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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Latroids and their representation by codes over modules


Author: Dirk Vertigan
Journal: Trans. Amer. Math. Soc. 356 (2004), 3841-3868
MSC (2000): Primary 05B35; Secondary 94B05, 16D90
Published electronically: July 24, 2003
MathSciNet review: 2058508
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Abstract: It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code is an evaluation of the Tutte polynomial of the matroid, and a standard identity relating the Tutte polynomials of dual matroids gives rise to a MacWilliams identity relating the weight enumerators of dual codes. More recently, codes over rings and modules have been considered, and MacWilliams type identities have been found in certain cases.

In this paper we consider codes over rings and modules with code duality based on a Morita duality of categories of modules. To these we associate latroids, defined here. We generalize notions of deletion, contraction, minors and duality, on both codes and latroids, and examine all natural relations among these.

We define generating functions associated with codes and latroids, and prove identities relating them, generalizing above-mentioned generating functions and identities.


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Additional Information

Dirk Vertigan
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918
Email: vertigan@math.lsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03367-1
PII: S 0002-9947(03)03367-1
Keywords: Linear code, ring, Artinian ring, finite ring, module, matroid, polymatroid, latroid, minor, minor class, duality, Morita duality, weight enumerator, Tutte polynomial, generating function, MacWilliams identity
Received by editor(s): July 15, 2002
Received by editor(s) in revised form: April 3, 2003
Published electronically: July 24, 2003
Additional Notes: The author’s research was partially supported by the National Security Agency, grant number MDA904-01-0014
Dedicated: Dedicated in memory of William T. Tutte, 1917-2002
Article copyright: © Copyright 2003 American Mathematical Society