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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Latroids and their representation by codes over modules
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by Dirk Vertigan PDF
Trans. Amer. Math. Soc. 356 (2004), 3841-3868 Request permission

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.
References
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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
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 3841-3868
  • MSC (2000): Primary 05B35; Secondary 94B05, 16D90
  • DOI: https://doi.org/10.1090/S0002-9947-03-03367-1
  • MathSciNet review: 2058508