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

CWI
CWI. Probability, Networks and Algorithms [PNA]
Networks and Optimization

Geelen, J., Gerards, B.& Kapoor, A. (1997). The excluded minors for GF(4)-representable matroids. In CWI. Probability, Networks and Algorithms [PNA] (R 9711). CWI.