Vertex counts for the unresolved case (r,n)=(4,8)

Determine the number of vertices of \Omega_{4,8} and the number of isomorphism classes of extremal matroids of rank 4 on 8 elements.

Background

The paper tabulates the numbers of vertices of \Omega_{r,n} and the isomorphism classes of extremal matroids for parameters through n=8. The entry corresponding to (r,n)=(4,8) is left unknown in both tables because the authors were unable to compute it.

References

We have been unable to compute the numbers for $(r,n)=(4,8)$.

— The polytope of all matroids  (2502.20157 - Ferroni et al., 27 Feb 2025) in Section 4, subsection “The vertices of \Omega_{r,n} and first examples of extremal matroids,” immediately before Table 1