On the matroids representable over \(GF(4)\)
One of the fundamental problems in matroid theory is the determination of the excluded minors for representability over the fields \(GF(q)\). This problem has only been solved when \(q\) is 2 or 3. This note considers this problem for \(q=4\). In particular, it is proved that a certain eight- element self-dual matroid is minor-minimal with the property of being nonrepresentable over \(GF(4)\). This matroid provides a counterexample to a conjecture of Kahn and Seymour that the excluded minors in this case consist of the six-point line and its dual, the non-Fano matroid and its dual, and one other six-element self-dual matroid.
scientific article; zbMATH DE number 3906509