Here I only address your representation-of-a-matroid algorithm question.
The paper,
Massimiliano Lunelli, Antonio Laface. "Representation of matroids." 2002. (arXiv abs link).
uses Gröbner bases of polynomials over the ring of integers to decide if a given matroid is representable. Here is a quote describing some results of using their algorithm:
![Lunelli][1]
Their code is