Abstract
We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are associated to linear codes, our definitions coincide with those for codes. Thus our Wei duality is a generalization of that for linear codes given by Schaathun. In the last part of the paper we show how some important chains of cycles of the matroids appearing, correspond to chains of component maps of minimal resolutions of the independence complex of the corresponding matroids. We also relate properties of these resolutions to chainedness and greedy weights of the matroids, and in many cases codes, that appear.
Highlights
For a linear code C over a finite field Fq an important way to characterize the code is to decribe its parameters, the word length n, the dimension k, and the minimum distance d
In [9] we described how the di are determined by certain properties of the matroid coming from any parity check matrix of the linear code
Codewords of linear codes being dependence relations between the column vectors of any parity check matrix of the code, it is natural to look at the matroid associated to a linear code
Summary
For a linear code C over a finite field Fq an important way to characterize the code is to decribe its parameters, the word length n, the dimension k, and the minimum distance d. A refinement of the minimum distance is the ordered set of the generalized Hammimg weights d1, . Dk, where di is the smallest support of any i-dimensional linear subcode of C, for i = 1, . The authors are grateful to the Department of Mathematics, IIT-Bombay for a stimulating stay, during which a substantial part of the present work was completed
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