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Code equivalence characterizes finite Frobenius rings
Author(s):
Jay
A.
Wood
Journal:
Proc. Amer. Math. Soc.
136
(2008),
699-706.
MSC (2000):
Primary 94B05;
Secondary 16D50, 16L60, 16P10.
Posted:
November 6, 2007
MathSciNet review:
2358511
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Abstract:
In this paper we show that finite rings for which the code equivalence theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. This result makes use of a strategy of Dinh and López-Permouth.
References:
-
- 1.
- G. E. Andrews, The theory of partitions, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013 (58:27738)
- 2.
- K. Bogart, D. Goldberg, and J. Gordon, An elementary proof of the MacWilliams theorem on equivalence of codes, Inform. and Control 37 (1978), no. 1, 19-22. MR 0479646 (57:19067)
- 3.
- H. Q. Dinh and S. R. López-Permouth, On the equivalence of codes over finite rings, Appl. Algebra Engrg. Comm. Comput. 15 (2004), no. 1, 37-50. MR 2142429 (2006d:94097)
- 4.
- -, On the equivalence of codes over rings and modules, Finite Fields Appl. 10 (2004), no. 4, 615-625. MR 2094161 (2005g:94098)
- 5.
- M. Greferath, A. Nechaev, and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl. 3 (2004), no. 3, 247-272. MR 2096449 (2005g:94099)
- 6.
- M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' equivalence theorem, J. Combin. Theory Ser. A 92 (2000), no. 1, 17-28. MR 1783936 (2001j:94045)
- 7.
- T. Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel) 76 (2001), no. 6, 406-415. MR 1831096 (2002b:16033)
- 8.
- V. L. Kurakin, A. S. Kuzmin, V. T. Markov, A. V. Mikhalev, and A. A. Nechaev, Linear codes and polylinear recurrences over finite rings and modules (a survey), Applied algebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999), Lecture Notes in Comput. Sci., vol. 1719, Springer, Berlin, 1999, pp. 365-391. MR 1846512 (2002h:94092)
- 9.
- T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294 (99i:16001)
- 10.
- F. J. MacWilliams, Error-correcting codes for multiple-level transmission, Bell System Tech. J. 40 (1961), 281-308. MR 0141541 (25:4945)
- 11.
- -, Combinatorial properties of elementary abelian groups, Ph.D. thesis, Radcliffe College, Cambridge, Mass., 1962.
- 12.
- T. Nakayama, On Frobeniusean algebras, I, Annals of Math. (2) 40 (1939), 611-633. MR 0000016 (1:3a)
- 13.
- -, On Frobeniusean algebras, II, Annals of Math. (2) 42 (1941), 1-21. MR 0004237 (2:344b)
- 14.
- J. H. van Lint and R. M. Wilson, A course in combinatorics, Cambridge University Press, Cambridge, 1992. MR 1207813 (94g:05003)
- 15.
- H. N. Ward and J. A. Wood, Characters and the equivalence of codes, J. Combin. Theory Ser. A 73 (1996), no. 2, 348-352. MR 1370137 (96i:94028)
- 16.
- J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555-575. MR 1738408 (2001d:94033)
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Additional Information:
Jay
A.
Wood
Affiliation:
Department of Mathematics, Western Michigan University, 1903 W. Michigan Ave., Kalamazoo, Michigan 49008--5248
Email:
jay.wood@wmich.edu
DOI:
10.1090/S0002-9939-07-09164-2
PII:
S 0002-9939(07)09164-2
Keywords:
Finite Frobenius rings,
Hamming weight,
equivalence theorem,
extension property
Received by editor(s):
February 6, 2007
Posted:
November 6, 2007
Communicated by:
Martin Lorenz
Copyright of article:
Copyright
2007,
American Mathematical Society
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