinet  zmath
    11. D. A. Mikhailov, “Unitary polylinear shift registers and their periods”, Discrete Math. Appl., 12:1 (2002), 15–44  mathnet  crossref  mathscinet  zmath
    12. E. V. Gorbatov, “Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences”, J. Math. Sci., 139:4 (2006), 6672–6707  mathnet  crossref  mathscinet  zmath
    13. V. P. Elizarov, “Solvable and locally closed modules and rings”, Discrete Math. Appl., 16:1 (2006), 29–37  mathnet  crossref  crossref  mathscinet  zmath  elib
    14. V. P. Elizarov, “Implications of a system of linear equations over a module”, Discrete Math. Appl., 17:2 (2007), 163–169  mathnet  crossref  crossref  mathscinet  zmath  elib
    15. Cayrel P.-L., Chabot Ch., Necer A., “Quasi-cyclic codes as codes over rings of matrices”, Finite Fields and Their Applications, 16:2 (2010), 100–115  crossref  mathscinet  zmath  isi
    16. V. L. Kurakin, “Semeistvo posledovatelnostei nad koltsom iz $8$ elementov s nizkoi korrelyatsiei”, Matem. vopr. kriptogr., 1:4 (2010), 85–109  mathnet  crossref
    17. V. L. Kurakin, “Semeistvo posledovatelnostei maksimalnogo perioda nad koltsom iz 8 elementov s nizkoi kross-korrelyatsiei”, Matem. vopr. kriptogr., 2:3 (2011), 47–73  mathnet  crossref
    18. V. P. Elizarov, V. L. Kurakin, “Factorially solvable rings”, Discrete Math. Appl., 23:3-4 (2013), 363–367  mathnet  crossref  crossref  mathscinet  elib  elib
    19. A. V. Mikhalev, A. A. Nechaev, “Tsiklovye tipy semeistv polilineinykh rekurrent i datchiki psevdosluchainykh chisel”, Matem. vopr. kriptogr., 5:1 (2014), 95–125  mathnet  crossref
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