  In 1961, Dirac showed that chordal graphs are exactly the graphs that can be
constructed from complete graphs by a sequence of clique-sums. In an earlier
paper, by analogy with Dirac's result, we introduced the class of
$GF(q)$-chordal matroids as those matroids that can be constructed from
projective geometries over $GF(q)$ by a sequence of generalized parallel
connections across projective geometries over $GF(q)$. Our main result showed
that when $q=2$, such matroids have no induced minor in $\{M(C_4),M(K_4)\}$. In
this paper, we show that the class of $GF(2)$-chordal matroids coincides with
the class of binary matroids that have none of $M(K_4)$, $M^*(K_{3,3})$, or
$M(C_n)$ for $n\geq 4$ as a flat. We also show that $GF(q)$-chordal matroids
can be characterized by an analogous result to Rose's 1970 characterization of
chordal graphs as those that have a perfect elimination ordering of vertices.
