Enumerating Toric-Colorable Seeds of Picard Number Five via Binary Matroids

We introduce a binary matroid approach to the enumeration of mod 2 toric-colorable seeds of fixed Picard number. We organize these matroids by their contraction category and enumerate weak pseudomanifold subcomplexes by a dynamic programming algorithm. The main computational step uses a Gray code traversal of the mod 2 kernel of the ridge-facet incidence matrix. As the main new result, we find that there are 198,846 mod 2 toric-colorable seeds of dimension four and Picard number five. We also check that they all are toric-colorable. Finally, the same framework independently reproduces the Picard number 4 enumeration of Choi, Jang, and Vall\'{e}e much faster than their previous method.