Please define what you mean by "matroidal network".  Do you wish to attach a graph to a matroid?

geometrically, A graphical representation of  matroid has some properties of network. The  existence of corresponding vector linear index codes linked to the matroid multilinear representations.in fact Matroidal Networks Associated with

Representable Matroids.

i want to find a new method on decomposition of binary codes via geometrically properties of network that is related to matroid.

A graph is defined by a vertex set and an adjacency relation. Please define both in this case of "matroidal network".