Geodesic
Finite quasi-Frobenius modules, applications to codes and linear recurrences
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 1, pp. 229-254.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{FPM_1995_1_1_a12,
     author = {A. A. Nechaev},
     title = {Finite {quasi-Frobenius} modules, applications to codes and linear recurrences},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {229--254},
     publisher = {mathdoc},
     volume = {1},
     number = {1},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1995_1_1_a12/}
}
TY  - JOUR
AU  - A. A. Nechaev
TI  - Finite quasi-Frobenius modules, applications to codes and linear recurrences
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 1995
SP  - 229
EP  - 254
VL  - 1
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_1995_1_1_a12/
LA  - ru
ID  - FPM_1995_1_1_a12
ER  - 
%0 Journal Article
%A A. A. Nechaev
%T Finite quasi-Frobenius modules, applications to codes and linear recurrences
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 1995
%P 229-254
%V 1
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_1995_1_1_a12/
%G ru
%F FPM_1995_1_1_a12
A. A. Nechaev. Finite quasi-Frobenius modules, applications to codes and linear recurrences. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 1, pp. 229-254. http://geodesic.mathdoc.fr/item/FPM_1995_1_1_a12/

[1] Atya M., Makdonald I., Vvedenie v kommutativnuyu algebru, Mir, M., 1972, 160 pp. | MR

[2] Ericson Th., Zinoviev W., Spherical codes, Manuscript

[3] Kuzmin A. S. Nechaev A. A., “Lineino predstavimye kody i kod Kerdoka nad proizvolnym polem Galua kharakteristiki 2”, Uspekhi mat. nauk, 49:5 (1994), 165–166 | MR

[4] Lambek I., Koltsa i moduli, Mir, M., 1971, 279 | MR | Zbl

[5] Mak-Vilyams F. D., Sloen N. D. A., Teoriya kodov, ispravlyayuschikh oshibki, Svyaz, M., 1979

[6] Nechaev A. A., “Lineinye rekurrentnye posledovatelnosti nad kommutativnymi koltsami”, Diskret. mat., 3:4 (1991), 105–127 | MR

[7] Nechaev A. A., “Lineinye rekurrentnye posledovatelnosti nad kvazifrobeniusovymi modulyami”, Uspekhi mat. nauk., 48:3 (1993), 197–198 | MR | Zbl

[8] Feis K., Algebra, koltsa, moduli, kategorii, 2, M., Mir, 1979 | MR

[9] Azumaya G., “A duality theory for injective modules (Theory of quasi-Frobenius modules)”, Amer. J. Math., 81:1 (1959), 249–278 | DOI | MR | Zbl

[10] Kuzmin A. S., Kurakin V. L., Mikhalev A. V., Nechaev A A., “Linear recurrences over rings and modules”, Contemporary Math. and it's Appl. Thematic survays, J. of Math. Science, 10 (1994) | MR | Zbl

[11] Kuzmin A. S., Nechaev A. A., “Error correcting codes on the base of linear recurring sequences over Galois rings”, IV-th Int. Workshop. Algebraic and Combinatorial Coding Theory, Proceedings (Novgorod), 1994, 132–135

[12] Morita K., “Duality for modules and its applications to the theory of rings with minimum condition”, Sci. Repts Tokyo Kyoiku Daigaku (A6, No 15, May), 1958, 83–142 | MR | Zbl

[13] Nechaev A. A., “Linear codes over finite rings and $QF$-modules”, IV-th Int. Workshop. Algebraic and Combinatorial Coding Theory, Proceedings (Novgorod), 1994, 154–157

[14] Wisbauer R., “Grundlagen der Modul-und Ringtheorie”, Munchen Verl. Reinhard Fischer (1988 VI), 596 | MR | Zbl

[15] Hammous A. R., Kumar P. V., Calderbrank A. R., Sloane N. J. A., Sole P., “The $nqz{Z}_4$ linearity of Kerdock”, Goethals and related codes, Preparata, 1993