There are exactly seven excluded minors for the class of GF(4)--representable matroids.

Matroids, geometric lattices (msc 05B35)
Logistics (theme 3)
CWI
CWI. Probability, Networks and Algorithms [PNA]
Networks and Optimization

Geelen, J, Gerards, A.M.H, & Kapoor, A. (1997). The excluded minors for GF(4)-representable matroids. CWI. Probability, Networks and Algorithms [PNA]. CWI.