On the Excluded Minors for Quaternary Matroids

This paper strengthens the excluded-minor characterization of GF(4)-representable matroids. In particular, it is shown that there are only finitely many 3-connected matroids that are not GF(4)-representable and that have no U2, 6-, U4, 6-, P6-, F−7-, or (F−7)*-minors. Explicitly, these matroids are all minors of S(5, 6, 12) with rank and corank at least 4, and P″8, the matroid that can be obtained from S(5, 6, 12) by deleting two elements, contracting two elements, and then relaxing the only pair of disjoint circuit-hyperplanes.