Let \cM be a minor-closed class of matroids that does not contain
arbitrarily long lines. The growth rate function, h:\bN\rightarrow \bN of
\cM is given by h(n) = \max(|M|\, : \, M\in \cM, simple, rank-$n$). The
Growth Rate Theorem shows that there is an integer c such that either:
h(n)β€cn, or (2n+1β)β€h(n)β€cn2, or there is a
prime-power q such that qβ1qnβ1ββ€h(n)β€cqn; this
separates classes into those of linear density, quadratic density, and base-q
exponential density. For classes of base-q exponential density that contain
no (q2+1)-point line, we prove that h(n)=qβ1qnβ1β for all
sufficiently large n. We also prove that, for classes of base-q exponential
density that contain no (q2+q+1)-point line, there exists k\in\bN such
that h(n)=qβ1qn+kβ1ββqq2β1q2kβ1β for all
sufficiently large n