  Mayhew and Royle (2008) showed that there are 564 excluded minors for the
class of GF(5)-representable matroids having at most 9 elements. We enumerate
the excluded minors for GF(5)-representable matroids having 10 elements: there
are precisely 2128 such excluded minors. In the process we find, for each $i
\in \{2,3,4\}$, the excluded minors for the class of
$\mathbb{H}_i$-representable matroids having at most 10 elements, and the
excluded minors for the class of $\mathbb{H}_5$-representable matroids having
at most 13 elements.
