Given a rank-$r$ binary matroid we construct a system of $O(r^3)$ linear equations in $O(r^3)$ variables that has a solution over GF$(2)$ if and only if the matroid is graphic.
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Academic Press
doi.org/10.1016/j.jctb.2013.07.00
Journal of Combinatorial Theory - Series B
Networks and Optimization

Geelen, J., & Gerards, B. (2013). Characterizing graphic matroids by a system of linear equations. Journal of Combinatorial Theory - Series B, 103(5), 642–646. doi:10.1016/j.jctb.2013.07.00