Higher support matroids

The purpose of this paper is to provide links between matroid theory and the theory of subcode weights and supports in linear codes. We describe such weights and supports in terms of certain matroids arising from the vector matroids associated to the linear codes. Our results generalize classical results by Whitney, Tutte, Crapo and Rota, Greene, and other authors. As an application of our results, we obtain a new and elegant dual correspondence between the bond union and cycle union cardinalities of a graph.