Latroids and their representation by codes over modules

Vertigan, Dirk

doi:10.1090/S0002-9947-03-03367-1

Latroids and their representation by codes over modules
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by Dirk Vertigan

Trans. Amer. Math. Soc. 356 (2004), 3841-3868

DOI: https://doi.org/10.1090/S0002-9947-03-03367-1

Published electronically: July 24, 2003

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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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